Analysis of the radial operation of distribution systems considering operation with minimal losses

Electrical Power and Energy Systems 67 (2015) 453–461

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Analysis of the radial operation of distribution systems considering operation with minimal losses Donizete Ritter, John F. Franco ⇑, Rubén Romero Universidade Estadual Paulista, Campus Ilha Solteira, SP, Brazil

a r t i c l e

i n f o

Article history: Received 5 November 2013 Received in revised form 1 December 2014 Accepted 5 December 2014 Available online 26 December 2014 Keywords: Distribution system optimization Distribution system reconfiguration Radiality constraints in distribution systems

a b s t r a c t Electric power distribution systems, and particularly those with overhead circuits, operate radially but as the topology of the systems is meshed, therefore a set of circuits needs to be disconnected. In this context the problem of optimal reconfiguration of a distribution system is formulated with the goal of finding a radial topology for the operation of the system. This paper utilizes experimental tests and preliminary theoretical analysis to show that radial topology is one of the worst topologies to use if the goal is to minimize power losses in a power distribution system. For this reason, it is important to initiate a theoretical and practical discussion on whether it is worthwhile to operate a distribution system in a radial form. This topic is becoming increasingly important within the modern operation of electrical systems, which requires them to operate as efficiently as possible, utilizing all available resources to improve and optimize the operation of electric power systems. Experimental tests demonstrate the importance of this issue. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction An electrical distribution system (EDS) is the part of the electrical system that is physically located in the area of consumption, i.e., that portion of the electrical system that starts at the distribution substation. Such system need adequate planning so that it can operate efficiently and reliably. Among the operation and expansion planning problems of distribution systems are the problems of reconfiguration of radial distribution systems and the problem of expansion planning of distribution systems. Both problems require that the optimal solution found be a radial topology. In this paper, we only analyze the problem of reconfiguration of distribution systems (RDS) but our conclusions can be extended to the distribution system expansion planning problem [1–3]. In the RDS problem, the purpose is usually to find a radial topology that allows the system to operate with minimal power losses. Radial topology was adopted by distribution companies some decades ago as the appropriate topology for operating a distribution system and especially for distribution systems with overhead circuits. The RDS problem is a complex problem related to planning the operation of distribution systems. For various technical reasons, there is a prevailing paradigm that a distribution system should operate in a radial configuration although the system has a meshed

⇑ Corresponding author. Tel.: +55 1837431236. E-mail address: [email protected] (J.F. Franco). http://dx.doi.org/10.1016/j.ijepes.2014.12.018 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

structure. The purpose is to find a radial topology that optimizes one or more objectives. The most common objective is the minimization of power losses. The EDS can be viewed as a graph with nodes and arcs, in which case, the purpose is to find the tree of the graph that allows the system to operate with minimal power losses, that is, a radial topology (tree) must be found that allows the system to operate with minimal losses and in an adequate way in relation to other operational constraints. This is a mixed integer nonlinear programming problem that is very difficult to solve. The technical reasons used to justify the radial operation of an EDS, include two reasons that are mentioned very often in the literature: (a) radial topology makes it possible to improve the coordination of protection and, (b) radial topology reduces short-circuit currents in substations. In the specialized literature there are many optimization techniques to solve the RDS problem. They can be divided into two major groups: (1) exact techniques, and (2) heuristics and metaheuristics. Exact techniques, such as branch-and-bound algorithms, were initially used only for relaxed models or using linearization techniques [1,17]. After complete models of these problems started to be used, heuristics and metaheuristics have been used very successfully to solve these problems in the last decades [4–16,18–20]. Some references regarding solution techniques for solving the RDS problem can be found in [1–13]. An important aspect of mathematical modeling is the efficient representation of the radiality constraint. For a long time, this was a controversial topic but in [2], a mathematical model that adequately represents

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the radiality constraint was presented along with the respective proof. In this study, an initial critique of to the paradigm that stipulates that the EDS must operate radially is presented. There is no accurate information about when this became the paradigm of operation but it was probably in the 1950s and 1960s. At that time, the application of optimization techniques to the operation of power systems was very limited. For example, an important 1975 article by Merlin–Back [1], notes that: ‘‘This paper proposes a method for determining a minimum-loss operating configuration in a meshed distribution system to be operated in spanning tree structure (the usual case for a distribution system)’’. Therefore, it can be concluded that by 1975, the paradigm of radial operation was already fully established. This paradigm has not been questioned even in the most recent papers. In [23] for example, the authors argue that distribution systems have a meshed structure but are usually operated as radial to assure effective coordination of their protection systems under emergency conditions and operational failures. The paper presents two new optimization models for the RDS problem: a mixedinteger conic programming model and a mixed-integer linear programming model. A radiality constraint is applied in both models. In [24,25], the authors consider that a network should be radial but there is no explanation for this as it can be seen in this excerpt: ‘‘The first constraint is that in the case of final solution, the electrical network should be a completely radial configuration. Both Kirchhoffs voltage and current laws must be observed to satisfy the AC load flow equations of the radial electrical system’’. In [26], there is some indication that the authors agreed this paradigm, because, for them, an essential criterion for the RDS is the conservation of the radial nature of the network, primarily to facilitate the coordination of protection. However, according to the authors, the methodology of probabilistic reliability evaluation described in their paper is not limited to radial distribution networks but is also valid for meshed systems. In this paper, we analyze the paradigm of radial operation and suggest new lines of research aimed at changing this paradigm. Experimental tests and a preliminary theoretical analysis show that radial topology is the worst operation topology if the goal is to operate the distribution system with minimal power losses. This observation, essentially experimental, is not compatible with modern proposals for the operation of electrical systems that propose optimal operation of all parts of the system. This study is limited to analyzing the operation of a distribution system and the paradigm of operation in radial topology. In future studies, we will analyze other aspects of the operation, especially changes and additional costs that may be incurred by operation proposals different that radial topology. Some of this research will include consideration of the economic and operating implications of non-radial operation on the increase of short-circuit currents in the substation and additional costs in the coordination of protection. The contributions of this paper are: 1. An experimental analysis that proves that radial topology is one of the worst topologies if the goal is for the system to operate with minimal power losses. 2. A preliminary theoretical analysis that shows that the initial radial topology is not the most appropriate if the goal is for the system to operate with minimal losses. 3. The presentation of a mathematical model and an optimization proposal that permits the analysis of the influence of a distribution system topology on power losses. 4. The initiation of a debate in the scientific community in order to consolidate or change the paradigm of operation of distribution systems, taking into account that a topology with a small number of loops can be interesting.

Mathematical modeling of the RDS problem We consider that traditional mathematical modeling of the RDS problem allows finding a radial topology with minimal losses. Thus, according to [2], the RDS problem for one substation is modeled as follows:

Xh

min v ¼

 i g ij xij V 2i þ V 2j  2V i V j cos hij

ð1Þ

ðijÞ2Xl

s:t:

X

P Si  P D i 

j2Xb

Q Si  Q Di 

ðxij Pij Þ ¼ 0 8i 2 Xb

X

j2Xb

V 6 Vi 6 V I2rij

þ

I2mij

6

ð2Þ

i

ðxij Q ij Þ ¼ 0 8i 2 Xb

8i 2 Xb

I2ij xij

ð3Þ

i

8ði; jÞ 2 Xl

xij 2 f0; 1g 8ði; jÞ 2 Xl X xij ¼ nb  1

ð4Þ ð5Þ ð6Þ ð7Þ

ðijÞ2Xl

in which Xl is the set of circuits; Xb is the set of nodes; Xbi is the set of connected nodes in node i ðXbi  Xb Þ; V is the minimum voltage magnitude; V is the maximum voltage magnitude; Iij is the maximum current flow in circuit ij; nb is the number of nodes ðnb ¼ jXb jÞ; P Di is the active power demand at node i; Q Di is the reactive power demand at node i; g ij is the conductance of circuit ij; bij is the susceptance of circuit ij; P ij is the active power flow that leaves node i toward node j; Q ij is the reactive power flow that leaves node i toward node j; Irij is the real component of the current flow in circuit ij; Imij is the imaginary component of the current flow in circuit ij; v represents the operational losses of the EDS; xij is the binary variable that determines if the circuit between nodes i and j is connected; V i is the voltage magnitude at node i; hij is the difference of phase angle between nodes i and j; P Si is the active power provided by substation at node i; Q Si is the reactive power provided by substation at node i; Furthermore, in the modeling above, the variable elements Pij and Q ij are obtained by (8) and (9).

Pij ¼ V 2i g ij  V i V j ðg ij cos hij þ bij sin hij Þ

ð8Þ

Q ij ¼ V 2i bij  V i V j ðg ij sin hij  bij cos hij Þ

ð9Þ

The objective function (1) represents the active power losses in the operation of the EDS. Constraints (2) and (3), along with (8) and (9), represent constraints related to Kirchhoff’s Laws for the AC model. Additionally, (4) represents the operational constraints on the voltage magnitude of nodes. The real and reactive components of the current in circuit ij in (5) are given by (10) and (11), respectively.

Irij ¼ g ij ðV i cos hi  V j cos hj Þ  bij ðV i sin hi  V j sin hj Þ

ð10Þ

Imij ¼ g ij ðV i sin hi  V j sin hj Þ þ bij ðV i cos hi  V j cos hj Þ

ð11Þ

The constraints presented in (6) represent the decision variables in the optimization problem and also account for the complexity of the problem given that xij are binary. Thus, for a feasible solution of the problem (1)–(11), the circuit between nodes i and j is connected if xij ¼ 1 and is not connected if xij ¼ 0. Additionally, only in substation PSi and Q Si do they have non-zero values. Finally (7) indicates that any feasible solution of the problem (1)–(11) must have exactly ðnb  1Þ circuits connected to the EDS. A detailed analysis of this condition is one of the main topics of this study.

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The mathematical model (1)–(11) is a mixed integer non-linear programming (MINLP) problem and can be solved using a solver for MINLP problems. It should be noted that according to the evidence presented in [2] every feasible solution as well as the optimal solution of the mathematical model (1)–(11) must be radial. In this context, the set of constraints (2) guarantees that every feasible solution must be connected since a path must exist between each load node of the distribution system and the substation. Additionally, (7) guarantees that each feasible solution must be formed by ðnb  1Þ circuits. Since each feasible solution of (1)–(11) must be connected and formed by ðnb  1Þ circuits, the graph corresponding to the electrical system is a spanning tree or, equivalently, a radial topology. For more details on the validity of the mathematical model and a generalization of the mathematical model for the case of several substations and transfer nodes (nodes without load) see [2]. In summary, if the mathematical model is solved using an optimization solver and this solver finds the optimal solution of the problem, then a radial topology must be found that allows the system to operate with minimal losses. Analysis of the mathematical model (1)–(11) leads to the conclusion that: (1) if this model is solved after eliminating (7) then a solver will find a connected optimal solution but it will not necessarily be radial, (2) this optimal solution will be found in order to determine the connected system with minimal power losses and it therefore should have the number of circuits necessary to meet these objectives, and, (3) this optimal solution must have at least ðnb  1Þ connected circuits and no more than the nl existing circuits. Thus, the central idea of this study is to experimentally verify the behavior of electrical systems whose data is found in the literature and illustrate the behavior of optimal solutions when (7) is eliminated from the mathematical model. Based on these results, some experimental conclusions can be formulated about optimal operation strategies of an EDS and to show the quality of the operation of the best radial topology when compared with the best connected topology, with the goal of achieving operation of the EDS with minimal power losses. Mathematical model used in the tests In this section, it is presented a proposal to find the optimal topology for an EDS that minimizes power losses by fixing the number of circuits that should be activated or connected in the operation of the system. It should be noted that in terms of proper connectivity, the EDS can operate with a minimum of activated circuits that is equal to ðnb  1Þ for a system of nb nodes and a maximum number of all the circuits activated. Consequently, tests can be done varying the number of circuits to be activated between those limits. The central idea of this section is to find the optimal solution for the RDS problem when the number of circuits to be connected is fixed. This number can be fixed between a minimum value of ðnb  1Þ circuits and a maximum of all the nl connected circuits. At the same time, by doing tests for all these cases, it can be verified how the optimal solution of the RDS problem varies when we fix the number of circuits that are connected. The tests presented in this paper show that if the goal is to reduce the power losses of an EDS, the radial configuration is the worst configuration. Additionally, a completely meshed topology (with all circuits operating) does not always represent the best configuration to operate with minimal losses. It will also be demonstrated that as the number of connected circuits in the topology of the reconfiguration of the system increases, the losses diminish. Another important aspect is the tendency and intensity of this reduction in power losses. To carry out the tests, we modified the mathematical modeling of problem (1)–(11) that is presented in the previous section and is

referred to as ‘‘traditional modeling’’ in this paper, in order to be able to control the number of circuits that can be connected. By doing this, we are able to find the best topology for a number of previously specified active circuits; this number is changed from the one related to the radial topology to the one related to a fully meshed topology. The new model, which is a modification of the traditional model (1)–(11), takes the following form:

min v ¼

  X g ij xij V 2i þ V 2j  2V i V j cos hij

ð12Þ

ðijÞ2Xl

s:t:

X

P Si  P D i 

j2Xb

Q Si  Q Di 

ðxij Pij Þ ¼ 0;

X

j2Xb

V 6 Vi 6 V

8i 2 Xb

ð13Þ

i

ðxij Q ij Þ ¼ 0;

8i 2 Xb

i

8i 2 Xb

I2rij þ I2mij 6 I2ij xij

ð14Þ

8ði; jÞ 2 Xl

xij 2 f0; 1g; 8ði; jÞ 2 Xl X xij ¼ nb þ k; 1 6 k 6 ðnl  nb Þ

ð15Þ ð16Þ ð17Þ ð18Þ

ðijÞ2Xl

where P ij and Q ij are given by (8) and (9), respectively. Irij and Imij are given by (10) and (11), respectively. nl is the number of candidate circuits to be connected to the system, and k is an integer value that assumes values from ð1Þ (radial configuration) up to ðnl  nb Þ (fully meshed topology). In the new model, by selecting the value of k; ðnl  nb þ 2Þ different tests for each EDS can be carried out and for each test, the best topology to minimize losses is found among all possible topologies with a specified number of circuits that should be activated. The model (12)–(18) is a generalization of the formulation (1)–(11). If the parameter k in (18) is equal to 1, then both models are equivalent and they can be used to solve the reconfiguration problem in which the solution is a radial configuration. The general formulation is used to study the operation of the distribution system under topologies different form the basic radial topology. Tests on known electrical systems In this section, in order to illustrate the proposal presented in this study, we present the results of analyses of various wellknown test systems in the literature, experimentally verifying that adding circuits to these distribution systems usually leads to a decrease in power losses. To carry out the tests, the mathematical model presented in Section ‘Mathematical model used in the tests’ was solved using the commercial solver KNITRO [21] expressed in the modeling language AMPL [22]. The numerical results were obtained using a PC Intel Core 2 Duo 4300, 1.79 GHz computer with 2 GB of RAM. KNITRO is a commercial solver designed for large problems with dimensions running into the hundred thousands. It is effective for solving linear, quadratic, and nonlinear smooth optimization problems, both convex and nonconvex. It is also effective for nonlinear regression problems with complementarity constraints and mixed-integer programming, particularly convex mixed integer nonlinear problems. KNITRO is highly regarded for its robustness and efficiency [21]. It implements a nonlinear branch and bound method for mixed-integer nonlinear programming problems. The voltage limit was fixed at 0.90 p.u. for all tests and the current limit was set for each circuit according to the data for each system. It was verified that the obtained solutions were feasible and all operating variables were within their limits. The solver found the solution within the default convergence tolerance (i.e., the absolute integrality gap stop tolerance was

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Substation

13

equal to 1.0e6). The computational efforts for each system are shown in the corresponding results table.

The data for the 14-node system is provided in [15]. This system has 14 nodes and 16 circuits and therefore it is possible to perform four different tests by fixing the number of activated circuits as 13 (radial topology), 14, 15 or 16 (meshed topology with all the circuits connected). An additional test was carried out by eliminating (18), i.e., leaving open the decision about the number of circuits that should be activated. Fig. 1 shows the optimal radial topology. The results of the tests are shown in Table 1. The first column shows the number of circuits connected in the obtained topology, R means radial topology and CE means that (18) was eliminated from the mathematical formulation. The table also shows the power losses, the circuits disconnected in the best solution found in each case (column 3), and the processing time. Table 1 presents important results including: (a) among all radial topologies that exist in the EDS, the one that produces the fewest power losses has circuits 1–10, 6–8 and 7–9 disconnected; (b) among all topologies with 14 circuits activated, the best has circuits 1–10 and 7–9 disconnected; (c) among all the topologies with 15 circuits activated, the best is the one with circuit 1–10 disconnected; and (d) the best topology aimed at reducing power losses is the topology with all circuits activated. Furthermore, the results in Table 1 show that radial topology produces the greatest power losses while the topology with all circuits connected has the smallest power losses. The best topology found (meshed with all circuits connected) is shown in Fig. 2. The results indicate that between these two extremes in operation there is a difference of 39.87 kW in power losses, which is a significant difference (8:55% less). In addition, there is a quasi-radial topology (with 2 circuits disconnected and therefore operating with only a single loop) that results in power losses of 430.03 kW. Consequently, this topology could represent an excellent operational proposal because it is very close to a radial topology but results in a reduction in the power losses of 36.10 kW (7.74%).

3

5 10 ...............1

11

14-node distribution system

4

.. 9 .. .. .. .. ..7 .. 12 . . . 8 . ... .... 6.......

2

PowerLosses: 466.13kW Fig. 1. Optimal radial topology for the 14-node test system.

Table 1 Results for the 14-node system. No. of circuits

Power losses (kW)

Disconnected circuits

Time (s)

13 (R) 14 15 16 CE

466.13 430.03 426.47 426.26 426.26

1–10, 6–8, 7–9 1–10, 7–9 1–10 – –

4.05 0.23 0.08 0.03 0.08

Substation

13

4

9 3

7

12

8 6 5

11

10

1

2

33-node distribution system

Power Losses: 426.26 kW

The data for the 33-node system is in [17]. This system has 33 nodes and 37 circuits and therefore we were able to perform six different tests by fixing the number of activated circuits as 32

Fig. 2. 14-node system: Best topology to reduce power losses.

SE

22

23

24

1

2

3

4

18

19

20

21

25

5

6

26

7

27

8

28

9

29

10

30

11

31

12

Fig. 3. Optimal radial topology for the 33-node test system.

32

13

14

15

16

17

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D. Ritter et al. / Electrical Power and Energy Systems 67 (2015) 453–461

therefore operating with only one loop) leads to power losses of 124.55 kW. Thus, this topology may also represent an excellent operation proposal because it provides a topology that is very close to a radial topology but with reduced power losses of 15.00 kW (10.75%). Another aspect that should be noted is that the test with a requirement for radial topology needs a much longer processing time in comparison with other tests. Additionally, for the 33-node system, the optimum solution, when the condition of radiality is relaxed, is a system with one disconnected circuit while the topology with all circuits activated has very similar power losses.

Table 2 Results for the 33-node system. No. of circuits

Power losses (kW)

Disconnected circuits

Time (s)

32 (R) 33 34 35 36 37 CE

139.55 124.55 123.82 123.43 123.25 123.29 123.25

6–7, 6–7, 8–9, 8–9, 8–9 – 8–9

4154.0 35.12 16.31 3.28 0.09 0.03 0.12

8–9, 13–14, 24–28, 31–32 8–9, 13–14, 31–32 13–14, 31–32 13–14

(radial topology), 33, 34, 35, 36 or 37 (meshed topology with all circuits connected). Also, an additional test was carried out by eliminating (18), i.e., leaving open the decision about how may circuits should be activated. Fig. 3 shows the optimal radial topology for this system. The results of the tests are shown in Table 2. The results in Table 2 show once again that radial topology is the topology that produces the largest power losses, but in this case, a topology with only one disconnected circuit (8–9) is the one that produces the smallest power losses. The results show that between these operational extremes there is a difference in power losses of 16.30 kW (11.68%). The difference in power losses of the best topology in comparison to the topology with all circuits connected is very small (0.04 kW). Once again, it can be observed that a quasi-radial topology (with four circuits disconnected and

84-node distribution system The data for the 84-node system is in [27]. This system has 84 nodes and 96 circuits and therefore we could perform 14 different tests by fixing the number of activated circuits as 83 (radial topology), 84, . . . , 95, or 96 (meshed topology with all the circuits connected). Fig. 4 shows the optimal radial topology for this system, while Fig. 5 shows the best topology to reduce the power losses. Table 3 shows the results obtained in tests with the 84-node system. For this system, a fully meshed configuration is not the best configuration in terms of power losses, while the configuration with 90 of a possible 96 circuits connected has the lowest power losses (the six circuits that should be disconnected are shown in Table 3).

0

2

1

3

5

4

6

7

8

9

55

10

54 64

11

12

53

52

51

50

49

48

47

63

62

61

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59

58

57

72

71

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68

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66

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77

56

13

to 43

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22 25

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29 83

39

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to 11 Fig. 4. Optimal radial topology for the 84-node test system.

82

81

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For this system, a smaller reduction in power losses was achieved (in the range of 2:09%) in comparison with the best radial solution, which once again resulted in the worst power losses. It was found that a topology with 87 connected circuits has power losses of 460.48 kW, representing a good solution for the operation of the system because this topology has few loops and a reduction of 2:00% in power losses. The processing time varied from 0.05 s. for the fully meshed topology to 2415 min. for the radial topology.

119-node distribution system The data for the 119-node system is in [28]. Since this system has 119 nodes and 133 circuits, we could perform 16 different tests fixing the number of activated circuits at 118 (radial topology), 119, . . . , 132 or 133 (meshed topology with all circuits connected). Table 4 shows the results obtained in tests with the 119-node system. For this system, the configuration with 130 circuits

0

2

1

4

3

6

5

7

8

9

55

10

54 64

11 12 to 43

53

52

51

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47

63

62

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22 25

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81

40

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to 11 Fig. 5. Best topology to reduce the power losses for the 84-node test system.

Table 3 Results for the 84-node system. No. of circuits

Power losses (kW)

Disconnected circuits

Time (s)

83 (R) 84 85 86 87 88 89 90 91 92 93 94 95 96 CE

469.88 465.25 462.41 461.23 460.48 460.22 460.09 460.06 460.11 460.30 460.27 460.30 460.34 462.68 460.06

6–7, 11–43, 12–13, 14–18, 16–26, 28–32, 33–34, 38–39, 41–42, 54–55, 61–62, 71–72, 82–83 6–7, 11–43, 12–13, 14–18, 16–26, 28–32, 33–34, 38–39, 41–42, 54–55, 60–61, 82–83 6–7, 11–43, 12–13, 14–18, 28–32, 33–34, 38–39, 40–42, 54–55, 61–62, 82–83 6–7, 11–43, 12–13, 14–18, 32–33, 38–39, 41–42, 54–55, 61–62, 82–83 5–55, 11–43, 12–13, 14–18, 32–33, 38–39, 41–42, 62–63, 82–83 5–55, 12–13, 14–18, 33–34, 38–39, 41–42, 62–63, 82–83 5–55, 12–13, 14–18, 37–38, 41–42, 62–63, 82–83 5–55, 12–13, 37–38, 40–42, 62–63, 82–83 5–55, 14–18, 41–42, 62–63, 82–83 5–55, 14–18, 62–63, 81–82 5–55, 14–18, 82–83 14–18, 82–83 82–83 – 5–55, 12–13, 37–38, 40–42, 62–63, 82–83

0.05 2 6 56 356 952 2145 2496 8469 15,469 34,710 65,821 98,587 144,968 3847

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D. Ritter et al. / Electrical Power and Energy Systems 67 (2015) 453–461 Table 4 Results for the 119-node system. No. of circuits

Power losses (kW)

Disconnected circuits

Time (s)

118 (R) 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 CE

853.61 843.60 839.71 836.38 833.23 829.44 824.29 824.19 822.26 821.43 819.98 820.11 819.96 820.04 820.26 820.50 819.96

23–24, 23–24, 23–24, 23–24, 23–24, 22–23, 22–23, 23–24, 22–23, 22–23, 22–23, 26–27, 22–23, 22–23, 26–27 – 22–23,

0.03 14 89 248 846 1248 1789 1894 2245 2143 2478 4567 14,780 58,762 189,670 623,145 18,947

25–26, 25–26, 25–26, 25–26, 25–26, 25–26, 26–27, 25–26, 25–26, 26–27, 25–26, 75–76, 26–27, 26–27

34–35, 39–40, 42–43, 43–54, 50–51, 58–59, 71–72, 74–75, 83–108, 86–105, 91–96, 97–98, 109–110 34–35, 39–40, 42–43, 43–54, 50–51, 58–59, 71–72, 74–75, 83–108, 86–105, 91–96, 97–98 34–35, 42–43, 43–54, 50–51, 58–59, 71–72, 74–75, 83–108, 86–105, 91–96, 98–99 34–35, 42–43, 43–54, 50–51, 60–61, 71–72, 73–74, 75–76, 83–108, 86–105 34–35, 43–44, 43–54, 50–51, 58–59, 71–72, 76–77, 83–108, 86–105 34–35, 38–39, 42–43, 58–59, 73–74, 75–76, 83–108, 86–105 42–43, 50–51, 58–59, 73–74, 75–76, 83–108, 86–105 42–43, 50–51, 59–60, 75–76, 83–108, 86–105 42–43, 58–59, 76–77, 83–108, 86–105 42–43, 75–76, 83–108, 86–105 75–76, 83–108, 86–105 83–108, 86–105 75–76

26–27, 75–76

12

13

15

14

16

17 24

2

23

11

10

18

19

7

8

20

5

6

22

26

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43

9

38

39

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41

42

32

33

34

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47

44

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51

47

48

30

3

31 28

0

21

4

48

53 52

29

36

37

55

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connected is the best configuration in terms of power losses. In this system, the best solution has a reduction of power losses of 3:94% compared to the best radial configuration, which again emerged as

the worst configuration. The topology with 124 circuits connected had power losses of 824.29 kW, representing a good solution for the operation of the system because it has few loops but a

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reduction of 3:43% in power losses. The processing time varied from 0.03 s. for the fully meshed topology to 10,835 min. for the radial topology. Fig. 6 shows the optimal radial topology for this system, while Fig. 7 shows the best topology to reduce the power losses. Results of tests with other well-known systems in the literature show a very similar behavior to the test results presented in this paper. Thus, experimentally, we observed the following tendencies: (a) radial topology is the type of topology that represents the highest levels of power losses when the number of circuits that must be connected in the optimization process is fixed; (b) the topology that has the lowest power losses is the one with all or almost all of the circuits connected; and (c) quasi-radial topologies (topologies with few loops) generally have power losses that are very close to the power losses of optimal topologies (topologies with all or almost all of the circuits connected). Therefore, an interesting proposal would be to operate with quasi-radial topologies. In order to develop a practical proposal for changing the operation scheme adopted for standard use in electrical systems, we must first examine the consequences of this proposed change on the operation of the system, most importantly in relation to the protection and increase of short-circuit currents that can produce a change in the logic of the standard of operation.

Preliminary analysis of the results An interesting research topic is to find a mathematical proof that shows that increasing the number of circuits connected in an EDS produces a reduction in power losses. This characteristic has been observed in experimental tests. It should be noted that radial topology is the topology with the smallest possible number of circuits connected in a distribution system. Thus, as the number of circuits is increased from a minimum of ðnb  1Þ circuits, power losses decrease and this trend is diminished in the final steps. Thus the optimal solution is a network with all or almost all circuits connected. Rigorous theoretical analysis is still ongoing on the tendency toward power loss reduction when the number of circuits connected in an EDS is increased. However, there are preliminary theoretical studies that demonstrate this tendency. In [1], it is proven that if the optimization problem of minimizing power losses in a system is solved subjected only to Kirchhoff’s Currents law, then optimality conditions for minimal power losses indicate that the network must satisfy Kirchhoff’s Voltage law when the impedance of the circuits is replaced by their resistance. In other words, the power flow problem (which satisfies the two Kirchhoff laws) necessarily minimizes the power losses of the system. To meet this

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goal, the power flows should be greater where the circuits have lower resistance since the intention is to minimize the sum of the square of the current multiplied by the resistance and, obviously, this proof has some simplifying assumptions, and therefore only indicates a trend. Intuitively, we also can conclude that if the number of circuits in a distribution system is increased then there is a tendency to reduce power losses. Suppose that there is a distribution system with k connected circuits with a value of power losses of pk . In this context, if an additional circuit is added to the system then, according to [1], we should see a redistribution of power flows with a trend towards increasing the current in circuits with less resistance. Since power losses depend on the square of the electric current, there is a tendency toward a reduction in power losses. Conclusions This paper has demonstrated that the paradigm of operating with a radial topology in EDS is the least effective if the goal is for the system to operate with minimal power losses. It was also noted that quasi-radial topologies (with few loops) have power losses very close to minimum power losses. Therefore, it is possible to operate distribution systems with quasi-radial topologies and thereby significantly reduce the power losses of the system. This issue should be examined in more detail, especially in relation to optimal operation of modern electric systems. Obviously, before there is a change in the operation paradigm, the impact on other aspects of system operation must be analyzed, especially in relation to the coordination of protection and shortcircuit currents at the substation. In future studies, we will develop these research topics and will carry out a rigourous mathematical analysis of the behavior of the distribution system when the number of circuits connected is increased. References [1] Merlin A, Back G. Search for minimum loss operational spanning tree configuration for an urban power distribution system. In: Proc 5th power system conf (PSCC), Cambridge; 1975. p. 1–18. [2] Lavorato M, Franco JF, Rider MJ, Romero R. Imposing radiality constraints in distribution system optimization problems. IEEE Trans Power Syst 2012;27(1): 172–80. [3] Gönen T. Electric power distribution systems engineering. New York: McGrawHill; 1986. [4] Schmidt H, Ida N, Kagan N, Guaraldo J. Fast reconfiguration of distribution systems considering loss minimization. IEEE Trans Power Syst 2005;20(3): 1311–9. [5] Gomes F, Carneiro S, Pereira J, Vinagre M, Garcia P, Araujo L. A new heuristic reconfiguration algorithm for large distribution systems. IEEE Trans Power Syst 2006;21(4):1616–23. [6] Lopez E, Opazo H, Garcia L, Bastard P. Online reconfiguration considering variability demand: applications to real networks. IEEE Trans Power Syst 2004;19(1):549–53.

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