Distribution Systems Planning Considering Distributed Generation and Energy Storage Systems
ALEJANDRO VALENCIA D´IAZ
UNIVERSIDAD TECNOL´OGICA DE PEREIRA
Faculty of Engineering - Electric Engineering program Master in Electrical Engineering
PEREIRA, RISARALDA, COLOMBIA
2020
Distribution Systems Planning Considering Distributed Generation and Energy Storage Systems Thesis submitted as a partial requirement to receive the grade of: Master in Electrical Engineering Advisor: Ram´on Alfonso Gallego Rend´on, Ph.D. Co-advisor: Ricardo Alberto Hincapi´e Isaza, Ph.D.
UNIVERSIDAD TECNOL´OGICA DE PEREIRA
Faculty of Engineering - Electric Engineering program Master in Electrical Engineering
PEREIRA, RISARALDA, COLOMBIA
2020
Dedication
• To my mother Mar´ıa Lida Diaz Nore˜na, my father Carlos Alfonso Valencia Ca˜nas, and
my brother Santiago Valencia Diaz whom supported me in all the difficult moments of this process and were always aware of my well-being.
The autor.
Acknowledgments
• To the advisor of this project Ph.D. Ram´on A. Gallego Rend´on and to the co-advisor
Ph.D. Ricardo A. Hincapi´e Isaza for giving me their support, trust, and guidance, to make this achievement possible.
• To all the professors of the Program of Electrical Engineering and Master in Electrical
Engineering for their valuable contributions to my professional development. The autor.
Abstract This thesis presents a new methodology for optimal integrated planning of medium and low voltage distribution systems with distributed generation (DG) in the low voltage network, and integration of energy storage systems (ESSs). This problem is formulated as a mixed integer non-linear problem and is solved using a simulated annealing algorithm (SA) with a novel neighborhood search method based on the Zbus matrix (NSZM). The NSZM method uses sensitivity factors based on the Zbus matrix for reducing the neighborhood size in order to find attractive solutions based on the electrical information of the Zbus matrix. Hence, the solution space is explored more efficiently in order to find a joint global solution that establishes a balanced benefit for the planning of both networks.
This methodology is compared to a bilevel approach used in the literature to verify its efficiency. The bilevel approach solved the optimal integrated planning of medium and low voltage distribution systems with a penetration of DG in the low voltage network using a real distribution system. The obtained results show the importance of considering both networks simultaneously in the planning of the electric distribution system, as well as the use of the Zbus matrix for sensitivity analysis, which allows finding answers with lower global costs. Subsequent to the validation of the methodology, the ESSs are integrated in the distribution system planning problem (DSP) of the both networks considering DG. The obtained results show the importance of integrating ESSs in the DSP problem due to the maximization of the profit from the energy purchase and sale. Furthermore, the results show the impact of considering ESSs in the MV network instead of LV network.
Contents
1
Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Problem Description
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3.1
DSP for MV networks
. . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3.2
DSP for LV networks . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3.3
Integrated DSP for MV and LV networks . . . . . . . . . . . . . . . .
6
1.4
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4.1
Research results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.5
Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2
Problem Formulation and Modeling
9
2.1
DSP for MV and LV networks . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2
DSP for MV and LV networks considering DGs
. . . . . . . . . . . . . . . .
13
i
Master’s Thesis: Contents
2.3
DSP for MV and LV networks considering DGs and ESSs . . . . . . . . . . .
15
3
Zbus Matrix
19
3.1
Constructive algorithm of the Zbus matrix . . . . . . . . . . . . . . . . . . .
20
3.2
Modifications of the Zbus matrix to reflect changes in the network . . . . . .
21
3.3
Topological information of the network provided by the Zbus matrix . . . . .
23
3.4
Electrical information of the network provided by the Zbus matrix . . . . . .
24
3.4.1
Impact of the DGs and ESSs on the active power losses of the network
25
3.5
Load flow based on the Zbus matrix . . . . . . . . . . . . . . . . . . . . . . .
26
4
Solution Methodology
28
4.1
Codification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.1.1
DSP considering DGs . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.1.2
DSP considering DGs and ESSs . . . . . . . . . . . . . . . . . . . . .
30
4.2
Initial configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.3
Neighborhood search method based on the Zbus matrix (NSZM) . . . . . . .
31
4.3.1
Neighborhood structure
. . . . . . . . . . . . . . . . . . . . . . . . .
32
4.3.2
Modifications of the Zbus matrix to reflect changes by the neighborhood structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.4
Evaluation of the configurations . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.4.1
Optimal Power Flow considering DGs and ESSs . . . . . . . . . . . .
36
ii
Master’s Thesis: Contents
4.4.2
Fitness Function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.5
Solution technique
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.5.1
Cooling scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.6
General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5
Application and Results
48
5.1
Description of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5.2
Integrated DSP of both networks considering DGs
. . . . . . . . . . . . . .
50
5.2.1
Validation of the methodology . . . . . . . . . . . . . . . . . . . . . .
51
5.2.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
5.3
Integrated DSP of both networks considering DGs and ESSs . . . . . . . . .
56
5.3.1
Validation of the decomposition method
. . . . . . . . . . . . . . . .
57
5.3.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
6
Conclusions and Future work
65
6.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
6.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
A Data Systems
76
A.1 Data of the distribution system . . . . . . . . . . . . . . . . . . . . . . . . .
76
A.2 Data of the modified distribution system . . . . . . . . . . . . . . . . . . . .
89
iii
Master’s Thesis: Contents B Final configurations of primary and secondary networks
91
B.1 Integrated DSP considering DGs
. . . . . . . . . . . . . . . . . . . . . . . .
91
B.2 Integrated DSP considering DGs and ESSs . . . . . . . . . . . . . . . . . . .
96
iv
List of Figures
3.1
Constructive algorithm of the Zbus matrix. . . . . . . . . . . . . . . . . . . .
21
4.1
Codification scheme for primary and secondary DSP considering DGs. . . . .
30
4.2
Codification scheme for primary and secondary DSP considering DGs and ESSs. 30
4.3
Zbus modification due to neighborhood criteria. . . . . . . . . . . . . . . . .
34
4.4
Pseudocode of the specialized SA. . . . . . . . . . . . . . . . . . . . . . . . .
43
4.5
Flowchart of the proposed methodology.
. . . . . . . . . . . . . . . . . . . .
47
5.1
Integrated distribution system. . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.2
Daily behavior of the load and DGs curves in pu.
. . . . . . . . . . . . . . .
50
5.3
Incumbent behavior for the four cases in millions of USD. . . . . . . . . . . .
55
5.4
Primary Network of the test system.
. . . . . . . . . . . . . . . . . . . . . .
57
5.5
Secondary Network of the test system.
. . . . . . . . . . . . . . . . . . . . .
58
5.6
Comparison of the generated power by the substations for the five cases.
. .
62
5.7
Incumbent behavior for the five cases in millions of USD. . . . . . . . . . . .
63
v
Master’s Thesis: List of Figures B.1 Case 1 - Primary Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
B.2 Case 1 - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
92
B.3 Case 2 - Primary Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
B.4 Case 2 - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
93
B.5 Case 3 - Primary Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
B.6 Case 3 - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
94
B.7 Case 4 - Primary Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
B.8 Case 4 - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
95
B.9 Case A - Primary Network.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
96
B.10 Case A - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
97
B.11 Case B - Primary Network.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
97
B.12 Case B - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
98
B.13 Case C - Primary Network.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
98
B.14 Case C - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
99
B.15 Case D - Primary Network.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
99
B.16 Case D - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
100
B.17 Case E - Primary Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
B.18 Case E - Secondary Networks. . . . . . . . . . . . . . . . . . . . . . . . . . .
101
vi
List of Tables
5.1
Comparison of cost of expansion in millions of USD. . . . . . . . . . . . . . .
53
5.2
Comparison of cost of expansion in millions of USD. . . . . . . . . . . . . . .
54
5.3
Candidate nodes for installing ESSs in the LV and MV networks.
. . . . . .
56
5.4
Comparison of the objective functions for the five cases in millions of USD. .
59
5.5
Comparison of the cost and profit in millions of USD. . . . . . . . . . . . . .
61
A.1 Candidate nodes for installing DTs and DGs. . . . . . . . . . . . . . . . . . .
77
A.2 Information of the wires used in the distribution system.
. . . . . . . . . . .
77
A.3 Upgrading costs of the wires in [USD/m]. . . . . . . . . . . . . . . . . . . . .
78
A.4 Elements information.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
A.5 Information of the circuits of the LV network.
. . . . . . . . . . . . . . . . .
78
A.6 Information of the feeders of the MV network. . . . . . . . . . . . . . . . . .
82
A.7 Nodal information of the LV network. . . . . . . . . . . . . . . . . . . . . . .
84
A.8 Nodal information of the MV network. . . . . . . . . . . . . . . . . . . . . .
87
vii
Master’s Thesis: List of Tables A.9 Load and DG curves information. . . . . . . . . . . . . . . . . . . . . . . . .
89
A.10 ESS information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
viii
Master’s Thesis: Nomenclature Nomenclature Chapter 3 Parameters
CEP
ij,p Fixed cost to expand the capacity of an existing primary feeder between nodes i −j, type p [$].
CES
ij,c Fixed cost to expand the capacity of an existing secondary circuit between nodes i −j, type c [$].
CESS
s Fixed cost to expand the capacity of an existing substation type s [$].
CNP
ij,p Fixed cost of a new primary feeder between nodes i −j, type p [$].
CNS
ij,c Fixed cost of a new secondary circuit between nodes i −j, type c [$].
CNSS
s Fixed cost of a new substation type s [$].
CNDT
d Fixed cost of a new DT type d [$].
CNDG
g Fixed cost of a new DG type g [$].
CNESS
b Fixed cost of a new ESS type b [$]. O&M fx b Fixed operation and maintenance cost of a new ESS type b [$/year]. k1 Factor which converts to present value. k2l Energy purchase cost [$/kWh] for each load level l. k3 Energy sale cost [$/kWh].
∆Tl Time duration of each load level l [h]. Imax p Maximum current limit of a primary wire type p [A]. Imax c Maximum current limit of a secondary wire type c [A]. NL Number of load levels.
Dyear Number of days of the year.
φb Charge/discharge slope of a ESS type b [%/kWh]. ηb Power injection/extraction efficiency of a ESS type b [%]. SoC0 i Initial state of charge of a ESS at node i [%]. SoCF i Final state of charge of a ESS at node i [%]. V P nom Nominal voltage of the primary network. V S nom Nominal voltage of the secondary network. ix
Master’s Thesis: Nomenclature a0 Coefficient corresponding to constant power. a1 Coefficient corresponding to constant current. a2 Coefficient corresponding to constant impedance. ZP ij,p Impedance of the primary feeder between nodes i −j, type p [Ω]. ZS ij,c Impedance of the secondary circuit between nodes i −j, type c [Ω]. GP ij Conductance of the primary feeder between nodes i −j [S]. BP ij Susceptance of the primary feeder between nodes i −j [S]. Smax d Maximum power limit of a DT type d [kVA]. Smax g Maximum power limit of a DG type g [kVA]. Smax s Maximum power limit of a substation type s [kVA]. Smaxc b Maximum extraction power limit of a ESS type b [kVA]. Smaxd b Maximum injection power limit of a ESS type b [kVA].
SPD
i,l Primary demand at node i, for a load level l [kVA].
SSD
i,l Secondary demand at node i, for a load level l [kVA]. V max i Maximum voltage limit at node i [kV].
V min i Minimum voltage limit at node i [kV].
SoCmax i Maximum state of charge of a ESS at node i [%]. SoCmin i Minimum state of charge of a ESS at node i [%]. Variables γEP ij,p Binary decision variable to expand the capacity of an existing primary feeder between nodes i −j, type p.
γES ij,c Binary decision variable to expand the capacity of an existing secondary circuit between nodes i −j, type c.
γESS i,s Binary decision variable to expand the capacity of an existing substation at node i, type s.
γNDT i,d Binary decision variable to install a new DT at node i, type d. γNDG i,g Binary decision variable to install a new DG at node i, type g. γNP ij,p Binary decision variable to install a new primary feeder between nodes i −j, type p.
γNS ij,c Binary decision variable to install a new secondary circuit between nodes i−j, type c.
x
Master’s Thesis: Nomenclature γNSS i,s Binary decision variable to install a new substation at node i, type s. γNESS i,b Binary decision variable to install a new ESS at node i, type b. ϕNESS i,l Binary decision variable for the operation state of the ESS at node i, for a load level l.
IP ij,l Current flow in primary branch i −j, for a load level l [A]. IS ij,l Current flow in secondary branch i −j for a load level l [A].
SDT
i,l Power injected to a DT at node i, for a load level l [kVA].
SDG
i,l Power injected by a DG at node i, for a load level l [kVA]. SS i,l Power injected by a substation at node i, for a load level l [kVA].
SESS
i,l Power injected or extracted by a ESS at node i, for a load level l [kVA].
SESSD
i,l Power injected by a ESS at node i, for a load level l [kVA].
SESSC
i,l Power extracted by a ESS at node i, for a load level l [kVA]. SoCi,l State of charge of the ESS at node i, for a load level l [%].
V BUS
i,l Bus voltage at node i, for a load level l [kV].
IBUS
i,l Bus current at node i, for a load level l [A].
ZIP P
i,l ZIP load model at primary node i, for a load level l.
ZIP S
i,l ZIP load model at secondary node i, for a load level l.
ZBUS
ij Zbus matrix [Ω].
Sets ΩEP Set formed by existing primary feeders.
ΩES Set formed by existing secondary circuits.
ΩESS
Set formed by existing substations.
ΩNDT
Set formed by new DTs.
ΩNDG
Set formed by new DGs.
ΩNESS
Set formed by new ESSs.
ΩNL Set formed by the load levels.
ΩNP Set formed by new primary feeders.
ΩNS Set formed by new secondary circuits.
ΩNSS
Set formed by new substations.
ΩPF Set formed by new and existing primary feeders. ΩPN Set formed by nodes of primary network.
xi
Master’s Thesis: Nomenclature ΩSC Set formed by new and existing secondary circuits. ΩSN Set formed by nodes of secondary circuits. ΩN Set formed by nodes of primary and secondary networks. ΩSS Set formed by new and existing substations.
ΩTDT
Set formed by types of DTs.
ΩTDG
Set formed by types of DGs.
ΩTESS
Set formed by types of ESSs.
ΩTP Set formed by types of primary feeders. ΩTS Set formed by types of secondary circuits.
ΩTSS
Set formed by types of substations. Chapter 4 Parameters k1 Factor which converts to present value. k2l Energy purchase cost [$/kWh] for each load level l. k3 Energy sale cost [$/kWh].
∆Tl Time duration of each load level l [h]. Sbase Base apparent power of the system [kVA]. NL Number of load levels.
Dyear Number of days of the year.
φi,b Charge/discharge slope of a ESS type b [%/kWh] at node i. ηi,b Power injection/extraction efficiency of a ESS type b at node i [pu]. SoC0 i Initial state of charge of a ESS at node i [%]. SoCF i Final state of charge of a ESS at node i [%]. V pu ref Magnitude of the voltage at node slack [pu]. θrad ref Angle of the voltage at node slack [rad]. a0 Coefficient corresponding to constant power. a1 Coefficient corresponding to constant current. a2 Coefficient corresponding to constant impedance. GP ij Conductance of the primary feeder between nodes i −j [pu]. BP ij Susceptance of the primary feeder between nodes i −j [pu]. xii
Master’s Thesis: Nomenclature P max i,g Maximum active power limit of a DG at node i, type g [pu]. P maxc i,b Maximum extraction limit of active power of a ESS at node i, type b [pu]. P maxd i,b Maximum injection limit of active power of a ESS at node i, type b [pu]. P D i,l Active demand at node i, for a load level l [pu]. QD i,l Reactive demand at node i, for a load level l [pu]. V max i Maximum voltage limit at node i [pu].
V min i Minimum voltage limit at node i [pu].
SoCmax i Maximum state of charge of a ESS at node i [%]. SoCmin i Minimum state of charge of a ESS at node i [%].
RBUS
ij Real part of the Zbus matrix [pu].
XBUS
ij Imaginary part of the Zbus matrix [pu]. Variables P DG i,l Active power injected by a DG at node i, for a load level l [pu]. P S i,l Active power injected by a substation at node i, for a load level l [pu]. QS i,l Reactive power injected by a substation at node i, for a load level l [pu].
P ESS
i,l Active power injected or extracted by a ESS at node i, for a load level l [pu].
P ESSD
i,l Active power injected by a ESS at node i, for a load level l [pu].
P ESSC
i,l Active power extracted by a ESS at node i, for a load level l [pu]. SoCi,l State of charge of the ESS at node i, for a load level l [%].
V BUS
i,l Bus voltage magnitude at node i, for a load level l [pu].
IBUS
i,l Bus current magnitude at node i, for a load level l [pu]. θV i,l Bus voltage angle at node i, for a load level l [rad]. θI i,l Bus current angle at node i, for a load level l [rad]. ZIPi,l ZIP load model in per unit representation at node i, for a load level l. Sets ΩNL Set formed by the load levels.
ΩN Set formed by nodes of primary and secondary networks.
ΩTDG
Set formed by types of DGs.
ΩTESS
Set formed by types of ESSs.
ΩDG Set formed by DGs nodes.
ΩB Set formed by ESSs nodes.
xiii
Chapter 1 Introduction
1.1
Motivation Distribution system planning (DSP) is a set of strategies that allows determining how many, where, and when elements can be installed in an electric network to satisfy the growing demand for a specific time horizon [1]. Traditionally, DSP is employed for upgrading existing elements and installing new electric circuits (medium and low voltage – MV/LV) and sources (substations and distribution transformers (DTs)) [2, 3]. Due to the combinatorial nature (NP-complete) of the DSP problem, both voltage levels (MV/LV) have been solved separately, allowing a reduced searching space but not leading to a joint global solution [4, 5]. In this context, several methodologies and different mathematical models have been proposed [6]. The number of papers regarding DSP for medium voltage [1, 2, 6–19] is higher than for low voltage [20–26].
Moreover, only a few of them consider both levels in an integrated way [3–5,27–29].
Nowadays, the electric power sector has been presented an energetic change due to the emergence of new technologies, such as distributed generators (DGs), energy storage systems
Master’s Thesis: Problem Description (ESSs), and electric vehicles (EVs), which have produced technical, economic, and environmental benefits [1,30,31]. For this reason, incorporating DGs and ESSs into the DSP problem is desired. There are more papers considering DGs and ESSs in the DSP for MV [30,32–37] than for LV [38–40]. For the integrated DSP of primary and secondary networks, only two have integrated DGs [5,29], and only integrates DGs, ESSs, and EVs [41].
1.2
Problem Description DSP is a complex mixed integer nonlinear optimization problem [29]. Classic optimization, and heuristic and metaheuristic methods have been used to solve DSP; classic optimization guarantees an optimal global solution but requires excessive computational effort due to the combinatorial complexity of DSP [4,5]. Heuristic and metaheuristic methods can find nearglobal solutions, reducing the computational effort [4,29]. As consequence, a metaheuristic is used to solve the DSP problem considering DGs and ESSs. This way, simulated annealing (SA) is applied because it has been employed in problems with similar mathematical complexity [9,42,43].
SA is a stochastic optimization algorithm that can converge asymptotically to the optimal global solution with probability one [44]. Although this may turn out to be computationally expensive, it is a valuable feature of the algorithm.
Nevertheless, quick answers can be obtained for the SA algorithm with adequate cooling schemes and reduced neighborhoods which may yield a bunch of near or even globally optimal solutions. These neighborhoods can be reduced by using sensitivity factors based on the Zbus matrix in order to reduced computational efforts and obtained good quality solutions.
The Zbus matrix reflects important electrical characteristics of the network due its relation between voltages and current injections in a system [45]. This electrical information can be used through sensitivity factors to reduce the solution space obtained by the neighborhood
Master’s Thesis: State of the Art structure (branch exchange, upgrading existing elements, and the location of DGs and DTs) in order to find good quality solutions in quick times.
The integration of DGs and ESSs in the DSP problem increases its mathematical complexity due to the variability and intermittency of the DGs and the optimal charge and discharge operation of the ESSs. Therefore, it is needed to develop more robust models in order to solve the DSP problem considering ESSs. These models need to reduced computational efforts for the solving of the optimal charge and discharge operation of the ESSs and considering multiple scenarios due to the variability and intermittency of the DGs.
1.3
State of the Art Despite the benefits provided by the Zbus matrix due its electrical information of the network, the Zbus matrix has not been used to solve the DSP problem. On the other hand, the Zbus has been used for different applications in the literature, such as load flow [46], reconfiguration of networks [47], and DGs location [48].
1.3.1
DSP for MV networks In the literature, different papers used several mathematical models, objective functions, time stages, and solution techniques for the planning of MV networks [6]. Some authors presented integer linear approximation models to find near optimal solutions, such as the proposed in [7]. In order to find real optimal solutions, mixed integer nonlinear programming (MINLP) models were presented and solved using metaheuristic solution techniques for large scale MV networks, such as the genetic algorithm (GA) proposed in [10]. Other authors used different metaheuristics to solve the DSP problem, such as simulated annealing (SA), tabu search (TS), and GA [9,11,13].
Master’s Thesis: State of the Art Multi planning stages are considered in the DSP of MV networks for a time horizon to considered growth in the demand. In [8], authors used GAs in the optimal multistage planning of MV networks. Multiobjective functions allow to consider non-supplied energy under contingencies in the DSP problem to improve reliability. In [15], authors proposed a multiobjective multistage DSP using TS.
More accurate linear models were proposed to guarantee global optimal solutions, such as the mixed integer linear programming (MILP) model presented in [12] and solved using OSL solver in GAMS. Other authors presented different MILP models, such as the proposed in [16–19].
Nowadays, new technologies such as DGs and ESSs are integrated to distribution systems due to their benefits. As consequence, some papers considered the location and sizing of DGs and ESSs in the DSP of MV networks [30,32–37]. Some authors used GAs to solve the DSP problem considering DGs [32,34]. Other authors used a bilevel approach which combines an improved GA and a MILP model to solve the DSP problem considering DGs and ESSs [35]. These technologies introduce the need of considering different scenarios due to the variability and intermittency of the DGs and charge/discharge schemes of the ESSs. As consequence, some authors considered different scenarios using a probabilistic MILP model, such as in [30]. In [36], the authors proposed a stochastic methodology to consider different scenarios. Other authors used a mixed-integer second-order cone programming (MISOCP) to consider DGs and ESSs, such as in [37].
1.3.2
DSP for LV networks Low voltage networks are three-phase and large size, due to these characteristics, the DSP problem becomes computationally expensive to solve. As consequence, most of the papers in the literature regarding DSP for low voltage are solved using metaheuristic techniques and
Master’s Thesis: State of the Art one time stage [6]. Some authors used evolutionary algorithms to solve the DSP problem in a large size LV network, such as in [20], and some others used to solve the DSP problem in a three-phase LV network, such as in [21].
Other authors used heuristic techniques with a micro optimization approach to solve the DSP of LV networks separately in small zones [22]. After micro optimization is done, the authors used TS with Voronoi’s diagram with a macro optimization approach to solve the DSP of the complete LV network with the results of the micro optimization [23]. In [24], authors also used TS to solved the DSP of a three-phase LV network.
Electric utilities have different information about their customers due to the socioeconomic strata. Besides, these companies establish different technical requirements for the LV networks. This information and requirements need to be considered in the DSP problem of LV networks [25,26]. In [25], the authors proposed a statistical methodology that uses a fractalbased algorithm to solve the DSP of a real UK distribution system. The authors considered all the information and requirements of the electric company. Other authors used the TS algorithm using diversified demand from the electric companies to solve the DSP of a LV network, such as in [26].
Nowadays, new technologies as DGs and ESSs are integrated to distribution systems but only DGs are considered into the DSP of LV networks. As consequence, some papers considered the location and sizing of DGs in the DSP for LV networks [38–40]. In [38], the authors used a GA to solve the DSP for a real three-phase LV network considering DGs and the reliability of the network. In [39], authors used a TS algorithm to solve the DSP for a three-phase LV network considering DGs and the reliability of the network. Other authors used clustering techniques to solve the DSP for a rural LV network, such as in [40].
Master’s Thesis: Contributions
1.3.3
Integrated DSP for MV and LV networks Integrated DSP for both networks leads to a joint global solution but becomes computationally expensive to solve. As consequence, most of the paper regarding this integrated DSP used metaheuristic techniques to solve the problem [3–5,28,29,41]. In [27], the authors used a MILP model to solve the DSP of both networks in an integrated way. Other authors used continuous constrained nonlinear programming methods to solve the integrated DSP problem, such as in [28].
In [4], the authors used a discrete particle swarm optimization (DPSO) method to solve the integrated DSP problem.
Other authors used a new metaheuristic technique called biogeography-based optimization (BBO) and compared its efficiency with GA and PSO, such as in [3].
Currently, new technologies such as DGs, ESSs and EVs are incorporated to distribution systems, but only three papers considered them in the integrated DSP for both networks [5, 29,41]. In [29], the authors used an imperialist competitive algorithm to solve the integrated DSP problem of both networks considering DGs.
Other authors used TS and a bilevel optimization approach to solve the integrated DSP problem considering DGs, such as in [5]. In [41], the authors used a general variable neighborhood search metaheuristic (GVNS), along with the Chu-Beasley Genetic Algorithm (CBGA) to solve the integrated DSP problem considering DGs, ESSs and EVs. Nevertheless, it is needed to develop more robust models in order to solve the integrated DSP problem of both networks considering DGs and ESSs.
1.4
Contributions The main contributions of this thesis are presented as follows:
Master’s Thesis: Contributions
• Proposal of a specialized simulated annealing algorithm which uses a novel neighbor-
hood search method based on the Zbus matrix for solving the integrated DSP problem of both networks (primary and secondary) as one system. The Zbus matrix establishes an electrical relation between the primary and secondary nodes, where a change in one network is reflected in the electrical characteristics of the other one. As consequence, the results show that a joint global solution is found.
• A novel neighborhood search method based on the Zbus matrix (NSZM) is used to
explore the solution space for the SA algorithm. The NSZM method uses sensitivity factors based on the Zbus matrix for reducing the neighborhood size in order to find attractive solutions in quick times. Hence, the solution space is explored more efficiently in order to find the optimal global solution.
• A new sensitivity analysis based on the Zbus matrix for solving the allocation and
sizing of DGs and ESSs is proposed. The difference in relation to other papers from the literature is that the impact of DGs and ESSs in the technical losses of both networks is measured in an easy way. As a consequence, the solution space is explored efficiently in order to find better solutions.
• A new nonlinear model based on the Zbus matrix is used for the optimal power flow to
evaluate operative conditions of the solutions considering ESSs and DGs. This nonlinear model finds the optimal operation of charge and discharge of the ESSs considering the operative conditions of the network and DGs.
• A new decomposition method is used to solve the optimal power flow considering ESSs
and DGs. This method allows to find the optimal operation of charge and discharge of the ESSs in quick times. Hence, the proposed methodology is robust and accurate when ESSs and DGs are considered.
Master’s Thesis: Structure of the Thesis
1.4.1
Research results
• Authors: Alejandro Valencia Diaz, Ram´on Alfonso Gallego Rend´on, Ricardo Alberto
Hincapi´e Isaza. Title: Use of energy storage systems in the optimal operation of distribution networks. Accepted in: Ciencia e Ingenier´ıa Neogranadina Vol. 29 N´um. 2
(2019).
• Authors: Alejandro Valencia Diaz, Ricardo Alberto Hincapi´e Isaza, Ram´on Alfonso
Gallego Rend´on. Title: Integrated planning of MV/LV distribution systems with DG using single solution based metaheuristics with a novel neighborhood search method based on the Zbus matrix. Under revision: International Journal of Electrical Power & Energy Systems.
1.5
Structure of the Thesis This thesis is organized as follows. The mathematical formulation of the problem is presented in Chapter 2. This Chapter describes three mathematical models: 1) the DSP for MV and LV networks, 2) the DSP for MV and LV networks considering DGs, and 3) the DSP for MV and LV networks considering DGs and ESSs. The second model is used to validate the solution methodology with the specialized literature. Chapter 3 describes the application of the Zbus matrix into the DSP problem. Chapter 4 presents the methodology based on the Zbus matrix used to solve the DSP problem considering DGs and ESSs. This Chapter presents the methodology used to solve the mathematical models presented in Chapter 2. The numerical results are presented in Chapter 5, where a test system is used to validate the presented methodology with results reported in the specialized literature. Chapter 6 presents the conclusions and future work.
Chapter 2 Problem Formulation and Modeling The mathematical formulation of the DSP is a mixed-integer nonlinear problem. It is nonlinear due to the multiplication of variables in the equations of Kirchhoff’s laws and the presence of the square of the voltage in the constant impedance term of the ZIP load model. Additionally, the model is mixed-integer because it has both integer and continuous variables (decision variables, current magnitudes, voltage levels, etc.). New technologies such as DGs and ESSs, introduce new constraints and variables to the DSP model. Furthermore, the objective function and some constraints are modified since these technologies change the paradigm of the DSP problem.
Three models are formulated in this chapter. The first model considers the DSP for MV and LV networks. This model considers in the objective function the inversion and operative costs of both networks; and presents the constraints that models the DSP problem. The second model considers the DGs in the DSP problem. This model introduces the inversion cost of new DGs in the objective function. Besides, a few constraints are modified and introduced from the first model. The third model considers DGs and ESSs in the DSP problem. This model introduces the inversion cost and the fixed operational and maintenance costs of new
Master’s Thesis: DSP for MV and LV networks ESSs. The operative cost in the objective function is changed by the profit from the energy purchase and sale. Furthermore, some constraints are modified and introduced from the second model.
2.1
DSP for MV and LV networks The first mathematical model is presented in (2.1)–(2.20). The primary and secondary networks are represented by a single-phase model. Eq. (2.1) is the objective function which minimizes the investment and operative costs of the primary and secondary networks. The objective function is the present value of eight terms. Terms 1 and 2 are the installation and upgrading costs of the new and existing primary feeders, respectively. Terms 3 and 4 are the installation and upgrading costs of the new and existing substations, respectively. Term 5 is the cost of installing new DTs. Terms 6 and 7 are the installation and upgrading costs of the new and existing secondary circuits, respectively. Term 8 is the operative cost of the networks (technical energy losses in primary feeders, secondary circuits, and distribution transformers).
min =
(
X ij∈ΩNP X p∈ΩT P
CNP
ij,p γNP ij,p + X ij∈ΩEP X p∈ΩT P
CEP
ij,pγEP ij,p+ X i∈ΩNSS X s∈ΩT SS
CNSS
s γNSS i,s + X i∈ΩESS X s∈ΩT SS
CESS
s γESS i,s + X i∈ΩNDT X d∈ΩT DT
CNDT
d γNDT i,d + X ij∈ΩNS X c∈ΩT S
CNS
ij,c γNS ij,c + X ij∈ΩES X c∈ΩT S
CES
ij,cγES ij,c + k1 NL X l=1 k2l Re X i∈ΩSS SS i,l − X i∈ΩP N
SPD
i,l ZIP P i,l + X i∈ΩSN
SSD
i,l ZIP S i,l ∆Tl
)
(2.1)
Subject to:
Master’s Thesis: DSP for MV and LV networks SS i,l = SDT i,l + SPD i,l ZIP P i,l + V BUS i,l
IBUS
i,l ∗ ∀i ∈ΩPN; ∀l ∈ΩNL
(2.2)
SDT
i,l
= SSD
i,l ZIP S i,l + V BUS i,l
IBUS
i,l ∗ ∀i ∈ΩSN; ∀l ∈ΩNL
(2.3)
ZIP P
i,l = a0 + a1
V BUS
i,l V P nom !
+ a2
V BUS
i,l V P nom !2 ∀i ∈ΩPN; ∀l ∈ΩNL
(2.4)
ZIP S
i,l = a0 + a1
V BUS
i,l V S nom !
+ a2
V BUS
i,l V S nom !2 ∀i ∈ΩSN; ∀l ∈ΩNL
(2.5)
V BUS
i,l = V P nom + X j∈ΩN
ZBUS
ij
IBUS
j,l ∀i ∈ΩPN; ∀l ∈ΩNL
(2.6)
V BUS
i,l = V S nom + X j∈ΩN
ZBUS
ij
IBUS
j,l ∀i ∈ΩSN; ∀l ∈ΩNL
(2.7)
IBUS
i,l = X ij∈ΩP F IP ij,l ∀i ∈ΩSS; ∀l ∈ΩNL
(2.8)
IP ij,l =
V BUS
i,l
−V BUS
j,l ZP ij,p ! γNP ij,p + γEP ij,p ∀ij ∈ΩPF; ∀l ∈ΩNL; ∀p ∈ΩTP
(2.9)
IS ij,l =
V BUS
i,l
−V BUS
j,l ZS ij,c ! γNS ij,c + γES ij,c ∀ij ∈ΩSC; ∀l ∈ΩNL; ∀c ∈ΩTS
(2.10)
Master’s Thesis: DSP for MV and LV networks
0 ≤
IP ij,l
≤ X p∈ΩT P γNP ij,p + γEP ij,p Imax p ∀ij ∈ΩPF; ∀l ∈ΩNL
(2.11)
0 ≤
IS ij,l
≤ X c∈ΩT S γNS ij,c + γES ij,c Imax c ∀ij ∈ΩSC; ∀l ∈ΩNL
(2.12)
0 ≤
SS i,l
≤ X s∈ΩT SS γNSS i,s + γESS i,s Smax s ∀i ∈ΩSS; ∀l ∈ΩNL
(2.13)
SDT
i,l
= V BUS
i,l X ki∈ΩP F IP ki,l − X im∈ΩP F IP im,l ∗ γNDT i,d ∀i ∈ΩNDT; ∀l ∈ΩNL
(2.14)
0 ≤
SDT
i,l
≤ X d∈ΩT DT γNDT i,d Smax d ∀i ∈ΩNDT; ∀l ∈ΩNL
(2.15)
V min i ≤
V BUS
i,l
≤V max i ∀i ∈ΩN; ∀l ∈ΩNL
(2.16)
X p∈ΩT P γNP ij,p + γEP ij,p ≤1 ∀ij ∈ΩPF
(2.17)
X c∈ΩT S γNS ij,c + γES ij,c ≤1 ∀ij ∈ΩSC
(2.18)
X s∈ΩT SS γNSS i,s + γESS i,s ≤1 ∀i ∈ΩSS
(2.19)
Master’s Thesis: DSP for MV and LV networks considering DGs X d∈ΩT DT γNDT i,d ≤1 ∀i ∈ΩNDT
(2.20)
The set of constraints is presented in (2.2)–(2.20). Eqs. (2.2) and (2.3) represent the nodal balance given by Kirchhoff’s laws for both networks. Eqs. (2.4) and (2.5) use the ZIP model for demands in both networks. Eqs. (2.6) and (2.7) use Ohm’s law to calculate the nodal voltage using the Zbus matrix. Eq. (2.8) uses the Kirchhoff’s first law to calculate the nodal currents in substations. Eqs. (2.9) and (2.10) calculate the currents in the primary feeders and secondary circuits, respectively. Eqs. (2.11), (2.12) and (2.13) are the operating limits of the primary feeders, secondary circuits, and substations, respectively. Eq. (2.14) determines the injected power in each DT. Eq. (2.15) is the operating limits of the DTs. Eq. (2.16) is the voltage limit in all nodes of both networks. Eqs. (2.17)–(2.20) ensure that only one type of wire, substation, and DT can be installed in the same place, respectively.
2.2
DSP for MV and LV networks considering DGs The mathematical formulation of the DSP considering DGs is similar to the previous one, only some constraints are modified and some are added due to the DGs. Eqs. (2.1) and (2.3) are modified and replaced by Eqs. (2.21) and (2.22), and Eqs. (2.23) and (2.24) are added. Eq. (2.21) is the objective function which minimizes the investment and operative costs of the primary and secondary networks considering DGs. The objective function is the present value of nine terms. Terms 1 to 8 are the same of Eq. (2.1), and term 9 is the installation cost of new DGs.
Master’s Thesis: DSP for MV and LV networks considering DGs min =
(
X ij∈ΩNP X p∈ΩT P
CNP
ij,p γNP ij,p + X ij∈ΩEP X p∈ΩT P
CEP
ij,pγEP ij,p+ X i∈ΩNSS X s∈ΩT SS
CNSS
s γNSS i,s + X i∈ΩESS X s∈ΩT SS
CESS
s γESS i,s + X i∈ΩNDT X d∈ΩT DT
CNDT
d γNDT i,d + X ij∈ΩNS X c∈ΩT S
CNS
ij,c γNS ij,c + X ij∈ΩES X c∈ΩT S
CES
ij,cγES ij,c + k1 NL X l=1 k2l Re X i∈ΩSS SS i,l − X i∈ΩP N
SPD
i,l ZIP P i,l + X i∈ΩSN
SSD
i,l ZIP S i,l ∆Tl+ X i∈ΩNDG X g∈ΩT DG
CNDG
g γNDG i,g
)
(2.21)
SDT
i,l + SDG i,l
= SSD
i,l ZIP S i,l + V BUS i,l
IBUS
i,l ∗ ∀i ∈ΩSN; ∀l ∈ΩNL
(2.22)
0 ≤
SDG
i,l
≤Smax g γNDG i,g ∀i ∈ΩNDG; ∀l ∈ΩNL; ∀g ∈ΩTDG
(2.23)
X g∈ΩT DG γNDG i,g ≤1 ∀i ∈ΩNDG
(2.24)
Eq. (2.22) represents the nodal balance given by Kirchhoff’s laws for LV network considering DGs. Eq. (2.23) is the operating limits of the DGs. Eq. (2.24) ensures that only one type of DG can be installed in the same place. The second mathematical model is presented in (2.21),(2.2),(2.22),(2.4)–(2.15),(2.23),(2.16)–(2.20),(2.24).
Objective function:
(
FObj = Eq. (2.21)
)
Master’s Thesis: DSP for MV and LV networks considering DGs and ESSs Subject to:
(
Eqs. (2.2), (2.22), (2.4) −(2.15), (2.23), (2.16) −(2.20), (2.24)
)
2.3
DSP for MV and LV networks considering DGs and ESSs The mathematical formulation of the DSP considering ESSs has some changes from the previous models. ESSs are integrated to maximize the profits from the energy purchase and sale due to the charge and discharge scheme of these. As consequence, the objective function and some constraints are modified, and some others are added. Eqs. (2.21), (2.2) and (2.22) are modified and replaced by Eqs. (2.25), (2.26) and (2.27). Eqs. (2.28)–(2.35) are added. Eq. (2.25) is the objective function which minimizes the investment and operative costs of the primary and secondary networks considering DGs and ESSs, and maximizes the profits from the energy purchase and sale. The maximize term can be converted to minimize by multiplying for minus one.
The objective function is the present value of nine terms. Terms 1 to 7, and 9 are the same of Eq. (2.21). Term 8 is the profit from the energy purchase and sale. Moreover, this term also considers the operative cost of the networks (technical energy losses in primary feeders, secondary circuits, distribution transformers, and energy storage systems). The value of this term is negative due to the change from maximize to minimize. Term 10 considered the installation cost and the fixed operational and maintenance costs of new ESSs.
Master’s Thesis: DSP for MV and LV networks considering DGs and ESSs min =
(
X ij∈ΩNP X p∈ΩT P
CNP
ij,p γNP ij,p + X ij∈ΩEP X p∈ΩT P
CEP
ij,pγEP ij,p+ X i∈ΩNSS X s∈ΩT SS
CNSS
s γNSS i,s + X i∈ΩESS X s∈ΩT SS
CESS
s γESS i,s + X i∈ΩNDT X d∈ΩT DT
CNDT
d γNDT i,d + X ij∈ΩNS X c∈ΩT S
CNS
ij,c γNS ij,c + X ij∈ΩES X c∈ΩT S
CES
ij,cγES ij,c + k1 NL X l=1 k2l X i∈ΩSS Re n SS i,l o ∆Tl − k3 Re X i∈ΩP N
SPD
i,l ZIP P i,l + X i∈ΩSN
SSD
i,l ZIP S i,l ∆Tl Dyear+ X i∈ΩNDG X g∈ΩT DG
CNDG
g γNDG i,g + X i∈ΩNESS X b∈ΩT ESS
CNESS
b + k1 O&M fx b γNESS i,b
)
(2.25)
SS i,l + SESS i,l
= SDT
i,l + SPD i,l ZIP P i,l + V BUS i,l
IBUS
i,l ∗ ∀i ∈ΩPN; ∀l ∈ΩNL
(2.26)
SDT
i,l + SDG i,l
+ SESS
i,l
= SSD
i,l ZIP S i,l + V BUS i,l
IBUS
i,l ∗ ∀i ∈ΩSN; ∀l ∈ΩNL
(2.27)
SESS
i,l
= SESSD
i,l
−SESSC
i,l ∀i ∈ΩNESS; ∀l ∈ΩNL
(2.28)
SoCi,l = SoCi,l−1 −φb
1
ηb
SESSD
i,l −ηbSESSC i,l !
∆Tl ∀i ∈ΩNESS; ∀l ∈ΩNL; ∀b ∈ΩTESS
(2.29)
SoCi,l = SoC0 i l = 0; ∀i ∈ΩNESS
(2.30)
Master’s Thesis: DSP for MV and LV networks considering DGs and ESSs SoCi,l = SoCF i l = NL; ∀i ∈ΩNESS
(2.31)
0 ≤SESSC
i,l ≤Smaxc b γNESS i,b ϕNESS i,l ∀i ∈ΩNESS; ∀l ∈ΩNL; ∀b ∈ΩTESS
(2.32)
0 ≤SESSD
i,l ≤Smaxd b γNESS i,b
1 −ϕNESS
i,l ∀i ∈ΩNESS; ∀l ∈ΩNL; ∀b ∈ΩTESS
(2.33)
SoCmin i ≤SoCi,l ≤SoCmax i ∀i ∈ΩNESS; ∀l ∈ΩNL
(2.34)
X b∈ΩT ESS γNESS i,b ≤1 ∀i ∈ΩNESS
(2.35)
Eqs. (2.26) and (2.27) represent the nodal balance given by Kirchhoff’s laws for MV and LV networks considering ESSs. Eqs. (2.28)–(2.35) model the integration of ESSs into the DSP model, where if ϕNESS i,l = 0 the ESSs are injecting power and if ϕNESS i,l = 1 the ESSs are extracting power. Eq. (2.28) considers the charge and discharge power of the ESSs as two different variables. Eq. (2.29) is the state of charge of the ESSs, this constraint considers the efficiency of charge and discharge of the ESSs. Eqs. (2.30) and (2.31) are the initial and final state of charge of the ESSs. Eqs. (2.32) and (2.33) are the operating limits of the charge and discharge power of the ESSs. Eq. (2.34) is the capacity limits of the ESSs. Eq. (2.35) ensures that only one type of ESS can be installed in the same place. The third mathematical model is presented in (2.25),(2.26)–(2.31),(2.4)–(2.15),(2.23),(2.32)–(2.34),(2.16)–(2.20),(2.24),(2.35). Objective function:
Master’s Thesis: DSP for MV and LV networks considering DGs and ESSs
(
FObj = Eq. (2.25)
)
Subject to:
(
Eqs. (2.26) −(2.31), (2.4) −(2.15), (2.23), (2.32) −(2.34), (2.16) −(2.20), (2.24), (2.35)
)
Chapter 3 Zbus Matrix The Zbus matrix represents the relation between current injections and voltages in a system. This matrix is formed by diagonal and off-diagonal elements that represent the electrical information of the network. The diagonal elements of Zbus are the equivalent impedances between each bus and the reference bus, which are the same as the Thevenin impedances of each bus [45]. The off-diagonal elements are called transfer impedances and define the ratio of change for the voltage at a certain bus caused by a current injection at another bus (i.e., bus k). As consequence, the Zbus matrix reflects important electrical characteristics of the network, such as information for the neighborhood structure (branch exchange, upgrading existing elements, and the location of DGs, ESSs and DTs to minimize operative costs). Besides its electrical information, the Zbus matrix also contains and reflects some topological characteristics of the network. Thus, the Zbus matrix is used as sensitivity factors for reducing the neighborhood size of the solution method (SA). Moreover, the Zbus matrix is used for the analysis of the impact of DGs and ESSs in the technical losses of both network and for the solution of load flow to evaluate operative conditions of the solutions. Therefore, the Zbus matrix is used as the main basis of the methodology proposed to solve the DSP problem.
Master’s Thesis: Constructive algorithm of the Zbus matrix
3.1
Constructive algorithm of the Zbus matrix The Zbus matrix can be found by inverting the Ybus matrix.
The disadvantage of this method is that inverting the Ybus matrix for large networks is computationally intensive. Therefore, it is better to build the Zbus by analyzing the relations between the currents and the voltages.
The constructive algorithm starts by adding branches from the reference node 0 until all the n nodes of the network are connected. In each step, a m × m partial Zbus matrix of the partial network of m buses and the reference node 0 is obtained. This procedure to find the Zbus matrix using a construction algorithm is described in [49]. This algorithm is only for radial networks and is shown in Fig. 3.1.
When a new element p −q is added to the partial network, a new bus q is incorporated to the partial network and the new resultant Zbus matrix is of dimension (m + 1) × (m + 1). To determine the new Zbus matrix requires only the calculation of the elements in the new row and column.
ZBUS
qi
= ZBUS
pi
ZBUS
iq
= ZBUS
ip i = 1, 2, ..., m; i̸ = q
(3.1)
ZBUS
= ZBUS
pq + zprimitive pq
(3.2)
ZBUS
qi = 0
ZBUS
iq = 0 i = 1, 2, ..., m; i̸ = q
(3.3)
ZBUS
qq = zprimitive pq
(3.4)
Master’s Thesis: Modifications of the Zbus matrix to reflect changes in the network The Eqs. (3.1) and (3.2) represents the adding of the element p −q in the partial bus impedance matrix.
The Eqs. (3.3) and (3.4) represents the adding of the element p −q in the partial bus impedance matrix when p is the reference node. Zbus matrix construction algorithm 1: begin 2:
while All nodes are not connected in the network do 3:
Add new element p −q 4:
if Node p is the reference node then 5:
Use Eq. (3.3); 6:
Use Eq. (3.4); 7:
else 8:
Use Eq. (3.1); 9:
Use Eq. (3.2); 10:
end if 11:
Partial Zbus matrix 12:
end while 13:
Final Zbus matrix 14: end Figure 3.1: Constructive algorithm of the Zbus matrix.
3.2
Modifications of the Zbus matrix to reflect changes in the network The Zbus matrix can be modified to reflect changes in the network without the need to build it again. These changes may be the addition or removal of elements and variation in the impedances of elements. These modifications are described in [49]. For adding an element, two options can happen: add a branch or a link. When a branch p −q is added, a new bus q is incorporated to the network adding a new row and column
Master’s Thesis: Modifications of the Zbus matrix to reflect changes in the network in the Zbus matrix. The Eqs. (3.1) and (3.2) are used to calculated the elements of the new row and column.
When a link p −q is added, no new bus is added to the network. Therefore, the dimensions of the Zbus matrix are unchanged, but all the elements in the matrix must be recalculated to incorporate the effect of the added link. To include the link in the Zbus matrix, the Eqs. (3.5) and (3.6) are used. When these equations are applied, a new row and column is added to the Zbus matrix, hence, a fictitious node l is added.
ZBUS
li
= ZBUS
pi
−ZBUS
qi
ZBUS
il
= ZBUS
ip
−ZBUS
iq i = 1, 2, ..., m; i̸ = l
(3.5)
ZBUS
ll
= ZBUS
pl
−ZBUS
ql + zprimitive pq
(3.6)
For recalculating the elements of the Zbus matrix, Kron’s reduction is used. This procedure was proposed by Kron [50], and is based on the elimination of a node in a matrix where the independent variable of this node is equal to zero. Eq. (3.7) represents the elimination of the row and column of the fictitious bus l by using the reduction of Kron.
ZBUS
modified = ZBUS n×n −Z
BUS
n×l Z
BUS
l×n
ZBUS
ll
(3.7)
For removing an element p −q, the next procedure is applied. Between p −q there is added, in parallel, a link whose impedance is equal to the negative of the impedance of the element p −q to be removed, then the procedure explained before to recalculate the Zbus matrix is applied.
The procedure to change the impedance of an element p −q is next. Between p −q there is added, in parallel, a link whose impedance is such that the equivalent impedance of the two
Master’s Thesis: Topological information of the network provided by the Zbus matrix elements in p −q is the desired value, then the elements of the Zbus matrix are recalculated. Eq. (3.8) gives the value of the impedance added in the link p −q to obtain the new value of impedance from the old value.
zchange pq = zold pq znew pq zold pq −znew pq
(3.8)
3.3
Topological information of the network provided by the Zbus matrix The Zbus matrix provides physical information of the network by comparing some elements of the matrix with each other. The physical information obtained from the Zbus matrix is the next: Identification of terminal nodes, nodes upstream from a specific node, nodes downstream from a specific node, and send and receipt nodes. This information is useful for the neighborhood structure used to solve the DSP problem.
The neighborhood structure is applied to explore the search solution space of the problem in order to find the optimal global solution. The neighborhood structure modifies the topology from the current solution in order to explore the solution space. When these changes in the topologies are applied, it is important to know the physical information of the network to keep making changes for the neighborhood structure. A mishandling of the neighborhood structure can lead to expensive computational efforts, forbidden topologies and bad quality solutions. Therefore, the Zbus matrix is used to know the physical information of the current solutions in order to optimize the neighborhood structure process. The physical information that can be obtained from the Zbus matrix is the next.
The procedure to know if a specific node is a terminal node is next. If the Eq. (3.9) is true for all i = 1, 2, ..., n, the node k is a terminal node.
Master’s Thesis: Electrical information of the network provided by the Zbus matrix
ZBUS
kk̸
= ZBUS
ki i = 1, 2, ..., n; i̸ = k
(3.9)
To know which nodes are upstream from a specific node, the next procedure is applied. If the Eq. (3.10) is true, the node i is a node upstream from the node k.
ZBUS
ii
= ZBUS
ki
(3.10)
The procedure to know which nodes are downstream from a specific node is next. If the Eq. (3.11) is satisfied, the node i is downstream from the node k.
ZBUS
kk
= ZBUS
ki
(3.11)
The procedure to identify which nodes are send and receipt nodes is next. If Eq. (3.12) is true, the node p is the receipt node and the node q is the send node. Otherwise, the node p is the send node and the node q is the receipt node.
ZBUS
pp
> ZBUS
(3.12)
3.4
Electrical information of the network provided by the Zbus matrix Due to the matrix form of the Zbus, it is easy to obtain in a quick way some electrical variables, such as the voltages. These are obtained by using the Eq. (3.13). The voltages
Master’s Thesis: Electrical information of the network provided by the Zbus matrix obtained are used to calculate in approximated way other variables such as the electric current in the branches and the power in the DTs.
V BUS = V slack + ZBUS S
DZIP
V slack + S DG V slack + S
ESS
V slack ∗
(3.13)
3.4.1
Impact of the DGs and ESSs on the active power losses of the network The Zbus matrix provides electrical information of the network in order to reduce the neighborhood size. This electrical information is obtained by using sensitivity factors in order to lead to the optimal global solution with reduced computational efforts. DGs and ESSs are power injections introduced in a distributed way in the network affecting the active power losses. Therefore, it is needed a sensitivity factor that measures the impact of a change in the bus current injected in a given bus k on the active power losses in the distribution system (i.e., ∂PLosses/∂IBUS k
).
The sensitivity factor applied in order to reduce the neighborhood structure of DGs and ESSs is demonstrated as follows. Network losses can be calculated using the Zbus matrix as shown in Eq. (3.14). From Eq. (3.14) the active network losses can be determined as shown in Eq. (3.15).
SLosses =
IBUST ZBUS
IBUS∗
(3.14)
PLosses = n X i=1 n X j=1 h
RBUS
ij
IBUS
i
IBUS
j cos (Θi −Θj) i
(3.15)
Master’s Thesis: Load flow based on the Zbus matrix Taking the derivative of PLosses, in Eq. (3.15), with respect to IBUS k , yields Eq. (3.16).
∂PLosses
∂IBUS
k = 2 n X j=1 h
RBUS
kj
IBUS
j cos (Θk −Θj) i
(3.16)
This sensitivity factor calculates the impact in the active network losses due different power injections of the DGs and ESSs. As consequence, this factor leads the neighborhood structure in order to find the best alternatives.
3.5
Load flow based on the Zbus matrix The load flow is essential for knowing the operating conditions of each generated configuration. As a consequence, a load flow based on the Zbus Gauss method described in [49] is proposed. This load flow is based upon the principle of superposition applied to the system bus voltages. The bus voltages depend on two different types of sources: the specified incoming bus voltage of the distribution substation, and the current injection which is generated by the loads. New types of sources like DGs and ESSs can be included such as PQ nodes. The steps of this load flow algorithm are described as follows. Step 1: Initialize bus voltage estimates and the Zbus matrix.
Step 2: Calculate the bus injection current using Eq. (3.17) for loads. Ik i = SD i ZIP k i V k i !∗ +
SDG
i V k i !∗ +
SESS
i V k i !∗
(3.17)
Step 3: Calculate the new bus voltages applying the superposition principle using Eq. (3.18). V k+1 BUS = V slack + ZBUSI k
BUS
(3.18)
Master’s Thesis: Load flow based on the Zbus matrix Step 4: Check for convergence using Eq. (3.18). If not converged go to step 2. max n |V k+1 BUS −V k
BUS|
o ≤Tolerance
(3.19)
Step 5: Calculate the rest of electric variables needed.
Chapter 4 Solution Methodology As was mentioned in Chapter 2, the optimization problem is a mixed-integer nonlinear programming problem, and to solve it, a simulated annealing (SA) algorithm with a new neighborhood search method based on the Zbus matrix (NSZM) is used. The NSZM method explores the solution space for the SA algorithm using a defined neighborhood structure and sensitivity factors based on the Zbus matrix. The proposed methodology starts with an initial configuration for both networks (medium and low voltage) obtained using a constructive heuristic algorithm. It is essential to highlight that only feasible topologies in the initial configuration are allowed. Afterward, the Zbus matrix of this solution is obtained using a constructive algorithm explained in Chapter 3 (Section 3.1).
The NSZM method chooses one criterion of the neighborhood structure in order to explore the solution space. In the process, the NSZM method uses the current Zbus matrix to obtain information provided by the sensitivity factors in order to choose a new solution. Afterward, the current Zbus matrix is modified to reflect the changes made by the neighborhood structure. Then, the stochastic mechanism of the SA algorithm controls the transition between the solutions. The acceptance of new configurations obeys the following criteria: topologies
Master’s Thesis: Codification with decreasing objectives are always accepted, whereas configurations with higher costs can be accepted or not with a certain probability. The possibility of accepting solutions with an elevated cost avoids getting trapped in local minima.
The main aspects of the SA algorithm and the new NSZM method used in this thesis will be discussed below.
4.1
Codification A codification scheme allows to represent the variables of a mathematical model into a vector. Therefore two different codification schemes are used in this thesis in order to represent two models: DSP of primary and secondary networks considering DGs and DSP of primary and secondary networks considering DGs and ESSs.
The codification employed for the primary and secondary DSP uses integer numbers, where each number (for all the elements) is associated to a different capacity. A zero indicates that the element is not installed.
4.1.1
DSP considering DGs The DSP of primary and secondary networks considering DGs is encoded using the codification scheme illustrated in Fig. 4.1. This vector is divided into five parts. The first part contains the locations and sizes of the existing and new primary feeders (size n1 + n2); the second part contains the locations and sizes of the existing and new secondary circuits (size n3 + n4); the third part involves the locations and capacities of the DTs (size n5); the fourth part contains the locations and sizes of the existing and new substations (size n6 + n7); and the fifth part contains the locations and capacities of the DGs (size n8).
Master’s Thesis: Codification Primary feeders Secondary circuits DTs Substations DGs
3
γEP
1,3
1
..
..
..
1
γEP n1,1 n1
2
γNP
1,2
1
..
..
..
4
γNP n2,4 n2
3
γES
1,3
1
..
..
..
1
γES n3,1 n3
2
γNS
1,2
1
..
..
..
3
γNS n4,3 n4
5
γNDT
1,5
1
..
..
..
1
γNDT n5,1 n5
2
γESS
1,2
1
..
..
..
1
γESS n6,1 n6
3
γNSS
1,3
1
..
..
..
2
γNSS n7,2 n7
3
γNDG
1,3
1
..
..
..
1
γNDG n8,1 n8 Figure 4.1: Codification scheme for primary and secondary DSP considering DGs.
4.1.2
DSP considering DGs and ESSs The DSP of primary and secondary networks considering DGs and ESSs is encoded using the codification scheme illustrated in Fig. 4.2. This vector is divided into five parts. The first part contains the locations and sizes of the existing and new primary feeders (size n1 + n2); the second part contains the locations and sizes of the existing and new secondary circuits (size n3 + n4); the third part involves the locations and capacities of the DTs (size n5); the fourth part contains the locations and sizes of the existing and new substations (size n6+n7); the fifth part contains the locations and capacities of the ESSs (size n8); and the six part contains the locations and capacities of the DGs (size n9).
Primary feeders Secondary circuits DTs
3
γEP
1,3
1
..
..
..
1
γEP n1,1 n1
2
γNP
1,2
1
..
..
..
4
γNP n2,4 n2
3
γES
1,3
1
..
..
..
1
γES n3,1 n3
2
γNS
1,2
1
..
..
..
3
γNS n4,3 n4
5
γNDT
1,5
1
..
..
..
1
γNDT n5,1 n5
...
...
Substations ESSs DGs
2
γESS
1,2
1
..
..
..
1
γESS n6,1 n6
3
γNSS
1,3
1
..
..
..
2
γNSS n7,2 n7
3
γNESS
1,3
1
..
..
..
2
γNESS n8,2 n8
4
γNDG
1,3
1
..
..
..
1
γNDG n8,1 n9 Figure 4.2: Codification scheme for primary and secondary DSP considering DGs and ESSs.
Master’s Thesis: Initial configuration
4.2
Initial configuration The initial configuration for both networks (medium and low voltage) is obtained using a constructive heuristic algorithm. This algorithm begins from the existing source (substations), and in each step, a new branch (primary feeder, DT, or secondary circuit) is connected to the system. When a branch is added, the operating limits are checked (voltage regulation and capacities of the elements). This strategy stops when all demand nodes are connected to the network. It is essential to highlight that only feasible topologies in the initial configuration are allowed.
4.3
Neighborhood search method based on the Zbus matrix (NSZM) This method explores the solution space using a defined neighborhood structure and sensitivity factors based on the Zbus matrix to find attractive solutions in quick times. The method uses an initial Zbus matrix to obtain a new Zbus matrix which represents a new solution for the SA algorithm. The NSZM method chooses one criterion of the neighborhood structure in order to explore the solution space. The criteria defined for the neighborhood structure uses the Eq. (3.13) and the sensitivity factor (SF) showed in Eq. (3.16) in order to choose a new solution. Afterward, the initial Zbus matrix is modified to reflect the changes made by the neighborhood structure.
The main aspects of the NSZM method are explained next.
Master’s Thesis: Neighborhood search method based on the Zbus matrix (NSZM)
4.3.1
Neighborhood structure The neighborhood structure of the NSZM method uses the next criteria: the upgrading of existing substations, primary feeders and secondary circuits, installation of new substations, DTs and DGs, and branch exchange in both networks. From a current solution, a set of topologies are generated using one of the criteria explained. These configurations are called neighbors (or a neighborhood), and because the large amount they are, can be reduced to a defined number (reduced neighborhood) using sensitivity factors. The Zbus matrix is used for the sensitivity factors (SFs) for reducing the size of the neighborhood and choosing one topology from the neighborhood. The neighborhood structure considered is explained below.
• Branch exchange. The branch with max {∆Vij} is selected to connect the link. Then,
the branch with min {Iij} of the loop formed is selected for removal. The currents of the loop are calculated using the Eq. (3.13).
• DT installation. A DT is chosen randomly from the three DTs with the largest capaci-
ties. If there are DTs disconnected nearby, one is chosen at random to be installed, and some loads are transferred from the first DT. Otherwise, the DT to be added is chosen at random from the disconnected DTs. Some loads need to be transferred to the new DT in order to comply radiality in the secondary circuits. Loads are transferred from the nearest existing DT to the new one. Loads are transferred by removing the circuit with min {Iij} of the loop formed. The currents of the loop are calculated using the Eq. (3.13). In the process, it is checked that the sizes of the DTs respect the operating limits.
• DT removal. The DT to be removed is chosen randomly from the three DTs with the
smallest capacities. The loads of this DT need to be transferred to another existing DT. The DT chosen for receiving these loads is the nearest one with less capacity. If there is more than one such, the DT that is less overloaded is chosen. If there is more than
Master’s Thesis: Neighborhood search method based on the Zbus matrix (NSZM) one DT that fulfills these conditions, the DT is chosen randomly from these. Loads are transferred by removing the circuit with min {Iij} of the loop formed. The currents of the loop are calculated using the Eq. (3.13).
• Upgrading of existing branches and DTs. Two alternatives are considered. In the first
one, an element is chosen randomly from the three most overloaded elements, and the element is replaced by one with a bigger size. In the second one, an element is chosen randomly from the three least overloaded elements, and the element is replaced by one with a smaller size. One of these alternatives is chosen randomly. Primary feeders, secondary circuits, and DTs are chosen for upgrading capacity. One of these elements is selected randomly. Applying these criteria should ensure that the sections upstream do not present a smaller-size wire in the primary feeders and secondary circuits.
• Installation and sizing of DGs and ESSs. Four alternatives are considered: addition
and removal of a DG or ESS and increasing and decreasing the size of a DG or ESS. One of these alternatives is chosen randomly. For each alternative, the sensitivity factor of Eq. (3.16) is applied, and the node which produces the minimum value is chosen. The sensitivity factor measures the impact of a change in the bus current injected in a given bus k on the active power losses in the distribution system (i.e., ∂PLosses/∂IBUS k
).
For the installation size of the DG or ESS, a random number from the types of DGs or ESSs is chosen and the sensitivity is applied.
Along the process, the neighborhood structure needs to know the physical information of the networks in order to make the changes proposed by the criteria. This information is obtained from the Zbus matrix as was explained in Chapter 3 (Section 3.3). After one of the criteria of the neighborhood structure is applied, the Zbus matrix is modified to reflect the changes made by the criterion.
Master’s Thesis: Neighborhood search method based on the Zbus matrix (NSZM)
4.3.2
Modifications of the Zbus matrix to reflect changes by the neighborhood structure The Zbus matrix can be modified to reflect changes in the network caused by the neighborhood structure without the need to build it again (see Fig. 4.3). These changes may be the addition or removal of elements (feeders or circuits) for branch exchange, the addition or removal of elements (DTs or circuits) for DTs location, and changes in the impedances of elements (primary feeders, DTs, and secondary circuits) for upgrading or degrading the capacity. These modifications are described in Chapter 3 (Section 3.2). Network k m Existing branch Zkm p q
0
(a) Branch exchange (b) Upgrading or degrading capacity Network k m Zkm p q Znew pq
0
−Zkm Zequivalent km = ∞ Network k m Zold km p q
0
Zchange km Zequivalent km = Znew km MV Network LV Networks A a Existing DT
ZDT
Aa Existing circuit Zcd b c d e f B MV Network LV Networks A a Existing DT
ZDT
Aa ZnewDT Be Zcd −Zcd Zequivalent Be = ∞ b c d e f B MV Network LV Networks A a Existing DT
ZDT
Aa Existing DT
ZDT
Be b c d e f B MV Network LV Networks A a Existing DT
ZDT
Aa
ZDT
Be
−ZDT
Be Zequivalent Be = ∞ Znew cd b c d e f B (c) DT installation (d) DT removal Figure 4.3: Zbus modification due to neighborhood criteria.
Master’s Thesis: Neighborhood search method based on the Zbus matrix (NSZM) To calculate the new Zbus matrix produced by the neighborhood structure, the procedure is the following. For a branch exchange, the new element is added as a link between nodes p−q (see Fig. 4.3a), and when it is added, the procedure explained before to recalculate the Zbus matrix is applied. After this, the element k −m of the loop formed by the adding element p −q is removed. Between k −m there is added, in parallel, a link whose impedance is equal to the negative of the impedance of the element k −m to be removed (see Fig. 4.3a), then the elements of the Zbus matrix are recalculated.
In per unit representation, the DTs are treated as lines.
As consequence, the procedure described for branch exchange is the same for DTs location (adding a DT or removing a DT). For DTs installation, the new DT is added as a link between nodes B −e (see Fig. 4.3c), and when it is added, the procedure explained before to recalculate the Zbus matrix is applied. After this, the circuit c −d of the loop formed by the adding element B −q is removed. Between c −d there is added, in parallel, a link whose impedance is equal to the negative of the impedance of the circuit c −d to be removed (see Fig. 4.3c), then the elements of the Zbus matrix are recalculated.
For the DTs removal, the procedure is quite the same. The new secondary circuit is added as a link between nodes c −d (see Fig. 4.3d), and when it is added, the procedure explained before to recalculate the Zbus matrix is applied. Afterward, the existing DT B−e is removed in order to comply radiality. Between B−e there is added, in parallel, a link whose impedance is equal to the negative of the impedance of the circuit B −e to be removed (see Fig. 4.3d), then the elements of the Zbus matrix are recalculated.
For upgrading or degrading the capacity of an element, the procedure is next. Between k−m there is added, in parallel, a link whose impedance is such that the equivalent impedance of the two elements in k −m is the desired value (see Fig. 4.3b), then the elements of the Zbus matrix are recalculated. Eq. (3.8) gives the value of the impedance added in the link k −m to obtain the new value of impedance from the old value.
Master’s Thesis: Evaluation of the configurations
4.4
Evaluation of the configurations During the procedure, the proposed configurations are evaluated using a load flow. When the DGs are considered into the DSP problem, the load flow used is the described in Section 3.5 due the DGs are treat as PQ nodes. Nevertheless, when the ESSs are considered into the DSP problem, it is needed an optimal power flow to determine the operation of the ESSs in order to maximize the profit from the energy purchase and sale. The optimal power flow (OPF) proposed in this thesis is explained next.
4.4.1
Optimal Power Flow considering DGs and ESSs The mathematical model proposed for the optimal power flow used in this thesis is based on the Zbus matrix. The constraints used in this OPF to represent the integration of ESSs were explained in the DSP mathematical model considering ESSs (Section 2.3). From that model, the Eqs. (2.28)–(2.34) are used and all the binary variables which reflect the inversion decisions of ESSs are eliminated. In Eqs. (2.32)–(2.33) the only binary variable left is ϕNESS i,l which represents the optimal operation of the ESSs, where if ϕNESS i,l = 0 the ESSs are injecting power and if ϕNESS i,l = 1 the ESSs are extracting power.
This variable makes the mathematical formulation of the OPF a mixed-integer nonlinear problem. Hence, the Eqs. (2.32)–(2.33) are linearized in the Eqs. (4.15)–(4.16) to convert the OPF model into a nonlinear problem.
The OPF mathematical model is presented in the Eqs. (4.1)–(4.20). Per unit representation is applied to the optimal power flow model in order to represent the MV and LV networks as one. The Eq. (4.1) is the objective function which maximizes the utilities from the energy purchase and sale. The set of constraints is presented in (4.2)–(4.20). Eqs. (4.2) and (4.3) represent the active and reactive power nodal balance given by Kirchhoff’s laws for both
Master’s Thesis: Evaluation of the configurations networks, respectively. Eq. (4.4) represents the ZIP model in per unit representation for demands in both networks. Eqs. (4.5) and (4.6) use Ohm’s law to calculate the real and imaginary part of the nodal voltage using the Zbus matrix.
Eqs. (4.7) and (4.8) define the value of the voltage in the slack node in magnitude and angle. Eqs. (4.9) and (4.10) use the Kirchhoff’s first law to calculate the real and imaginary part of the nodal currents in substations. Eq. (4.11) is the state of charge of the ESSs, this constraint considers the efficiency of charge and discharge of the ESSs. Eqs. (4.12) and (4.13) are the initial and final state of charge of the ESSs. Eqs. (4.14)–(4.17) ensure that the ESSs are not extracting and injecting power at the same time. Eq. (4.14) considers the charge and discharge power of the ESSs as two different variables. Eqs. (4.15) and (4.16) are the operating limits of the charge and discharge power of the ESSs. Eq. (4.18) is the capacity limits of the ESSs. Eq. (4.19) is the operating limits of the DGs. Eq. (4.20) is the voltage limit in all nodes of both networks. max = k1Dyear NL X l=1 k3 X i∈ΩN P D i,lZIPi,l −k2l X i∈ΩSS P S i,l ∆Tl Sbase
(4.1)
s.t.
P S i,l + P DG i,l
+ P ESS
i,l =P D i,lZIPi,l + V BUS i,l
IBUS
i,l cos θV i,l −θI i,l ∀i ∈ΩN; ∀l ∈ΩNL
(4.2)
QS i,l = QD i,lZIPi,l + V BUS i,l
IBUS
i,l sin θV i,l −θI i,l ∀i ∈ΩN; ∀l ∈ΩNL
(4.3)
ZIPi,l = a0 + a1V BUS i,l + a2
V BUS
i,l 2 ∀i ∈ΩN; ∀l ∈ΩNL
(4.4)
Master’s Thesis: Evaluation of the configurations
V BUS
i,l cos θV i,l
=V BUS
slack,l cos θV slack,l + X j∈ΩN
IBUS
i,l h
RBUS
ij cos θI j,l −
XBUS
ij sin θI j,l i ∀i ∈ΩN; ∀l ∈ΩNL; i̸ = slack
(4.5)
V BUS
i,l sin θV i,l
=V BUS
slack,l sin θV slack,l + X j∈ΩN
IBUS
i,l h
RBUS
ij sin θI j,l +
XBUS
ij cos θI j,l i ∀i ∈ΩN; ∀l ∈ΩNL; i̸ = slack
(4.6)
V BUS
i,l = V p.u ref i = slack; ∀l ∈ΩNL
(4.7)
θV i,l = θrad ref i = slack; ∀l ∈ΩNL
(4.8)
IBUS
i,l cos θI i,l = X ij∈ΩP F GP ij h
V BUS
i,l cos θV i,l
−V BUS
j,l cos θV j,l i − BP ij h
V BUS
i,l sin θV i,l
−V BUS
j,l sin θV j,l i ∀i ∈ΩSS; ∀l ∈ΩNL
(4.9)
IBUS
i,l sin θI i,l = X ij∈ΩP F BP ij h
V BUS
i,l cos θV i,l
−V BUS
j,l cos θV j,l i + GP ij h
V BUS
i,l sin θV i,l
−V BUS
j,l sin θV j,l i ∀i ∈ΩSS; ∀l ∈ΩNL
(4.10)
SoCi,l = SoCi,l−1 −φi,b
1
ηi,b
P ESSD
i,l −ηi,bP ESSC i,l !
∆Tl Sbase ∀i ∈ΩB; ∀l ∈ΩNL; ∀b ∈ΩTESS
(4.11)
Master’s Thesis: Evaluation of the configurations SoCi,l = SoC0 i l = 0; ∀i ∈ΩB
(4.12)
SoCi,l = SoCF i l = NL; ∀i ∈ΩB
(4.13)
P ESS
i,l
= P ESSD
i,l
−P ESSC
i,l ∀i ∈ΩB; ∀l ∈ΩNL
(4.14)
P ESSC
i,l
−P ESSD
i,l ≤P maxc i,b ∀i ∈ΩB; ∀l ∈ΩNL; ∀b ∈ΩTESS
(4.15)
P ESSD
i,l
−P ESSC
i,l ≤P maxd i,b ∀i ∈ΩB; ∀l ∈ΩNL; ∀b ∈ΩTESS
(4.16)
P ESSC
i,l
≥0; P ESSD
i,l ≥0 ∀i ∈ΩB; ∀l ∈ΩNL
(4.17)
SoCmin i ≤SoCi,l ≤SoCmax i ∀i ∈ΩB; ∀l ∈ΩNL
(4.18)
0 ≤P DG
i,l ≤P max i,g ∀i ∈ΩDG; ∀l ∈ΩNL; ∀g ∈ΩTDG
(4.19)
V min i
≤V BUS
i,l ≤V max i ∀i ∈ΩN; ∀l ∈ΩNL; i̸ = slack
(4.20)
Nonlinear problems such as the OPF presented are computational expensive in large scale
Master’s Thesis: Evaluation of the configurations networks. Linearization methods can reduce computational efforts but are still computational expensive due the large amount of new constraints added. As consequence, the OPF model proposed in Eqs. (4.1)–(4.20) is solved using a new decomposition method. This decomposition method splits in two steps. The first step considers that the cost of the energy purchase is bigger than the cost of the technical energy losses. Thus, the network losses are eliminated from the model presented in Eqs. (4.1)–(4.20). The Eqs. (4.1)–(4.2) are modified and replaced by Eqs. (4.21) and (4.22). Moreover, Eqs. (4.3)–(4.10) and (4.20) are eliminated. With these modification the nonlinear model is converted into a linear model. max = k1 NL X l=1 k3 X i∈ΩN P D i,l −k2l X i∈ΩSS P S i,l ∆Tl Sbase
(4.21)
X i∈ΩN P S i,l + P DG i,l
+ P ESS
i,l −P D i,l = 0 ∀l ∈ΩNL
(4.22)
The linear model is presented in (4.21),(4.22),(4.11)–(4.19). In this linear model, the electrical variables and the reactive balance are removed. Eq. (4.21) is the objective function which maximizes the utilities from the energy purchase and sale. Eq. (4.22) represents the active power global balance given by Kirchhoff’s laws for both networks.
Objective function:
(
FObj = Eq. (4.21)
)
Subject to:
(
Eqs. (4.22), (4.11) −(4.19)
)
Master’s Thesis: Evaluation of the configurations The linear model results are the optimal operation of the ESSs. Afterward, the second step uses the load flow explained in Section 3.5 to consider the network losses due the operation of ESSs. The EESs are treated as nodes PQ. The results of this load flow are the objective function described in Eq. (4.1) and the operative conditions of the network. This decomposition solution method only works for feasible solutions, but this is not a problem since unfeasible solutions are not desirable.
4.4.2
Fitness Function The load flow results are the network losses to be summed in the objective function, and the operating limits to determine the feasible solutions. When the ESSs are considered, the load flow result is the profit from the energy purchase and sale instead of the network losses. The objective function is used by the solution technique in order to compare the solutions and only feasible solutions need to be accepted. Under this premise, unfeasible solutions are penalized in their respective objective function. The sum of the objective function plus the penalty costs of the respective violated constraints is called the fitness function (Ffit), and is obtained as follows.
Ffit =Eq. (2.1) + VMV fpVMV ΨVMV + VLV fpVLV ΨVLV + fpIMV ΨIMV + fpILV ΨILV + fpSMV ΨSMV + fpSLV ΨSLV
(4.23)
The factors fpV , fpI and fpS are associated to the penalties for violation of voltage limits, and overloads in elements and sources (branches, DTs or substations). The terms with the subscripts MV and LV refer to the primary and secondary network, respectively. These factors are multiplied by a binary decision variable (Ψ). If any constraint is violated, this
Master’s Thesis: Solution technique variable is one, otherwise it is zero. The voltages penalties are also multiplied by the magnitude of the voltage violated (V ). The units of these factors ensure that each term is expressed in monetary units.
4.5
Solution technique SA is a stochastic optimization procedure that can converge asymptotically to the optimal global solution with probability one. A stochastic mechanism controls the transition between two successive configurations in the SA algorithm. The acceptance of new configurations obeys the following criteria: topologies with decreasing objectives are always accepted, whereas configurations with higher costs can be accepted or not with a certain probability. The possibility of accepting solutions with an elevated cost avoids getting trapped in local minima.
In this thesis, is proposed a specialized SA algorithm that uses the NSZM method to explore the solution space in order to provide attractive solutions that are evaluated by the stochastic mechanism of the SA algorithm. The pseudocode of the SA is shown in Fig. 4.4.
Master’s Thesis: Solution technique SA’s pseudocode 1: begin 2:
Generate the initial configuration 3:
Initialize: Incumbent, T0, N0, k 4:
while stop criterion is not reached do 5:
for i = 1 to Nk do 6:
Obtain a solution from the NSZM method 7:
Evaluate objective function 8:
if best solution is improved then 9:
Accept best solution; 10:
else 11:
if exp ∆Fo Tk > random [0, 1] then 12:
Accept worst solution; 13:
else 14:
Do not accept worst solution; 15:
end if 16:
end if 17:
if solution accepted is better than the incumbent then 18:
Update incumbent; 19:
end if 20:
end for 21:
Tk+1 = α Tk 22:
Nk+1 = β Nk 23:
k = k + 1 24:
The initial solution is now the incumbent solution 25:
end while 26:
Print results 27: end Figure 4.4: Pseudocode of the specialized SA. Simulated annealing models the process of annealing in metallurgy. This technique involves heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. Heating and cooling the material affects both the temperature and the thermodynamic free energy. At a given temperature, SA sequentially moves from one configuration to the next until thermal equilibrium is reached.
Master’s Thesis: Solution technique The efficiency of the algorithm regarding both the quality of the final solutions as well as the number of iterations will depend on the choice of the parameters of the cooling scheme.
4.5.1
Cooling scheme The procedures used in the calculation of the parameters are based on the idea of thermal equilibrium and are detailed below. The cooling scheme is defined by the following four parameters.
Initial temperature T0 The initial temperature T0 is determined in such a way that the number of configurations with higher costs that are accepted does not surpass a certain limit. If too many topologies with elevated costs are accepted, the search space will be guided to unattractive regions. Under this premise, the computational time would be increased, trying to return to attractive regions. The initial temperature is calculated as follows [51]:
T0 = µ −ln (Φ)F (x0)
(4.24)
In Eq. (4.24), Φ (%/100) is the probability of accepting a solution that is worse, by a determined µ (%/100), than the objective function of the initial solution F (x0), (i.e., solutions that are µ = 1% worse than the cost of the initial solution would be accepted with a probability of Φ = 13%).
Number of transitions Nk at temperature Tk The number of transitions Nk should be such that a state of thermal near-equilibrium will be reached at a given temperature Tk. The number of transitions and the rate of change of the
Master’s Thesis: Solution technique temperature between two consecutive temperature levels are closely related. This way, if the temperature steps are too big, thermal equilibrium would be reached only with a high value of Nk; and if the temperature steps are too small, thermal equilibrium would be reached only with a low value of Nk. The initial number of transitions N0 can be defined as the number of variables of the problem. To calculate the value of Nk for each temperature level, Eq. (4.25) is applied, where β is a constant greater than or equal to one.
Nk+1 = β Nk
(4.25)
Rate of change of the temperature The rate of cooling has a direct effect on the number of iterations required at each temperature level. Eq. (4.26) calculates the rate of change of the temperature, where α is a constant varying between 0.8 and 0.99.
Tk+1 = α Tk
(4.26)
Final temperature Tf The stopping criterion is determined in such a way that at the optimal point the expected improvement in the objective function becomes negligible. Two stopping criteria are used: the first one assumes a fixed number of temperature levels Tk for the cooling process (a number between 6 and 50); and the second criterion stops when the incumbent solution is not improved for a predefined number of temperature levels.
Master’s Thesis: General methodology
4.6
General methodology The procedure begins with the initial configuration, which is obtained as was explained in Section 4.2.
After that, the Zbus matrix is obtained for the initial configuration as was explained in Chapter 3 (see Section 3.1).
Then, the cooling scheme and incumbent are initialized. The cooling scheme is calculated as was explained in subsection 4.5.1 to start with the SA algorithm. The NSZM method uses the initial Zbus matrix in order to obtain a new solution for the SA algorithm (see Section 4.3). This new configuration represented by the Zbus matrix is evaluated using a load flow in order to determine its operative conditions. If the DSP problem only considers DGs, the load flow explained in Section 3.5 is used, but if the DSP problem considers DGs and ESSs, the decomposition method for the optimal power flow explained in Subsection 4.4.1 is used. Afterward, the objective function of this solution is determined by using the fitness function explained in Subsection 4.4.2. Then, the stochastic mechanism of the SA algorithm controls the transition between the solutions. The acceptance of new configurations obeys the following criteria: topologies with decreasing objectives are always accepted, whereas configurations with higher costs can be accepted or not with a certain probability. The possibility of accepting solutions with an elevated cost avoids getting trapped in local minima. Then, if the solution is better than the incumbent, the incumbent is updated. This procedure is repeated until Nk transitions are completed. For the next temperature level, Tk and Nk are updated as was explained in subsection 4.5.1. After a temperature level is completed, the initial Zbus matrix for the next temperature level is the incumbent Zbus matrix. The procedure ends when the stop criterion is reached (see subsection 4.5.1). Fig. 4.5 shows the flowchart of the proposed methodology.
Master’s Thesis: General methodology Start Read database Generate the initial configuration Build the Zbus matrix Initialize:
• i = 1
• cooling scheme:
T0, N0, k = 0
• incumbent
Obtain a solution from the NSZM method Evaluate the new solution Improves the current solution?
exp OFold−OFnew Tk > random [0, 1] Accept best solution Accept worst solution Don’t accept worst solution Improves the incumbent?
Update incumbent i ≤Nk Is stop criterion met?
i = i + 1
• i = 1
• Tk+1 = α Tk
• Nk+1 = β Nk
• k = k + 1
• The initial solution is now
the incumbent solution Print results End Yes No No Yes Yes No Yes No No Yes Figure 4.5: Flowchart of the proposed methodology.
Chapter 5 Application and Results
5.1
Description of the system To validate the proposed methodology, the real distribution system of Fig. 5.1 is used. In this figure, existing and proposed branches are represented by solid and dashed lines respectively. Existing and proposed substations are represented by squares.
The black points are the primary nodes, the white circles are the secondary nodes, and the white circles with a black point inside are nodes shared by both networks. The data system is different when ESSs are considered.
Master’s Thesis: Description of the system SS1
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Figure 5.1: Integrated distribution system.
The full system database when only DGs are considered can be found in Appendix A.1, and the full system database when DGs and ESSs are considered can be found in Appendix A.2. The battery type used for the ESSs is Sodium-Sulfur (NaS). The efficiency and the lifetime of this battery are ηb = 90% and 20 years, respectively [52]. The deep of discharge (DoD), the energy capital cost, and the operation and maintenance cost (O&M) are 100%, 320 [USD/kWh], and 80 (USD/kW-year), respectively [53]. Fig. 5.2 shows the behavior of the load and DGs curves used. The DGs curve used is a typical daily curve from a photovoltaic generator in the andean zone. Fig. 5.2 is used when DGs and ESSs are considered together.
Master’s Thesis: Integrated DSP of both networks considering DGs
1
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Load curve level l pu Load curve
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1
DGs curve level l pu DGs curve Figure 5.2: Daily behavior of the load and DGs curves in pu.
5.2
Integrated DSP of both networks considering DGs To solve this problem, the methodology explained in Chapter 4 is used. The cooling scheme factors are 1%, 5%, 1.1, and 0.95 for µ, Φ, β, and α, respectively. The values of the penalty factors are defined as follows. Factors fpVMV and fpVLV are equal to 1000. Factors fpIMV and fpILV are dynamic values equivalent to the cost of the next upgraded wire times 100. Factor fpSMV is a dynamic value equivalent to the difference between the cost of the existing substation and the next upgraded substation, and factor fpSLV is a dynamic value equivalent to the cost of the next upgraded DT. The two stopping criteria used are: 60 temperature levels for the cooling process, and 15 temperature levels if the best solution found is not improved. The test system used is shown in Fig. 5.1 and the full system database is found in Appendix A.1.
In order to verify the efficiency of the proposed methodology in this paper, four cases are studied: (1) integrated planning without DGs (case 2 in [5]), (2) integrated planning with
Master’s Thesis: Integrated DSP of both networks considering DGs DG in the LV network (case 3 in [5]), (3) integrated planning without DGs, considering the copper losses in DTs, and (4) integrated planning with DG in the LV network, considering the copper losses in DTs. The two cases analyzed in [5], do not consider the copper losses of the DTs in the objective function since the DTs are the conflicting variables in the bilevel approach proposed. Cases (3) and (4) consider the copper losses of the DTs in the objective function since the methodology proposed in this thesis integrates the two networks into one. The DGs are modeled as PQ nodes and their possible locations are based on the sensitivity analysis explained in Subsection 3.4.1.
5.2.1
Validation of the methodology To validate the proposed methodology, the results are compared with the ones reported in [5], using the same design aspects than they employed. It is necessary to clarify that in [5] the primary and secondary networks are modeled as one-single phase and three-phase models, respectively, and in the model presented in this paper both networks (primary and secondary) are modeled by an one-single phase representation. Therefore, before applying the proposed methodology to the test system, an evaluation of the final configuration presented in [5] is previously performed using the methodology proposed in this paper. In [5] are proposed three cases of study, where case 2 is the integrated planning without DGs. The published results for this case are 1.789, 0.337, and 0.184 (millions of dollars), for the objective function, and costs of the energy technical losses in the MV and LV networks, respectively. When the final configuration of case 2 in [5] is evaluated with the integrated mathematical model proposed in this paper, the results are 1.773, 0.335, and 0.169 (millions of dollars) for the same aspects (objective function, and the costs of the energy technical losses in the MV and LV networks). From both results, can be seen that the error between the two objective functions is 0.89%.
These values allow to conclude that although the
Master’s Thesis: Integrated DSP of both networks considering DGs representation of both voltage levels proposed in this article is one-single phase, the results are quite reliable and approximate to those obtained under a three-phase representation.
5.2.2
Results The algorithm was implemented in Matlab (2017), using an Intel®Core i5-3470 8 GB RAM PC. The CPU time for Cases 1, 2, 3 and 4 are 10160 s (2.82 h), 11451 s (3.18 h), 7806.3 s (2.16 h) and 9732.7 s (2.70 h), respectively. The solutions for Cases 2 and 3 reported in [5] are obtained using a bilevel integrated planning for both networks. Case 2 does not consider DGs and Case 3 does consider DGs. The incumbent behavior for the objective function of all cases is shown in Fig. 5.3.
A comparison of these results is shown in Tables 5.1 and 5.2 in terms of present value, where the term ETL means the costs of the energy technical losses. Figs. B.1–B.8 show the best solutions found for the four studied cases. To facilitate the visualization of the obtained topologies, the primary and secondary networks of each case are presented in separate figures. In these figures, the primary and secondary branches are represented by solid lines. For all cases of the two networks, the number in parentheses is associated to the type of wire for each branch; branches without a number have type 1 wire. The DTs are represented by black triangles and their types are presented by an underlined number. In all four cases, the existing substation was not upgraded, and a new type 1 substation was installed. The installation nodes of the DGs and their types for Case 2 are: 23 (type 1), 38 (type 1), 43 (type 1), 88 (type 1), 95 (type 1), 103 (type 1), 110 (type 1), 130 (type 1), 131 (type 1), and 136 (type 2). For Case 4 they are: 23 (type 2), 38 (type 1), 43 (type 1), 88 (type 1), 95 (type 1), 103 (type 1), 110 (type 1), 130 (type 2), 131 (type 1), and 136 (type 2). In all cases the solutions are feasible.
Master’s Thesis: Integrated DSP of both networks considering DGs Table 5.1: Comparison of cost of expansion in millions of USD. Cost Description Reported in [5] Reported in this thesis Case 1 Case 2 Case 1 Case 2 Fixed Substations
0.336
0.336
0.336
0.336
MV feeders
0.410
0.397
0.393
0.376
LV circuits
0.255
0.254
0.283
0.270
DT
0.265
0.225
0.258
0.215
DG —
0.032
—
0.026
Total
1.267
1.246
1.270
1.223
Variable ETL in MV
0.337
0.307
0.207
0.194
ETL in LV
0.184
0.183
0.164
0.183
Total
0.521
0.490
0.371
0.377
Total cost
1.789
1.737
1.640
1.599
The consolidated results from Table 5.1 show that the total costs found are better than those reported in [5]. The results from Table 5.2 show that the lowest costs were obtained in Cases 2 and 4, which have a penetration of DG.
Master’s Thesis: Integrated DSP of both networks considering DGs Table 5.2: Comparison of cost of expansion in millions of USD.
Cost Description Case 1 Case 2 Case 3 Case 4 Fixed Substations
0.336
0.336
0.336
0.336
MV feeders
0.393
0.376
0.375
0.358
LV circuits
0.283
0.270
0.283
0.271
DT
0.258
0.215
0.254
0.237
DG —
0.026
—
0.029
Total
1.270
1.223
1.248
1.231
Variable ETL in MV
0.207
0.194
0.208
0.196
ETL in LV
0.164
0.183
0.174
0.178
ETL in DTs — —
0.178
0.129
Total
0.371
0.377
0.560
0.502
Total cost
1.640
1.599
1.808
1.733
Case 1 achieved a lower variable cost (USD 0.490 M) than that reported for Case 1 in [5] (USD 0.521 M) using almost the same fixed cost. Under this premise, the integrated model proposed can find a topology with lower global cost. Case 2 has lower fixed and variable costs (1.223 and 0.377 millions of USD) than Case 2 reported in [5] (1.256 and 0.490 millions of USD). As a consequence, the sensitivity analysis proposed for the installation of DGs is highly recommended.
Cases 3 and 4 are more detailed because they consider the energy technical losses in the DTs. Note that the penetration of DG in the LV network allows installing DTs, LV circuits, and MV feeders with smaller sizes than Case 3. In addition, the energy technical losses in MV, LV and DTs are also affected, which is reflected in the lowest operative cost. The lower costs are obtained because the location and sizing of the DTs enhances the power
Master’s Thesis: Integrated DSP of both networks considering DGs flow circulation between both systems (MV and LV networks), which decreases the technical losses and the investment costs in the elements of both networks. For all cases, the solutions found have different topologies.
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Temperature level Tk Incumbent Case 1
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Temperature level Tk Incumbent Case 3
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Temperature level Tk Incumbent Case 4 Figure 5.3: Incumbent behavior for the four cases in millions of USD.
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs
5.3
Integrated DSP of both networks considering DGs and ESSs To solve this problem, the methodology explained in Chapter 4 is used. The cooling scheme factors are 0.5%, 2.365%, 1.1, and 0.95 for µ, Φ, β, and α, respectively. The values of the penalty factors are defined as follows. Factors fpVMV and fpVLV are equal to 660000. Factors fpIMV and fpILV are equivalent to the cost of the current wire times 10000. Factor fpSMV is a dynamic value equivalent to the difference between the cost of the existing substation and the next upgraded substation, and factor fpSLV is a dynamic value equivalent to the cost of the current DT times 60000. The two stopping criteria used are: 60 temperature levels for the cooling process, and 15 temperature levels if the best solution found is not improved. The test system used is shown in Fig. 5.1 and the full system database is found in Appendix A.2. The sensitivity factor of the Eq. (3.16) is used to reduce the candidate nodes for the ESSs installation. The reduced candidate nodes obtained by using this sensitivity factor are shown in Table 5.3.
Table 5.3: Candidate nodes for installing ESSs in the LV and MV networks. Network Nodes LV
129, 111, 109, 127, 130, 113, 30, 11,
132, 135, 137, 116, 8, 106, 48
MV
157, 139, 158, 168, 109, 142, 104, 156,
106, 159, 111, 169, 146, 160, 151
In order to analyze the benefits of the integration of the ESSs into the DSP problem, the following five cases are studied. Case (A) is the integrated planning without DGs and ESSs. Cases (B) and (C) are: (B) integrated planning with DGs and ESSs in the LV network and (C) integrated planning with DGs in the LV network and ESSs in the MV network. These
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs two cases consider the losses of the ESSs due the efficiency (ηb = 90%). Cases (D) and (E) are the same as (B) and (C) but these cases do not consider the losses of the ESSs (ηb = 100%).
5.3.1
Validation of the decomposition method To validate the proposed decomposition method used to solve the OPF considering DGs and ESSs, the test system of Figs. 5.4 and 5.5 is used. The decomposition method is compared with the nonlinear model presented in Subsection 4.4.1. The database of the system is presented in Appendix A.2.
SS1
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Figure 5.4: Primary Network of the test system.
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs
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Figure 5.5: Secondary Network of the test system.
The Table 5.4 shows the comparison between both methods. Case (I) do not considered DGs and ESSs. Five cases are used in order to compared the efficiency of the decomposition method. Case (II) has the next DGs and ESSs nodes: 6 (type 4), 43 (type 3), 54 (type 2), 103 (type 4), 110 (type 4), and 131 (type 3) are nodes with DGs; and 111 (type 4), 127 (type 3), 113 (type 4), 11 (type 4), 116 (type 1), and 48 (type 4) are nodes with ESSs. Case (III) has the next DGs and ESSs nodes: 23 (type 1), 38 (type 2), 62 (type 3), 71 (type 3), 95 (type 4), 103 (type 4), 130 (type 3), and 131 (type 2) are nodes with DGs; and 139 (type 4), 109 (type 4), 104 (type 3), 106 (type 1), 169 (type 1), 146 (type 1), and 151 (type 2) are nodes with ESSs. Cases (II) and (III) consider the losses of the ESSs due the efficiency (ηb = 90%). Case (IV) has the next DGs and ESSs nodes: 6 (type 3), 43 (type 4), 54 (type 1), 88 (type 2), 117 (type 3), and 136 (type 4) are nodes with DGs; and 111 (type 3), 127 (type 4), 30 (type 2), 11 (type 4), 137 (type 3), and 48 (type 3) are nodes with ESSs. Case (V) has the
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs next DGs and ESSs nodes: 6 (type 2), 23 (type 2), 38 (type 4), 62 (type 2), 71 (type 3), 95 (type 1), 103 (type 3), 130 (type 4), and 131 (type 1) are nodes with DGs; and 168 (type 3), 142 (type 2), 104 (type 3), 106 (type 3), 111 (type 2), 146 (type 1), and 160 (type 3) are nodes with ESSs. Cases (IV) and (V) do not consider the losses of the ESSs (ηb = 100%). Table 5.4: Comparison of the objective functions for the five cases in millions of USD. Case I Case II Case III Case IV Case V decomposition OF
13.6169
16.1772
16.0903
16.3539
16.5507
method CPU time [s]
0.1092
1.0393
0.8212
0.3296
0.3388
nonlinear OF
13.6175
16.1773
16.0898
16.3543
16.5507
model CPU time [s]
59.5858
1014.9358
375.7602
2513.6399
541.5253
Error between
0.0043
0.0003
0.0032
0.0025
0.0003
the OFs [%] The two methods were implemented using an interface between Matlab (2017b) and GAMS (24.5.4) and an Intel®Core i5-4460S 12 GB RAM PC. The solvers CPLEX and KNITRO were used for the linear and nonlinear model, respectively. The results from Table 5.4 verify the efficiency of the proposed decomposition method. Table 5.4 shows that the computational effort is reduced with the decomposition method and the objective function results are the same. Thus, the results obtained show that the decomposition method achieves the optimal solution in less time.
5.3.2
Results The algorithm was implemented using an interface between Matlab (2017b) and GAMS (24.5.4) and an Intel®Core i5-4460S 12 GB RAM PC. The solver CPLEX is used to solve the linear problem of the decomposition method. The CPU time for Cases A, B, C, D and E are
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs 7999.87 s (2.22 h), 212339.99 s (58.98 h), 210917.60 s (58.59 h), 203203.75 s (56.45 h), and 200664.16 s (55.74 h), respectively. It is to be expected that these times could be reduced if a production-grade routine such as C++ is applied and a better optimization tool interface is implemented. In any case, planning studies do not require real-time simulations, thus the above timings are considered adequate. The incumbent behavior for the objective function of all cases is shown in Fig. 5.7.
A comparison of these results is shown in Table 5.5 in terms of present value, where the term ETL means the costs of the energy technical losses. Figs. B.9–B.18 show the best solutions found for the five studied cases. To facilitate the visualization of the obtained topologies, the primary and secondary networks of each case are presented in separate figures. In these figures, the primary and secondary branches are represented by solid lines. For all cases of the two networks, the number in parentheses is associated to the type of wire for each branch; branches without a number have type 1 wire. The DTs are represented by black triangles and their types are presented by an underlined number. In all five cases, the existing substation was not upgraded, and a new type 1 substation was installed. For all the cases, new DGs and ESSs type 4 were installed in all the candidate nodes. In all cases the solutions are feasible. For all cases, the solutions found have different topologies.
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs Table 5.5: Comparison of the cost and profit in millions of USD.
Cost Description Case A Case B Case C Case D Case E Fixed Substations
0.336
0.336
0.336
0.336
0.336
MV feeders
0.574
0.571
0.543
0.568
0.539
LV circuits
0.411
0.512
0.512
0.507
0.514
DT
0.341
0.324
0.284
0.325
0.280
DG —
0.094
0.094
0.094
0.094
ESS
—
1.920
1.920
1.920
1.920
O&M of the ESS —
1.362
1.362
1.362
1.362
Total cost
1.662
5.119
5.051
5.112
5.044
Variable ETL in MV, LV
0.730
0.725
0.739
0.746
0.738
and DTs ETL in ESSs —
1.064
1.064
0.000
0.000
Energy purchase
85.563
78.136
77.998
76.982
76.810
Energy sale
99.167
99.273
99.166
99.285
99.165
Profit
13.604
21.138
21.168
22.303
22.355
Total profit
11.943
16.019
16.117
17.191
17.310
Case A shows that the only way to maximize the profit from the energy purchase and sale when no DGs and ESSs are considered is minimizing the costs of the energy technical losses in the network and the inversion costs. Thus, the objective value obtained is (USD 11.943 M). Cases B and C show that incorporating DGs and ESSs into the DSP problem improve the profit from the energy purchase and sale but increase the inversion costs. Nevertheless, the objective functions obtained (16.019 and 16.117 millions of USD) show that integrate these elements is desirable. These Cases also show that is better to include the ESSs into the MV network rather than the LV network. Cases D and E show the same behavior than Cases B and C even if the losses of the ESSs are not considered. If the ESSs had not losses,
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs the profit would have improved (17.191 and 17.310 millions of USD). Despite the Case B is the only one who improved the costs of the energy technical losses from the reference Case A, the objective functions of the rest of cases are better than Case B. Moreover, the costs of the energy technical losses for Cases C, D and E are just a little bit bigger than Case A.
1
5
10
15
20
24
4
6
8
10
Load curve level l Power generated [MW] Case A Case B (a)
1
5
10
15
20
24
4
6
8
10
Load curve level l Power generated [MW] Case A Case C (b)
1
5
10
15
20
24
2
4
6
8
10
Load curve level l Power generated [MW] Case A Case D (c)
1
5
10
15
20
24
2
4
6
8
10
Load curve level l Power generated [MW] Case A Case E (d) Figure 5.6: Comparison of the generated power by the substations for the five cases.
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs
1
10
20
30
40
50
60
−11.94 −11.92 −11.9 Temperature level Tk Incumbent Case A
1
10
20
30
40
50
60
−16 −15.5 −15 −14.5 Temperature level Tk Incumbent Case B
1
10
20
30
40
50
60
−16 −15.5 −15 Temperature level Tk Incumbent Case C
1
10
20
30
40
50
60
−17 −16.5 −16 −15.5 Temperature level Tk Incumbent Case D
1
10
20
30
40
50
60
−17.2 −17 −16.8 −16.6 Temperature level Tk Incumbent Case E Figure 5.7: Incumbent behavior for the five cases in millions of USD.
Master’s Thesis: Integrated DSP of both networks considering DGs and ESSs Fig. 5.6 shows the behavior of the power generated by the substations and compares the Cases B, C, D, and E which have DGs and ESSs with Case A which has not. Figs. 5.6a–5.6d show that for Cases B, C, D, and E the load curve is modified due DGs and ESSs and that the new peak hour is moved at the hour 22. Figs. 5.6a and 5.6b show that the behavior of the load curve seems to be the same but is different from Figs. 5.6c and 5.6d. As consequence, it is shown that the losses of the ESSs impact the load curve behavior in order to maximize the profit from the energy purchase and sale. Figs. 5.6c and 5.6d show that the behavior of the load curve seems to be the same. Although the behavior seems to be the same for these cases the values are a little bit different due the losses of the network. When no DGs and ESSs are considered in Case A, the peak hour is presented at the hour 20 with a power value of 10.499 MW. When DGs and ESSs are considered in Cases B, C, D, and E this peak value is reduced to 8.508 MW, 8.488 MW, 8.509 MW, and 8.488 MW, respectively. Due this value is reduced, a new peak hour is presented at the hour 22 in Cases B, C, D, and E with a power value of 8.688 MW, 8.682 MW, 8.689 MW, and 8.682 MW, respectively. In order to measure the efficiency of electrical energy usage due the DGs and ESSs the load factor for all cases is calculated. For Cases A, B, C, D, and E the load factor obtained is 0.638, 0.713, 0.712, 0.703, and 0.702, respectively. As consequence, the increase in this value shows that the integration of ESSs and DGs into the DSP problem is desirable.
Chapter 6 Conclusions and Future work
6.1
Conclusions This thesis has presented a new methodology which uses a simulated annealing algorithm with a novel neighborhood search method based on the Zbus matrix (NSZM) for the optimal integrated planning of medium and low voltage distribution systems considering distributed generation (DG) and energy storage systems (ESSs). The NSZM method uses a defined neighborhood structure combined with sensitivity factors based on the Zbus matrix in order to find attractive solutions in quick times for the simulated annealing algorithm. The methodology has been validated and tested using a real distribution system of the literature. The results obtained with the proposed methodology are better than the reported in the literature. As consequence, it is demonstrated that the use of a defined neighborhood structure combined with sensitivity factors based on the Zbus matrix for exploring the solution space for a metaheuristic algorithm leads to good quality solutions in relative quick times. In this thesis, the distribution system planning (DSP) problem of primary and secondary networks was studied with and without DGs. When DGs are considered in the DSP problem,
Master’s Thesis: Future Work it has been demonstrated that a reduction in investment and operative costs is achieved. Furthermore, the results obtained with the proposed sensitivity analysis based on the Zbus matrix for solving the allocation and sizing of DGs are better than the reported in the literature. Therefore, it is demonstrated that the use of the Zbus matrix in the analysis of the impact of DGs in the technical losses of both networks is desirable to solve the DSP problem considering DGs.
After this, the DSP problem of primary and secondary networks was studied with and without DGs and ESSs.
When DGs and ESSs are considered in the DSP problem, it has been demonstrated that the efficiency of electrical energy usage is improved and the profits from the energy purchase and sale are increased. The results obtained show that it is better to include ESSs into the MV network rather than into the LV network. Moreover, the results show that the decomposition method used for the optimal operation of the ESSs comes to the optimal solution. Hence, the robustness and effectiveness of the methodology proposed to solve the DSP problem considering DGs and ESSs is verified.
The Zbus matrix is a mathematical model which establishes an electrical relation between the primary and secondary nodes. Therefore, a change in one network is reflected in the electrical characteristics of the other one. Under this premise, the Zbus matrix is used in the methodology proposed in this thesis to solve the integrated DSP problem of primary and secondary networks, and its validity has been demonstrated when the results are compared with those reported in the literature.
6.2
Future Work From the attained results and the drawbacks found along the process, the following topics could be explored.
Master’s Thesis: Future Work
• Regarding the proposed methodology based on the Zbus matrix, the employment of a
three-phase model for the primary and secondary networks.
• Regarding the proposed methodology considering DGs, the employment of different
technologies for DGs and the analysis of the conflict when the utility is not the owner of the DGs.
• Regarding the proposed methodology considering DGs and ESSs, the integration of
electric vehicles into the DSP problem considering different scenarios is desirable.
• A more efficient implementation could be developed. The usage of a different meta-
heuristic combined with the NSZM method to improve the solutions and reduce processing time. Furthermore, these times could be reduced if a production-grade routine such as C++ is applied and a better optimization tool interface is implemented.
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Appendix A Data Systems A.1 Data of the distribution system The real distribution system of Fig. 5.1 is proposed in [5]. This distribution system integrates the primary and secondary networks. The primary distribution network has 48 existing nodes,
1 existing substation (type 2) and 51 existing feeders (type 1). To supply the new power
demand, 60 new feeders and 1 new substation can be installed. In order to supply the 138 new secondary demand nodes, there are proposed 33 new DTs, 15 new DGs, and 147 new secondary circuits. Additionally, 5 types of substations, 6 types of wires for primary and secondary, 8 types of DTs, and 4 types of DGs are considered.
The nominal voltage of this system is 13.2 kV (line to line) for the primary network and
440 V (line to neutral) for the secondary network. The maximum voltage regulation for the
primary and secondary systems is 10%. The planning horizon is 20 years. The load duration curve is discretized in three load levels: 100%, 60%, and 30% of peak demand, with durations of 1000, 6760, and 1000 h respectively. The ZIP load model coefficients are a0 = 0.2 and a2 = 0.8. The discount rate is 10% and the energy cost is 0.15 USD/kWh. The full system
Master’s Thesis: Data of the distribution system database can be also find in [?].
The candidate nodes for the installation of DTs and DGs are presented in Table A.1. Table A.1: Candidate nodes for installing DTs and DGs. Element Nodes DT
2, 8, 11, 16, 30, 33, 37, 45, 48, 51, 56, 59, 64,
80, 83, 87, 91, 94, 97, 104, 106, 109, 111, 113,
116, 118, 122, 124, 127, 129, 132, 135, 137
DG
6, 23, 38, 43, 54, 62, 71, 88, 95, 103, 110, 117, 130, 131, 136
Table A.2: Information of the wires used in the distribution system. MV Network LV Network Type R X Amp USD/m R X Amp USD/m [ohm/km] [ohm/km] [ohm/km] [ohm/km]
1
0.52
0.22
205
26
1.04
0.45
150
14
2
0.32
0.14
275
40
0.65
0.28
180
20
3
0.26
0.12
305
47
0.52
0.22
205
26
4
0.18
0.1
390
57
0.32
0.14
275
40
5
0.14
0.08
460
64
0.26
0.12
305
47
6
0.12
0.07
600
72
0.18
0.1
Master’s Thesis: Data of the distribution system Table A.3: Upgrading costs of the wires in [USD/m]. MV Network LV Network Type
1
2
3
4
5
6
1
2
3
4
5
6
1
0
10
19
26
36
43
0
4
10
22
31
38
2
—
0
5
12
22
29
—
0
4
16
25
32
3
— —
0
5
15
22
— —
0
10
19
26
4
— — —
0
5
12
— — —
0
5
12
5
— — — —
0
5
— — — —
0
5
6
— — — — —
0
— — — — —
0
Table A.4: Elements information.
Substations DTs DGs Type kVA
USD
R HV X HV kVA
USD
kW
USD
[ohm] [ohm]
1
7000
336000
99.704
142.894
30
3177.57
50
2500
2
10000
672000
61.0916
98.7976
45
3953.07
75
3750
3
20000
1344000
33.7638
73.9706
75
5502.69
100
5000
4
30000
2016000
21.2014
49.89
112.5
7439.72
125
6250
5
40000
2688000
15.1782
43.915
150
9376.74
— —
6
— —
9.9467
29.3356
225
11053.72
— —
7
— —
7.1148
25.149
300
16806
— —
8
— —
5.151
18.9131
400
22408
— — Table A.5: Information of the circuits of the LV network. From To km Existing From To km Existing
1
2
0.1440
0
27
28
0.1824
0
2
3
0.1440
0
28
18
0.1800
Master’s Thesis: Data of the distribution system Table A.5 continued from previous page. From To km Existing From To km Existing
2
4
0.1344
0
18
39
0.1440
0
2
5
0.1440
0
3
29
0.1440
0
5
6
0.1056
0
29
30
0.1560
0
5
7
0.1632
0
30
31
0.1920
0
7
8
0.1440
0
31
32
0.1248
0
8
9
0.1560
0
32
33
0.1632
0
9
10
0.1440
0
33
24
0.1800
0
10
11
0.1800
0
33
34
0.1378
0
11
12
0.1800
0
34
35
0.1800
0
12
13
0.1800
0
35
36
0.1195
0
13
14
0.1800
0
36
37
0.1800
0
14
15
0.1800
0
37
38
0.1920
0
15
16
0.1800
0
38
39
0.1800
0
16
17
0.1440
0
38
40
0.1440
0
18
17
0.1008
0
3
41
0.1440
0
4
19
0.1056
0
41
42
0.1584
0
19
20
0.0864
0
42
43
0.1560
0
20
21
0.0173
0
42
44
0.1560
0
21
22
0.1344
0
44
45
0.1008
0
22
23
0.1800
0
45
46
0.1522
0
23
24
0.1800
0
46
47
0.1848
0
23
25
0.1800
0
47
48
0.1800
0
25
26
0.1608
0
48
49
0.1440
0
26
27
0.1800
0
49
50
0.1800
0
50
51
0.1728
0
73
74
0.1800
Master’s Thesis: Data of the distribution system Table A.5 continued from previous page. From To km Existing From To km Existing
51
52
0.1440
0
74
75
0.1824
0
52
53
0.1440
0
75
76
0.1800
0
54
53
0.1800
0
76
78
0.1440
0
40
54
0.1440
0
38
79
0.1440
0
18
66
0.1344
0
79
80
0.1560
0
16
55
0.1632
0
80
81
0.1920
0
55
56
0.1440
0
81
82
0.1248
0
56
57
0.1560
0
82
83
0.1632
0
57
58
0.1440
0
83
77
0.1800
0
58
59
0.1800
0
83
84
0.1378
0
59
60
0.1800
0
84
85
0.1800
0
60
61
0.1800
0
85
86
0.1195
0
61
62
0.1800
0
86
87
0.1800
0
62
63
0.1800
0
87
88
0.1920
0
63
64
0.1800
0
88
78
0.1800
0
64
65
0.1440
0
88
89
0.1440
0
76
65
0.1008
0
54
90
0.1560
0
66
67
0.1056
0
90
91
0.1008
0
67
68
0.0864
0
91
92
0.1522
0
68
69
0.0173
0
92
93
0.1848
0
69
70
0.1344
0
93
94
0.1800
0
70
71
0.1800
0
94
95
0.1440
0
71
77
0.1800
0
95
96
0.1800
0
71
72
0.1800
0
96
97
0.1728
0
72
73
0.1608
0
97
98
0.1440
Master’s Thesis: Data of the distribution system Table A.5 continued from previous page. From To km Existing From To km Existing
98
99
0.1440
0
119
120
0.1080
0
100
99
0.1800
0
120
121
0.1080
0
89
100
0.1440
0
121
122
0.1080
0
101
102
0.1080
0
122
123
0.1620
0
102
103
0.1080
0
123
124
0.1620
0
103
104
0.1080
0
124
125
0.1620
0
104
105
0.1620
0
125
126
0.1620
0
105
106
0.1620
0
126
127
0.1620
0
106
107
0.1620
0
127
128
0.1080
0
107
108
0.1620
0
128
129
0.1080
0
108
109
0.1620
0
129
130
0.1080
0
109
110
0.1080
0
130
113
0.1080
0
110
111
0.1080
0
127
131
0.1080
0
111
112
0.1080
0
131
132
0.1080
0
112
113
0.1080
0
132
133
0.1620
0
113
114
0.1620
0
133
134
0.1620
0
114
115
0.1620
0
134
135
0.1620
0
115
116
0.1620
0
135
136
0.1620
0
116
117
0.1620
0
136
137
0.1620
0
117
118
0.1620
0
137
138
0.1080
0
118
101
0.1080
0
138
122
0.1080
0
118
119
0.1080
0
— — — —
Master’s Thesis: Data of the distribution system Table A.6: Information of the feeders of the MV network. From To km Existing From To km Existing SS1
139
0.226404
1
16
18
0.2448000
0
139
140
0.226404
1
64
76
0.2448000
0
140
141
0.153672
1
2
23
0.6580800
0
141
142
0.288000
1
23
18
0.8832000
0
142
143
0.153672
1
18
71
0.6580800
0
140
144
0.258480
1
71
76
0.8832000
0
144
145
0.153672
1
2
3
0.1440000
0
145
146
0.096000
1
23
33
0.3600000
0
146
147
0.360000
1
18
38
0.3240000
0
146
148
0.153672
1
71
83
0.3600000
0
148
149
0.307344
1
76
88
0.3240000
0
149
150
0.153672
1
3
30
0.3000000
0
150
151
0.312000
1
30
33
0.4800000
0
150
152
0.192000
1
33
37
0.6172800
0
152
153
0.307344
1
37
38
0.1920000
0
153
154
0.288000
1
38
80
0.3000000
0
154
155
0.153672
1
80
83
0.4800000
0
155
156
0.192000
1
83
87
0.6172800
0
SS1
157
0.258480
1
87
88
0.1920000
0
157
158
0.258480
1
3
42
0.3024000
0
158
159
0.258480
1
38
54
0.2880000
0
159
160
0.258480
1
88
100
0.2880000
0
160
161
0.153672
1
42
45
0.2568000
0
161
162
0.153672
1
45
48
0.5169600
0
162
163
0.240000
1
48
51
0.4968000
Master’s Thesis: Data of the distribution system Table A.6 continued from previous page. From To km Existing From To km Existing
163
164
0.226404
1
51
54
0.4680000
0
163
165
0.120000
1
54
91
0.2568000
0
165
166
0.360000
1
91
94
0.5169600
0
166
167
0.360000
1
94
97
0.4968000
0
SS1
168
0.240000
1
97
100
0.4680000
0
168
169
0.240000
1
94
183
0.2300000
0
169
170
0.374880
1
100
176
0.2100000
0
170
171
0.258480
1
88
167
0.4699636
0
171
172
0.096000
1
76
164
0.4516364
0
172
173
0.153672
1
64
147
0.4254545
0
172
174
0.258480
1
104
106
0.3236763
0
174
175
0.120000
1
106
109
0.4855145
0
175
176
0.192000
1
109
111
0.2157842
0
176
177
0.153672
1
111
113
0.2157842
0
177
178
0.192000
1
113
116
0.4855145
0
178
179
0.096000
1
116
118
0.3236763
0
179
180
0.120000
1
118
104
0.4315684
0
180
181
0.153672
1
118
122
0.4315684
0
181
182
0.153672
1
122
124
0.3236763
0
182
183
0.096000
1
124
127
0.4855145
0
183
184
0.120000
1
127
129
0.2157842
0
184
185
0.120000
1
129
113
0.2157842
0
185
186
0.153672
1
127
132
0.2157842
0
SS2
8
0.340000
0
132
135
0.4855145
0
5
8
0.307200
0
135
137
0.3236763
Master’s Thesis: Data of the distribution system Table A.6 continued from previous page. From To km Existing From To km Existing
5
2
0.144000
0
137
122
0.2157842
0
8
11
0.480000
0
156
132
0.1000000
0
11
16
0.900000
0
153
135
0.3400000
0
16
56
0.307200
0
152
137
0.3490909
0
56
59
0.480000
0
SS2
109
0.2500000
0
59
64
0.900000
0
— — — — Table A.7: Nodal information of the LV network. Node kVA Existing Node kVA Existing
1
0.8550
0
70
3.5900
0
2
8.7300
0
71
5.4400
0
3
17.1675
0
72
8.2900
0
4
8.7300
0
73
9.9100
0
5
23.4675
0
74
9.9100
0
6
7.9650
0
75
6.6700
0
7
29.2950
0
76
0.1900
0
8
43.5150
0
77
0.1900
0
9
43.5150
0
78
6.6700
0
10
43.5150
0
79
5.4400
0
11
57.7350
0
80
35.7750
0
12
43.5150
0
81
5.4400
0
13
43.5150
0
82
5.4400
0
14
32.4225
0
83
5.4400
0
15
43.0875
0
84
9.9100
0
16
29.2950
0
85
9.9100
0
17
0.8550
0
86
9.9100
Master’s Thesis: Data of the distribution system Table A.7 continued from previous page. Node kVA Existing Node kVA Existing
18
0.8550
0
87
9.2900
0
19
24.4800
0
88
8.8900
0
20
24.4800
0
89
1.8100
0
21
24.4800
0
90
1.7700
0
22
16.1550
0
91
8.0900
0
23
24.4800
0
92
9.6700
0
24
0.8550
0
93
9.6700
0
25
37.3050
0
94
12.8300
0
26
44.5950
0
95
9.6700
0
27
44.5950
0
96
9.6700
0
28
30.0150
0
97
9.6700
0
29
24.4800
0
98
5.2150
0
30
160.9875
0
99
4.9300
0
31
24.4800
0
100
0.1900
0
32
24.4800
0
101
15.4700
0
33
24.4800
0
102
17.2800
0
34
44.5950
0
103
22.6600
0
35
44.5950
0
104
17.2800
0
36
44.5950
0
105
30.9700
0
37
41.8050
0
106
15.4700
0
38
40.0050
0
107
38.6700
0
39
30.0150
0
108
57.4300
0
40
8.1450
0
109
57.4300
0
41
36.4050
0
110
57.4300
0
42
22.1850
0
111
76.2100
Master’s Thesis: Data of the distribution system Table A.7 continued from previous page. Node kVA Existing Node kVA Existing
43
7.9650
0
112
57.4300
0
44
7.9650
0
113
57.4300
0
45
36.4050
0
114
42.8000
0
46
43.5150
0
115
56.8700
0
47
43.5150
0
116
38.6700
0
48
57.7350
0
117
15.4700
0
49
43.5150
0
118
15.4700
0
50
43.5150
0
119
32.3200
0
51
43.5150
0
120
32.3200
0
52
23.4675
0
121
32.3200
0
53
22.1850
0
122
21.3300
0
54
0.8550
0
123
32.3200
0
55
6.5100
0
124
15.4700
0
56
9.6700
0
125
49.2500
0
57
9.6700
0
126
58.8700
0
58
9.6700
0
127
58.8700
0
59
12.8300
0
128
39.6200
0
60
9.6700
0
129
32.3200
0
61
9.6700
0
130
212.5000
0
62
7.2050
0
131
32.3200
0
63
9.5750
0
132
32.3200
0
64
6.5100
0
133
32.3200
0
65
0.1900
0
134
58.8700
0
66
1.9400
0
135
58.8700
0
67
5.4400
0
136
58.8700
Master’s Thesis: Data of the distribution system Table A.7 continued from previous page. Node kVA Existing Node kVA Existing
68
5.4400
0
137
55.1900
0
69
5.4400
0
138
52.8100
0
Table A.8: Nodal information of the MV network. Node kVA Existing Node kVA Existing
139
387.197
1
186
121.994
1
140
121.994
1
5
0
0
141
73.479
1
23
0
0
142
387.197
1
18
0
0
143
121.994
1
71
0
0
144
18.370
1
76
0
0
145
73.479
1
3
0
0
146
387.197
1
38
0
0
147
293.911
1
88
0
0
148
18.370
1
42
0
0
149
121.994
1
54
0
0
150
73.479
1
100
0
0
151
387.197
1
2
0
0
152
293.911
1
8
0
0
153
121.994
1
11
0
0
154
18.370
1
16
0
0
155
73.479
1
30
0
0
156
387.197
1
33
0
0
157
121.994
1
37
0
0
158
387.197
1
45
0
0
159
73.479
Master’s Thesis: Data of the distribution system Table A.8 continued from previous page. Node kVA Existing Node kVA Existing
160
18.370
1
51
0
0
161
121.994
1
56
0
0
162
18.370
1
59
0
0
163
121.994
1
64
0
0
164
293.911
1
80
0
0
165
18.370
1
83
0
0
166
293.911
1
87
0
0
167
387.197
1
91
0
0
168
121.994
1
94
0
0
169
293.911
1
97
0
0
170
18.370
1
104
0
0
171
73.479
1
106
0
0
172
387.197
1
109
0
0
173
121.994
1
111
0
0
174
387.197
1
113
0
0
175
18.370
1
116
0
0
176
73.479
1
118
0
0
177
387.197
1
122
0
0
178
121.994
1
124
0
0
179
73.479
1
127
0
0
180
18.370
1
129
0
0
181
73.479
1
132
0
0
182
73.479
1
135
0
0
183
18.370
1
137
0
0
184
73.479
1
SS1
Master’s Thesis: Data of the modified distribution system Table A.8 continued from previous page.
Node kVA Existing Node kVA Existing
185
387.197
1
SS2
0
0
A.2 Data of the modified distribution system Some data of the real distribution system of Fig. 5.1 proposed in Appendix A.1 is modified and added in order to incorporated the ESSs.
The nominal voltage of this system is 13.2 kV (line to line) for the primary network and
440 V (line to line) for the secondary network. The maximum voltage regulation for the
primary and secondary systems is 10%. The planning horizon is 20 years. The ZIP load model coefficients are a0 = 0.2 and a2 = 0.8. The discount rate is 10%. The energy sale cost is 0.2 USD/kWh. The Tables A.1–A.8 show the rest of the database system. Table A.9: Load and DG curves information.
Load level [l] Load curve [pu] Energy purchase DG curve [pu] cost [USD/kWh]
1
0.489130
0.084
0.0002
2
0.423913
0.080
0.0002
3
0.423913
0.080
0.0002
4
0.402174
0.075
0.0002
5
0.402174
0.075
0.0002
6
0.402174
0.075
0.0400
7
0.500000
0.085
0.0800
8
0.521739
0.090
0.0800
9
0.608696
0.105
0.1600
Master’s Thesis: Data of the modified distribution system Table A.9 continued from previous page Load level [l] Load curve [pu] Energy purchase DG curve [pu] cost [USD/kWh]
10
0.652174
0.135
0.4000
11
0.673913
0.145
0.7200
12
0.717391
0.185
1.0000
13
0.717391
0.185
0.6000
14
0.652174
0.135
0.2400
15
0.630435
0.125
0.1200
16
0.608696
0.105
0.0800
17
0.586957
0.098
0.0800
18
0.695652
0.175
0.0800
19
0.978261
0.305
0.0003
20
1.000000
0.325
0.0003
21
0.934783
0.285
0.0003
22
0.869565
0.275
0.0003
23
0.826087
0.265
0.0003
24
0.543478
0.100
0.0003
Table A.10: ESS information. Type Capacity kW
USD
O&M fx b φb [kWh] [USD/year] [%/kWh]
1
100
33.333
32000
2666.667
1.000
2
200
66.667
64000
5333.333
0.500
3
300
100
96000
8000.000
0.333
4
400
133.333
128000
10666.667
0.250
Appendix B Final configurations of primary and secondary networks B.1 Integrated DSP considering DGs SS1
157
158
159
160
161
161
162
163
165
164
18
23
2
139
140
144
145
146
147
64
56
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
104
116
118
124
122
137
135
3
30
33
37 38
80
167 166
42
45
48
51
94
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(4)
(2)
(2)
(2)
(2)
(2)
(4)
(4)
(4)
(4)
(3)
(3)
(2)
(3)
(2)
(4)
(4)
Figure B.1: Case 1 - Primary Network.
Master’s Thesis: Integrated DSP considering DGs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(2)
(3)
(3)
(3)
(6)
(6)
(3)
(5)
(3)
(3)
(5)
(4)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(6)
(2)
(2)
(2)
6
5
6
7
6
6
6
5
5
5
3
4
3
4
4
3
6
5
4
4
4
3
4
6
7
3
5
6
Figure B.2: Case 1 - Secondary Networks. SS1
157
158
159
160
161
161
162
163
165
164
18
23
2
139
140
144
145
146
147
64
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
116
118
124
122
137
3
30
33
37 38
80
167 166
42
45
48
51
54
91
94
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(4)
(2)
(2)
(2)
(2)
(2)
(4)
(4)
(2)
(2)
(2)
(3)
(2)
(3)
(5)
Figure B.3: Case 2 - Primary Network.
Master’s Thesis: Integrated DSP considering DGs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(3)
(3)
(3)
(4)
(4)
(3)
(2)
(2)
(3)
(2)
(3)
(2)
(2)
(2)
(2)
(2)
(6)
(3)
(2)
(2)
(2)
5
4
6
6
6
6
4
4
5
4
4
4
2
1
4
5
5
4
4
4
5
3
5
6
5
4
Figure B.4: Case 2 - Secondary Networks. SS1
157
158
159
160
161
161
162
163
165
164
18
23
2
139
140
144
145
146
147
64
56
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
104
116
118
122
137
135
3
30
33
37 38
80
167 166
42
45
48
51
94
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(4)
(2)
(2)
(2)
(4)
(4)
(4)
(4)
(4)
(2)
(2)
(4)
(4)
Figure B.5: Case 3 - Primary Network.
Master’s Thesis: Integrated DSP considering DGs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(3)
(3)
(3)
(4)
(4)
(3)
(5)
(3)
(2)
(5)
(3)
(3)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(5)
(6)
(2)
(2)
6
5
6
6
6
6
6
5
6
5
3
4
4
4
4
3
6
5
4
5
4
4
6
7
4
5
6
Figure B.6: Case 3 - Secondary Networks. SS1
157
158
159
160
161
161
162
163
165
164
18
23
2
139
140
144
145
146
147
64
56
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
116
118
124
122
137
3
30
33
37 38
80
167 166
42
45
48
51
94
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(4)
(2)
(2)
(2)
(4)
(4)
(4)
(4)
(4)
(2)
(2)
(3)
(4)
Figure B.7: Case 4 - Primary Network.
Master’s Thesis: Integrated DSP considering DGs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(3)
(3)
(3)
(4)
(4)
(3)
(2)
(2)
(3)
(2)
(3)
(2)
(2)
(2)
(2)
(2)
(6)
(4)
(2)
(2)
(2)
6
5
6
6
6
6
6
4
6
6
3
3
4
2
4
5
6
5
4
5
4
3
6
6
5
6
Figure B.8: Case 4 - Secondary Networks.
Master’s Thesis: Integrated DSP considering DGs and ESSs B.2 Integrated DSP considering DGs and ESSs SS1
157
158
159
160
161
161
162
163
165
164
76
71
18
23
2
139
140
144
145
146
147
64
59
56
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
104
116
118
124
122
137
135
3
30
33
37 38
80
83
87 88
167 166
42
45
48
51
54
91
94
97
100
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(5)
(4)
(4)
(4)
(2)
(2)
(5)
(5)
(5)
(5)
(5)
(3)
(2)
(2)
(6)
(4)
(4)
(3)
(2)
(5)
Figure B.9: Case A - Primary Network.
Master’s Thesis: Integrated DSP considering DGs and ESSs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(5)
(6)
(2)
(2)
(3)
(3)
(6)
(3)
(2)
(6)
(6)
(6)
(6)
(6)
(3)
(6)
(6)
(3)
(3)
(6)
(6)
(3)
(3)
(6)
(6)
(2)
(3)
(2)
(3)
(3)
(3)
(3)
(3)
(4)
(3)
(2)
(3)
(5)
(5)
(5)
(3)
(5)
(6)
(2)
(2)
(2)
(4)
(5)
(3)
(6)
(2)
(6)
(5)
(5)
(5)
7
6
7
7
7
7
7
6
6
6
2
3
3
3
4
2
2
2
5
3
6
6
5
6
5
5
4
7
8
6
4
6
2
Figure B.10: Case A - Secondary Networks. SS1
157
158
159
160
161
161
162
163
165
164
76
71
18
23
2
139
140
144
145
146
147
64
59
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
104
116
118
124
122
137
135
3
30
33
37 38
80
83
87 88
167 166
42
45
48
51
54
91
94
97
100
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(5)
(4)
(4)
(4)
(2)
(2)
(5)
(5)
(5)
(5)
(5)
(4)
(2)
(2)
(2)
(6)
(3)
(5)
(4)
(3)
(5)
Figure B.11: Case B - Primary Network.
Master’s Thesis: Integrated DSP considering DGs and ESSs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(5)
(6)
(6)
(6)
(2)
(3)
(3)
(6)
(3)
(2)
(6)
(6)
(6)
(6)
(6)
(3)
(6)
(6)
(4)
(3)
(6)
(6)
(3)
(3)
(6)
(6)
(2)
(3)
(6)
(6)
(6)
(3)
(3)
(3)
(3)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(2)
(3)
(5)
(6)
(5)
(3)
(5)
(6)
(6)
(2)
(2)
(2)
(4)
(5)
(3)
(6)
(6)
(6)
(5)
(6)
(5)
6
6
7
6
7
6
6
5
6
5
3
4
2
4
4
4
4
4
6
6
6
6
6
4
4
3
7
8
6
6
6
1
Figure B.12: Case B - Secondary Networks. SS1
157
158
159
160
161
161
162
163
165
164
76
71
18
23
2
139
140
144
145
146
147
64
59
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
104
116
118
124
122
137
135
3
30
33
37 38
80
83
87 88
167 166
42
45
48
51
54
91
94
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(5)
(4)
(4)
(4)
(2)
(2)
(5)
(5)
(5)
(5)
(5)
(3)
(2)
(2)
(2)
(6)
(4)
(2)
(2)
(4)
Figure B.13: Case C - Primary Network.
Master’s Thesis: Integrated DSP considering DGs and ESSs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(5)
(6)
(6)
(6)
(3)
(3)
(6)
(3)
(2)
(6)
(6)
(6)
(6)
(6)
(3)
(6)
(6)
(4)
(3)
(6)
(6)
(3)
(3)
(6)
(6)
(2)
(3)
(6)
(6)
(6)
(3)
(3)
(3)
(3)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(2)
(3)
(5)
(6)
(5)
(3)
(5)
(6)
(6)
(2)
(2)
(2)
(4)
(5)
(3)
(6)
(6)
(6)
(5)
(6)
(4)
6
5
6
6
6
6
6
5
6
5
3
4
3
4
4
4
4
4
3
6
6
5
5
4
4
4
6
7
5
3
6
Figure B.14: Case C - Secondary Networks. SS1
157
158
159
160
161
161
162
163
165
164
76
71
18
23
2
139
140
144
145
146
147
64
59
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
104
116
118
124
122
137
135
3
30
33
37 38
80
83
87 88
167 166
42
45
48
51
54
91
94
97
100
168
169
170
171
173
172
174
175
176
177
178
179
180
181
182
183
184
185
186
(5)
(4)
(4)
(4)
(2)
(2)
(5)
(5)
(5)
(5)
(4)
(3)
(2)
(2)
(2)
(6)
(3)
(5)
(4)
(3)
(5)
Figure B.15: Case D - Primary Network.
Master’s Thesis: Integrated DSP considering DGs and ESSs
76
78
75
74
73
72
71
77
70
69
68
67
66
18
39
28
27
26
25
23
24
22
21
20
19
4
2
1
64
65
63
62
61
60
59
58
57
56
55
16
17
15
14
13
12
11
10
9
8
7
5
6
132
131
127
128
129
130
113
112
111
110
109
108
107
106
105
104
114
115
116
117
118
126
125
124
123
122
137
136
135
133
134
103
102
101
119
120
121
138
3
41
29
30
31
32
33
34
35
36
37
38
40
79
80
81
82
83
84
85
86
87
88
89
43
42
44
45
46
47
48
49
50
51
52
53
54
90
91
92
93
94
95
96
97
98
99
100
(6)
(6)
(6)
(2)
(3)
(3)
(6)
(3)
(2)
(6)
(6)
(6)
(6)
(6)
(3)
(6)
(6)
(4)
(3)
(6)
(6)
(3)
(3)
(6)
(6)
(2)
(3)
(6)
(6)
(6)
(3)
(3)
(3)
(3)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(2)
(3)
(5)
(6)
(5)
(3)
(5)
(6)
(6)
(2)
(2)
(2)
(4)
(5)
(3)
(6)
(6)
(6)
(5)
(6)
(5)
6
6
7
6
7
6
6
6
6
5
2
4
2
4
4
4
4
4
6
6
6
6
6
4
4
3
7
8
6
6
6
2
Figure B.16: Case D - Secondary Networks. SS1
157
158
159
160
161
161
162
163
165
164
76
71
18
23
2
139
140
144
145
146
147
64
59
16
11
8
5
141 142 143
148 149 150
151
152
153
154
155
156
132
127
129
113
111
109
SS2
106
104
116
118
124
122
137
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Figure B.17: Case E - Primary Network.
Master’s Thesis: Integrated DSP considering DGs and ESSs
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Figure B.18: Case E - Secondary Networks.
Cita: Valencia Díaz, Alejandro (2020), Distribution Systems Planning Considering Distributed Generation and Energy Storage Systems, Universidad Tecnológica de Pereira, p. N. https://hdl.handle.net/11059/12229