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A power restoration strategy for the distribution network based on the weighted ideal point method
Electrical Power and Energy Systems 63 (2014) 1030–1038
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A power restoration strategy for the distribution network based on the weighted ideal point method Jing Ma ⇑, Wei Ma, Dong Xu, Yang Qiu, Zengping Wang State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China
a r t i c l e
i n f o
Article history: Received 17 August 2013 Received in revised form 13 June 2014 Accepted 6 July 2014 Available online 30 July 2014 Keywords: Power recovery Multi-objective optimization Rough set theory Ideal point method
a b s t r a c t A power restoration strategy for the distribution network based on the weighted ideal point method is proposed in this paper. First, with the power loss, voltage quality and load balancing as the reconfiguration goal, the comprehensive evaluation function is established. And then, the rough set theory is used to calculate the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. Simulation tests on the 33-bus and 69-bus radial distribution networks verify that, the proposed method of using multi-objective function instead of single objective function to gain the optimal solution is more comprehensive in different considerations and more in line with the requirement of the actual distribution network power restoration. Furthermore, the uncertainty problem in traditional weight setting due to dependence on experience can be avoided. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Power restoration in the distribution network refers to the restoring of power supply to the lost load in the non-fault area by means of network reconfiguration after a fault occurs. Power restoration should meet the constraints of the node voltage and line current, as well as taking into account the economic and reliable operation of the distribution network after the fault recovery. Therefore, power restoration is a multi-objective optimization problem with multiple constraints [1–3]. Currently there are many methods to establish and solve the power restoration objective function and these methods can be sorted into three major categories. (1) Mathematical optimization methods [4–7]. This category of methods is based on a complete mathematical theory, and is not dependent on the initial structure of the distribution network. However, these methods have the ‘dimension calamity’ problem, thus are only applicable to power restoration of systems with relatively small scale and complexity. (2) Heuristic search methods [8–10]. This category of methods is widely used for fault restoration in the distribution network, and is mainly based on switch operation, according to the search mode guided by the heuristic rules. These methods are real-time, and ⇑ Corresponding author. Address: No. 52 Mailbox, North China Electric Power University, No. 2 Beinong Road, Changping District, Beijing 102206, China. Tel.: +86 10 80794899 (work), mobile: +86 15801659769. E-mail address: [email protected] (J. Ma). http://dx.doi.org/10.1016/j.ijepes.2014.07.017 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
have good versatility and practicability. However, the search result is easily affected by the initial state, thus the algorithm stability is not favorable. (3) Artificial intelligence methods [11–16]. This category of methods is effective in solving specific power restoration problems. However, parameters such as the penalty coefficient, the initial particle swarm, and the mutation probability are needed. These parameters will directly affect the calculation speed and convergence of the algorithm. And currently there is no clear theoretical basis as to what values the parameters should apply. Continuous grope is needed concerning the specific problem. The advantage of the above methods is that the power restoration problem can be expressed accurately in the form of objective function with constraints However, since power restoration is a complicated optimization problem with multiple objectives, and the sub-objectives are conflictingly inter-connected, it is difficult to describe the comprehensive objective function. Besides, the weight setting is often reliant on subjective experience, which makes the solving process complicated. In view of the above problems, a power restoration strategy for the distribution network based on the weighted ideal point method is proposed in this paper. First, with the power loss, voltage quality and load balancing as the reconfiguration goal, the comprehensive evaluation function is established. And then, the rough set theory is used to determine the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. Simulation tests on the 33-bus and 69-bus radial distribution networks verify
J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038
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Nomenclature U min j U max j Imax Imax i Ii D1 D2 C1 C2 C3 m Pi Qi Ui ri ki Pimax Um Ue S U A V
lower bound of the RMS value of the voltage at node j. upper bound of the RMS value of the voltage at node j. line current constraint matrix the maximum current allowed in line i the current on line i node voltage constraint line current constraint power loss function voltage quality function load balancing function number of lines active power on the end of line i reactive power on the end of line i voltage of the end node of line i resistance of line i state variable of line i maximum capacity of line i measured voltage vector of the system rated voltage vector of the system knowledge representation system universe of discourse finite set of properties value range of A
f C D posQ(P)
u(Ci) cC(D)
cCCi ðDÞ Y wi Hi(+) Hi() Ci(x) gj(+) gj(+) Li(+) Li() Ti
decision function condition property set decision property set the set of elements in U that can be accurately divided to U/P through Q importance degree of property Ci dependence degree of the decision property set D on the condition property set C dependence degree of the decision property set D on the condition property set C–Ci comprehensive evaluation function weight coefficient of the sub-objective function ideal point vector of the ith index of object H anti-ideal point vector of the ith index of object H value of the ith index positive evaluation function negative evaluation function the distance from the evaluation function to the ideal point the distance from the evaluation function to the anti-ideal point the weighed ideal point close degree
that, the proposed method of using multi-objective function instead of single objective function to gain the optimal solution is more comprehensive in different considerations and more in line with the requirement of the actual distribution network power restoration. Furthermore, the uncertainty problem in traditional weight setting due to dependence on experience can be avoided.
where Imax represents the maximum current allowed in line I; m is i the number of lines. The line current constraint is as follows:
Constraints and objective functions
Power loss function
After the fault is located and separated accurately, there will be lost load in the downstream of the fault area. First, all the contact switches in the original network that may help to restore power to the lost load are virtually closed. It should be noted that, this operation may cause branches to interconnect and form loops (the route between two different sources is also equivalent to a loop). On this basis, all the breakable branches in the loops constitute of an operable switch set. And then, virtual breaking of the switches in the operable switch set is done separately, and power flow of the resulting network is calculated to determine whether each virtual breaking meets the constraints of the node voltage and line current. With all the constraints met, the optimal breaking scheme can then be determined by the optimization of the multi-objective function of power loss, voltage quality and load balancing.
The node voltage amplitude constraint is as follows:
U min j
ð1Þ
U max j
where and are the lower bound and upper bound of the RMS value of the voltage at node j. In order to meet the thermal stability constraint of the distribution network lines, the line current constraint matrix Imax is defined according to the maximum current allowed in each line: max Imax ¼ ½Imax ; ; Imax 1 ; ; Ii m
T
ð3Þ
where Ii is the current of line i.
The objective function of the distributed system power loss is:
C1 ¼
m X P2i þ Q 2i i¼1
U 2i
! r i ki
ð2Þ
ð4Þ
where m is the number of lines; Pi and Qi are the active and reactive power of line i; Ui is the voltage of line i; ri is the resistance of line i; ki is the state variable of line i, with ki = 0 representing that line i is open and ki = 1 representing that line i is closed. The smaller the power loss function C1 is, the higher the power utilization rate is, and the more energy saving and economic the distribution network proves to be. Voltage quality function The objective function of the power quality is:
C 2 ¼ kU m U e k
Constraints of node voltage and line current
U min U j U max j j
Ii Imax i
ð5Þ
where Um and Ue are the measured voltage vector and rated voltage vector of the system. The smaller the voltage quality function C2 is, the closer the bus voltage is to the rated value, and the better the voltage quality is. Load balancing function Load balancing refers to the ability of the distribution system to match the load distribution with the power supply capacity of each line. It aims to improve operational efficiency of the lines and power supply capability of the distributed system, at the same time reducing power loss and lowering the risk of overloading.
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The function with a balanced load distribution as the optimization goal C3 can be expressed as:
C3 ¼
2 n 1X Pi n i¼1 Pmax i
ð6Þ
where n is the number of closed lines. Pmax is the maximum i capacity of line i. The smaller the load balancing function C3 is, the more balanced the load rate is for each line, and the higher the equipment utilization rate is, the bigger the overall power supply capacity is.
where the lower approximation set R-(X) represents the set of elements under knowledge R that definitely belong to X; the upper approximation set R(X) represents the set of elements under knowledge R that possibly belong to X; [x]R represents the equivalence class divided according to equivalence relation R that includes x. Suppose knowledge P and knowledge Q are both equivalence relations in theory domain U, and the Q-positive domain of P is defined as:
posQ ðPÞ ¼
[
Q ðXÞ
ð8Þ
X2U=P
Power restoration strategy for the distribution network based on the weighted ideal point method First, the comprehensive evaluation function is established. And then, the rough set theory is used to calculate the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. The rough set theory The rough set theory is a mathematical tool to deal with fuzzy and indefinite information, as well as qualitative and quantitative factors, which is widely applied in the power system [17–19]. Compared with other methods to deal with uncertainty problem (for example, data mining methods based on the probability theory/the fuzzy theory/the evidence theory), the rough set theory has a biggest advantage that no previous experience is needed other than the data set to be dealt with. In this paper, the rough set theory is used to determine the weight coefficients of the objective functions. By transforming the weight solving problem to the property importance evaluation problem in the rough set, the weights of the objective functions can be calculated using the dependence degree and property importance evaluation method in the rough set theory. Thus, the uncertainty problem in traditional weight setting due to dependence on previous experience can be avoided.
posQ(P) represents the set of elements in U that can be accurately divided to U/P through Q. (1) Knowledge dependence degree In order to illustrate the dependence between knowledge P and knowledge Q, the dependence degree of P on Q is defined as:
cQ ðPÞ ¼
cardðposQ ðPÞÞ cardðUÞ
ð9Þ
where card() is the set cardinality. Obviously, 0 6 cQ(P) 6 1, and the closer cQ(P) is to 1, the more dependent P is on Q. (1) Property importance degree In the decision table, the importance of each condition property Ci to the decision property set D differs. If Ci is deleted, then the dependence degree of D on the condition property C–Ci is defined as:
cCCi ðDÞ ¼
cardðposCC i ðDÞÞ cardðUÞ
ð10Þ
From (9) and (10), it can be seen that the importance degree of the condition property Ci to D is:
uðC i Þ ¼ cC ðDÞ cCCi ðDÞ
ð11Þ
The bigger u(Ci) is, the more important Ci is in the condition property set C.
(1) Basic concepts The knowledge representation system is S = (U, A, V, f), where U is a non-zero finite set of objects, which is called universe of discourse; A is a non-zero finite set of properties; V is the value range of A; f:U A ? V is an decision function which assigns an information value to each property of each object. Data in the knowledge representation system are expressed in relationships. When A = C[D and C\D = U, C is called the condition property set and D is called the decision property set. The knowledge representation system with both the condition property and decision property is called the decision table, which is a special and important kind of knowledge representation system. The theoretical foundation of RS is the classification mechanism, which is also interpreted as the equivalence relation (also called knowledge). The equivalence relations are used to divide a particular space. Suppose R is an equivalence relation in theory domain U, and U is divided by R into dis-intersected sub sets Ei, denoted as U/R = {E1, E2,. . ., Ek}, where Ei and the empty set are called the basic sets. For X # U, if X cannot be expressed accurately by the union of basic sets, then X is called a rough set. The upper and lower approximation sets of X are defined as follows:
R ðXÞ ¼ [fx 2 Uj½xR # Xg R ðXÞ ¼ [fx 2 Uj½xR \ X–Ug
ð7Þ
Power restoration strategy comprehensive evaluation function and calculation of the weight coefficients Comprehensive evaluation function According to ‘Constraints and objective functions’, there are three sub-objective functions of the power restoration strategy in this paper. Thus, the weight vector of the evaluation functions is W = [w1, w2, w3], which meets the following conditions:
8 3 > < Xw ¼ 1 i > i¼1 : 0 wi 1
ð12Þ
The comprehensive evaluation function Y is then established:
Y ¼ w1 C 1 þ w2 C 2 þ w3 C 3
ð13Þ
Weight coefficients of the sub-objective functions In order to calculate the weights of the sub-objective functions, the relational data model is first established. Taking the sub-objective functions as the condition properties, the condition property set C = {C1, C2,C3} is formed. And taking the constraints as the decision properties, the decision property set D = {D1, D2} is formed. Supposing there are k evaluation objects, by taking the sub-objective function and constraints of the tth evaluation object as a piece of information, information ut = {ct1, ct2,. . .,ctp, dt1, dt2,. . .,dtq} is
J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038
defined and then theory domain U = {u1, u2,. . .,ut,. . .uk} is formed. The information table consisting of uk is the relational data model of the evaluation objects. The weight coefficients of the sub-objective functions can be calculated with the following steps:
8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 > h i > uX p ffiffiffiffiffiffiffiffiffiffiffiffi > C i;j Hi;j ðþÞ 2 > t > L w Y j ðþÞ ¼ j ðþÞ ¼ i > ðþÞ H > i;j < i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 > > h i pffiffiffiffiffiffiffiffiffiffiffiffi uX > C i;j Hi;j ðÞ 2 > t > L ðÞ ¼ ðÞ w Y ¼ > j j i > Hi;j ðÞ :
1033
ð20Þ
i¼1
(1) Calculate the dependence degree of the decision property set D on the condition property set C: cC(D); (2) Calculate the dependence degree of the decision property set D on the condition property set C–Ci: cCC i ðDÞ; (3) Calculate the importance degree of property Ci: u(Ci); (4) Calculate the weight coefficients of the sub-objective functions according to (14).
wi ¼
uðC i Þ 3 X
ði ¼ 1; 2; 3Þ
ð14Þ
uðC j Þ
j¼1
Thus, the weighed ideal point close degree can be calculated as follows:
Tj ¼
Lj ðÞ Lj ðþÞ þ Lj ðÞ
ð21Þ
where the bigger Tj is, i.e. the closer the evaluation function is to the ideal point and the farther it is from the anti-ideal point, the more reasonable the solution of the multi-objective function is. And the maximum Tj is the optimal solution of the comprehensive evaluation function.
Optimal solution of the comprehensive evaluation function Graph theory for maintaining the radially of the system The evaluation index basically takes two forms: the income type index and the loss type index. The bigger the income type index is, the better the evaluation result is; the bigger the loss type index is, the worse the evaluation result is. On this basis, the ideal point and anti-ideal point can be defined as following: For the income type index:
Hi ðþÞ ¼ max C i
ð15Þ
Hi ðÞ ¼ min C i For the loss type index:
Hi ðþÞ ¼ min C i
ð16Þ
Hi ðÞ ¼ max C i
where Hi(+) and Hi() are the ideal point vector and anti-ideal point vector of the ith index of object H; Ci is the actual value of this index, i = 1, 2, 3. The weighted ideal point evaluation function is the distance from the index to the ideal point and anti-ideal point, i.e.
kC i Hi ðþÞk ! min
ð17Þ
kC i Hi ðÞk ! max
The closer the index solution is to the ideal point and the farther it is from the anti-ideal point, the better the index solution is. According to (17), the positive evaluation function and negative evaluation function are defined:
8 h i C Hi;j ðþÞ 2 > < g j ðþÞ ¼ i;jH ðþÞ i;j h i2 > : g ðÞ ¼ C i;j Hi;j ðÞ j Hi;j ðÞ
ð18Þ
To ensure the radially of the system, the Graph Theory approach could be applied [20]. First, the Depth First Search method is used to search for the operable switch set corresponding to the loops in the distribution network. Second, by breaking any line in the operable switch set, the loop could be opened and the radially of the system could be guaranteed. Take the distribution network in Fig. 1 for example, the formation of the operable switch set is illustrated as follows. First, by the Depth First Search method, the two independent loops in Fig. 1 could be found: Lloop1 and Lloop2. According to the graph theory, an independent loop is formed by the closing of some link branches, and the breaking of any link branch will open the loop. Therefore, the set of link branches that form the independent loops are called the operable switch set. Obviously, the operable switch set corresponding to Lloop1 is {1–2, 2–3, 3–4, 4–5, 5–7, 7–8, 8–9, 1–9}, and the operable switch set corresponding to Lloop2 is {2–3, 3–4, 4–10, 10–11, 11–13, 13–14, 14–15, 2–15}. After the operable switch sets corresponding to the loops are found, break any line in the operable switch sets, and the radially of the system in Fig. 1 could be guaranteed.
Simulation verification To test the proposed algorithm, 12.66 kV standard 33-bus and 69-bus radial distribution systems are used. The upper bound and lower bound of the node voltage RMS value are 1.05p.u and 0.95p.u respectively. The line limits of the 33-bus and 69-bus radial distribution test systems can be found in [20].
where Ci,j represents the jth element of Ci. Hi,j(+) and Hi,j() represent the jth element of Hi(+) and Hi() respectively. Applying (14), (18) and (13), the weighted ideal point evaluation function can be obtained:
8 3 h i X > C Hi;j ðþÞ 2 > > Y j ðþÞ ¼ wi i;jH ðþÞ > i;j < i¼1 3 h i > X > C Hi;j ðÞ 2 > > wi i;jH ðÞ : Y j ðÞ ¼ i¼1
ð19Þ
i;j
Then the distance from the evaluation function to the ideal point and anti-ideal point can be defined using the Euclidean distance:
Fig. 1. A connected graph derived from one distribution network with all tie breakers closed.
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J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038 Table 2 Power flow calculation result for IEEE 33-bus system.
Fig. 2. 33-bus system.
Table 1 Information data table. Object
C1
C2
C3
D1
D2
u1 u2 .. . ut .. . uk
C11 C21 .. . Ct1 .. . Ck1
C12 C22 .. . Ct2 .. . Ck2
C13 C23 .. . Ct3 .. . Ck3
D11 D21 .. . Dt1 .. . Dk1
D12 D22 .. . Dt2 .. . Dk2
33-bus radial distribution network The electrical wiring of the 33-bus test system is shown in Fig. 2, which includes 33 nodes, 32 lines and 5 contact switches. The base load is 4369.35 kVA (see Table 1). (1) Power flow calculation Suppose the fault occurs on both line 9–10 and line 14–15, the fault location system [21] will identify the fault and relevant switches will be tripped. Thus the load in the downstream of the fault area will lose power supply. The lost load set is defined as:
K 1 ¼ f10; 11; 12; 13; 14g K 2 ¼ f15; 16; 17; 18g
ð22Þ
When the lost load is known, connection switches BRK12–22, BRK9–14 and BRK18–33 will close automatically to restore power supply to the lost load. According to the principle of restoring as much load as possible, if the line current constraints are met, by closing switch BRK12–22 all the lost load in K1 can be restored to power supply quickly. However, if the line current constraints are not met, some load on this power supply route need to be cut according to the importance degree of the load. As for the lost load in K2, by closing switches BRK9–14 and BRK18–33, they can all be restored to power supply. However, the following loop will result: Lloop ¼ f6—7; 7—8; 8—9; 9—15; 15—16; 16—17; 17—18; 18—33; 33 —32;32—31;31—30; 30—29; 29—28; 28—27; 27—26; 26—6g
ð23Þ
In order to maintain the radiation-shaped structure of the distribution system, the virtual switches in the operable switch set need to be virtually operated and flow calculation needs to be carried out on the network after the virtual breaking. And then, it needs to be decided if each virtual breaking meets the node voltage constraint D1 and line current constraint D2, at the same time the values of the power loss function C1, the voltage quality function C2 and the load balancing function C3 are calculated, shown in Table 2.
Scheme
Breaking switches
C1 (kW)
C2 (kV)
C3
D1
D2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6–7 7–8 8–9 9–15 15–16 16–17 17–18 18–33 33–32 32–31 31–30 30–29 29–28 28–27 27–26 26–6
213 198 186 186 193 186 202 193 199 186 186 198 205 213 213 231
6.2 5.3 4.9 4.7 4.7 4.4 4.2 4.3 4.4 4.4 4.4 5.3 5.8 6.2 6.2 6.9
0.21 0.19 0.18 0.18 0.18 0.17 0.17 0.16 0.16 0.17 0.17 0.19 0.20 0.22 0.22 0.23
0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0
0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0
Note: in the constraints of node voltage and line current, ‘0’ means the constraints are not met and ‘1’ means the constraints are met.
(2) Weight coefficient In the decision table, the decision property set and the condition property set are D = {D1 the node voltage constraint, D2 the line current constraint} and C = {C1 the power loss, C2 the voltage quality, C3 the load balancing} respectively. (1) The dependence degree of the decision property set D on the condition property set C It can be seen from Table 2 that, the division of theory domain U by D is U/D = {Y1, Y2, Y3, Y4}, where Y1 = {1, 15, 16}, Y2 = {2, 3, 12, 13, 14}, Y3 = {4}, Y4 = {5,6,7,8,9,10,11}. The division of theory domain U by C is U/C = {X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12}, where X1 = {1}, X2 = {2, 12}, X3 = {3}, X4 = {4}, X5 = {5}, X6 = {6, 10, 11}, X7 = {7}, X8 = {8}, X9 = {9}, X10 = {13}, X11 = {14, 15}, X12 = {16}. The sub set Y1 represents the set of breaking schemes that meet neither the node voltage constraint nor the line current constraint. The C positive domain of D is:
posC ðDÞ ¼ fX 1 ; X 2 ; X 3 ; X 4 ; X 5 ; X 6 ; X 7 ; X 8 ; X 9 ; X 10 ; X 12 g
ð24Þ
¼ f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 16g According to (9), in the decision table the dependence degree of D on C is:
cC ðDÞ ¼
cardðposC ðDÞÞ 14 ¼ ¼ 0:8750 cardðUÞ 16
ð25Þ
(2) The dependence degree of the decision property set D on the condition property set C–Ci Take the power loss as an example for analysis. The division of theory domain U by C–C1 is U/C–C1 = {X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11}, where X1 = {1}, X2 = {2, 12}, X3 = {3}, X4 = {4, 5}, X5 = {6, 10, 11}, X6 = {7}, X7 = {8}, X8 = {9}, X9 = {13}, X10 = {14, 15}, X11 = {16}.
posCC 1 ðDÞ ¼ fX 1 ; X 2 ; X 3 ; X 5 ; X 6 ; X 7 ; X 8 ; X 9 ; X 11 g ¼ f1; 2; 3; 6; 7; 8; 9; 10; 11; 12; 13; 16g
ð26Þ
According to (10), the dependence degree of D on the condition property set without the power loss C–C1 is:
cCC1 ðDÞ ¼
cardðposCC 1 ðDÞÞ 12 ¼ ¼ 0:7500 cardðUÞ 16
ð27Þ
Similarly, it can be calculated that the dependence degrees corresponding to the voltage quality and the load balancing are 0.7500 and 0.8125 respectively.
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J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038
(3) The importance degree of property Ci According to (11), the importance degrees of the power loss, the voltage quality and the load balancing are:
8 > < uðC 1 Þ ¼ cC ðDÞ cCC 1 ðDÞ ¼ 0:1250 uðC 2 Þ ¼ cC ðDÞ cCC2 ðDÞ ¼ 0:1250 > : uðC Þ ¼ c ðDÞ c ðDÞ ¼ 0:0625 3
C
69-bus radial distribution network
ð28Þ
CC 3
(4) The weight coefficients of the sub-objective functions According to (14), the weight coefficients of the power loss, the voltage quality and the load balancing can be calculated, being 0.4000, 0.4000 and 0.2000 respectively. The dependence degree, importance degree and weight coefficient of the evaluation indexes are listed in Table 3.
The evaluation functions of power loss, voltage quality and load balancing are all loss-type indexes. The smaller their value is, the more reasonable the breaking scheme and the resulting network prove to be. The schemes that meet both the node voltage and line current constraints constitute of the feasible scheme set. According to (16), the ideal point and anti-ideal point vectors of the evaluation indexes can be calculated:
The electrical wiring of the 69-bus test system is shown in Fig. 3, which includes 69 nodes, 68 lines and 5 contact switches. The total active load is 3802.2 kW, and total reactive load is 2694.6 kVar. Suppose that fault occurs on line 13–14 and line 20–21 at the same time. The fault locating system [21] identifies the faults and relevant switches are tripped immediately. Therefore, power supply to the downstream load of the fault location is lost. Define the lost load set as:
(3) Close degree and scheme rating
Lloop ¼
this index is considered, there will be two optimal schemes. The multi-index optimization in the proposed method, however, is quick to distinguish that scheme 8 is better than scheme 9.
K 1 ¼ f14; 16; 17; 18; 20g
When the lost load is determined, the contact switches BRK15–46, BRK13–21 and BRK27–65 will close automatically to restore power supply to the lost load. For the lost load in K1, fast power restoration could be achieved by closing switch BRK15–46. For the lost load in K2, fast power restoration could be achieved by closing switch BRK13–21 and switch BRK27–65. However, the following loop is formed:
9—10; 10—11; 11—12; 12—13; 13—21; 21—22; 22—23; 23—24; 24—25; 25—26; 26—27; 27—65; 65—54; 64—63; 63—62; 62—61; 61—60; 60—59; 59—58; 58—58; 57—56; 56—55; 55—54; 54—53; 53—9
8 H1 ðþÞ ¼ ½186; 186; . . . ; 186116 > > > > > H1 ðÞ ¼ ½213; 231; . . . ; 231116 > > > < H2 ðþÞ ¼ ½4:4; 4:4; :::; 4:4116 > > H2 ðÞ ¼ ½6:9; 6:9; . . . ; 6:9116 > > > > > H3 ðþÞ ¼ ½0:16; 0:16; . . . ; 0:16116 > : H3 ðÞ ¼ ½0:23; 0:23; . . . ; 0:23116
ð29Þ
And then by (17) and (20), the weighted ideal point evaluation function of the feasible schemes can be calculated. Finally, the weighted ideal point close degree can be gained according to (21). The schemes in the feasible scheme set are rated according to the value of their close degree, shown in Table 4. Comparison between Tables 4 and 2 is shown in Table 5, where ‘0’ means the node voltage and line current constraints cannot be both met; ‘1’ represents the optimal scheme; ‘2’ represents the sub-optimal scheme; ‘3’ represents the 3rd grade scheme, and so on. It can be seen from Table 5 that, the optimal scheme obtained by the proposed method (corresponding to the biggest close degree of the optimal evaluation function) is scheme 8, i.e. breaking line 18–33. In this scheme, the load balancing index is the optimal, while the power loss index and the voltage quality index both reach the sub-optimal, i.e. rating ‘2’. However, the sub-optimal values of the two indexes are both very close to the optimal values. It can also be seen that, concerning the load balancing index, both scheme 8 and scheme 9 reach the optimal. Therefore, if only Table 3 Dependence degree, importance degree and weight coefficient of the indexes. Evaluation function
Power loss
Voltage quality
Load balancing
Dependence degree Importance degree Weight coefficient
0.7500 0.1250 0.4000
0.7500 0.1250 0.4000
0.8125 0.0625 0.2000
ð30Þ
K 2 ¼ f21; 22; 24; 26; 27g
ð31Þ
In order to keep the network structure radial, virtual breaking of the lines in Lloop are conducted, and flow calculation is carried out in the post-breaking network. And then, decide whether each virtual breaking meets the node voltage constraint D1 and line current constraint D2. At the same time, the power loss function C1, voltage quality function C2 and the load balance function C3 are calculated, shown in Table 6. It can be seen from Table 6 that, the division of the decision attribute set D on domain U is U/D = {Y1, Y2, Y3}, where Y1 = {1, 2}, Y2 = {3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 22, 23, 24, 25}, Y3 = {11, 12, 13, 14, 15, 16, 17}. Y3 represents the set of power restoration schemes that meet both the node voltage constraint and the line current constraint. The division of the condition attribute set C on domain U is U/C = {X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22}, where X1 = {1}, X2 = {2}, X3 = {3}, X4 = {4, 5}, X5 = {6}, X6 = {7, 8}, X7 = {9}, X8 = {10}, X9 = {11}, X10 = {12}, X11 = {13}, X12 = {14, 15}, X13 = {16}, X14 = {17}, X15 = {18}, X16 = {19}, X17 = {20}, X8 = {21}, X19 = {22}, X20 = {23}, X21 = {24}, X22 = {25}. The dependence degree of the decision attribute set D on the condition attribute set C is C ðDÞÞ cC ðDÞ ¼ cardðpos ¼ 1, which means the decision attribute set D is cardðUÞ
totally dependent on the condition attribute set C. According to the rough set theory, the dependence degree, importance degree and weight coefficient of the power loss, voltage quality and the load balance can be obtained, shown in Table 7. According to (14) and (18) to (13), the weighted ideal point evaluation function of each scheme is calculated. Finally, the weighted ideal point fitness degree could be calculated according to (21). And then, the grade of each scheme in the feasible scheme set is evaluated according to the fitness degree. Relevant calculation results and evaluation results are shown in Table 8. It can be seen from that, the optimal scheme obtained with the proposed
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Table 4 Evaluation function, close degree and scheme rating for 33-bus system. Scheme
Breaking line
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6–7 7–8 8–9 9–15 15–16 16–17 17–18 18–33 33–32 32–31 31–30 30–29 29–28 28–27 27–26 26–6
Evaluation function Li(+)
Li()
0.3444 0.1901 0.1193 0.0937 0.0967 0.0411 0.0612 0.0282 0.0535 0.0411 0.0411 0.1901 0.2734 0.3567 0.3567 0.4764
0.0898 0.1889 0.2413 0.2555 0.2469 0.2851 0.2849 0.2935 0.2806 0.2851 0.2851 0.1889 0.1365 0.0832 0.0832 0
Close degree Tj
Constraints met?
Scheme rating
0.2067 0.4985 0.6691 0.7315 0.7184 0.8740 0.8232 0.9124 0.8398 0.8740 0.8740 0.4985 0.3330 0.1891 0.1891 0
No No No No Yes Yes Yes Yes Yes Yes Yes No No No No No
0 0 0 0 5 2 4 1 3 2 2 0 0 0 0 0
Note: ‘0’ means the node voltage and line current constraints cannot be both met; ‘1’ represents the optimal scheme; ‘2’ represents the sub-optimal scheme; ‘3’ represents the 3rd grade scheme, and so on.
Table 5 Calculation results and rating evaluation for 33-bus system. Scheme
Breaking switches
C1 (kW)
C1 rating
C2 (kV)
C2 rating
C3
C3 rating
Close degree Tj
Tj rating
Constraints met?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6–7 7–8 8–9 9–15 15–16 16–17 17–18 18–33 33–32 32–31 31–30 30–29 29–28 28–27 27–26 26–6
213 198 186 186 193 186 202 193 199 186 186 198 205 213 213 231
0 0 0 0 2 1 5 2 4 1 1 3 0 0 0 0
6.2 5.3 4.9 4.7 4.7 4.4 4.2 4.3 4.4 4.4 4.4 5.3 5.8 6.2 6.2 6.9
0 0 0 0 4 3 1 2 3 3 3 6 0 0 0 0
0.21 0.19 0.18 0.18 0.18 0.17 0.17 0.16 0.16 0.17 0.17 0.19 0.20 0.22 0.22 0.23
0 0 0 0 3 2 2 1 1 2 2 0 0 0 0 0
0.2067 0.4985 0.6691 0.7315 0.7184 0.8740 0.8232 0.9124 0.8398 0.8740 0.8740 0.4985 0.3330 0.1891 0.1891 0
0 0 0 0 5 2 4 1 3 2 2 0 0 0 0 0
No No No No Yes Yes Yes Yes Yes Yes Yes No No No No No
Fig. 3. 69-bus test system.
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J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038 Table 6 Power flow calculation result for 69-bus system. Scheme
Breaking line
C1 (kW)
C2 (kV)
C3
D1
D2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
9–10 10–11 11–12 12–13 13–21 21–22 22–23 23–24 24–25 25–26 26–27 27–65 65–64 64–63 63–62 62–61 61–60 60–59 59–58 58–57 57–56 56–55 55–54 54–53 53–9
216 212 184 158 158 146 146 146 143 143 146 143 136 125 125 125 124 213 230 230 230 230 234 239 240
8.2 8.0 7.0 6.1 6.1 5.8 5.6 5.6 5.4 5.3 5.4 5.4 5.0 5.0 5.0 4.9 4.9 6.5 6.8 7.1 7.3 7.5 7.8 8.1 8.3
0.24 0.24 0.23 0.22 0.22 0.21 0.21 0.21 0.21 0.21 0.21 0.20 0.21 0.21 0.21 0.21 0.22 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Note: for the node voltage and line current constraints, ‘0’ represents the constraints are not satisfied and ‘1’ represents the constraints are satisfied.
method is scheme 16, i.e. breaking line 61–62. In scheme 16, the indexes of voltage quality and load balance reach the optimal solution, and the power loss index reaches the sub-optimal. Comparison with power restoration strategy based on genetic algorithm Take the 69-bus system for example, comparison is made between the proposed method and power restoration strategy based on genetic algorithm. In the fault restoration scheme based on genetic algorithm, each switch is viewed as a gene on the chromosome. The binary encoding method is used, i.e. when the switch is ‘off’, the corresponding gene is 0, and when the switch is ‘on’, the corresponding gene is 1. The initial population is 50. First, it is determined whether all the power restoration schemes are radial in network structure, and those with non-radial network structure are deleted. And then, the fitness degrees of the fault restoration schemes with radial network structure are calculated. During the genetic operation process, the roulette wheel selection method is used, and the crossover algorithm applies single-point crossover. The crossover probability is 0.7, and the mutation probability is 0.5. After the genetic operation, the fault restoration effect is shown in Table 9. It can be seen that, when applied for power restoration, the proposed method is superior to the genetic algorithm in lowest node voltage and flow calculation times. Analysis of computation time
Table 7 Dependence degree, importance degree and weight coefficient of the indexes for 69bus system. Evaluation function
Power loss
Voltage quality
Load balancing
Dependence degree Importance degree Weight coefficient
0.9200 0.0800 0.2857
0.8800 0.1200 0.4286
0.9200 0.0800 0.2857
In terms of computation time, the computation time of the proposed power restoration scheme is from when the loop is formed to when the optimal scheme is obtained. During this process, the computation time is mainly the flow calculation time, which equals to the time needed for each flow calculation multiplied by the flow calculation times. The time needed for each flow calculation is dependent on the power network itself, and the flow calculation times are dependent on the number of elements in the
Table 8 Evaluation function, close degree and scheme rating for 69-bus system. Scheme
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Breaking line
9–10 10–11 11–12 12–13 13–21 21–22 22–23 23–24 24–25 25–26 26–27 27–65 65–64 64–63 63–62 62–61 61–60 60–59 59–58 58–57 57–56 56–55 55–54 54–53 53–9
Evaluation function Li( + )
Li()
0.6025 0.5717 0.3899 0.2237 0.2237 0.1555 0.1358 0.1358 0.1090 0.1014 0.1190 0.1057 0.0597 0.0302 0.0302 0.0271 0.0535 0.4675 0.5467 0.5665 0.5808 0.5959 0.6330 0.6740 0.6943
0.0679 0.0784 0.1728 0.2650 0.2650 0.3054 0.3158 0.3158 0.3310 0.3365 0.3267 0.3379 0.3633 0.3794 0.3794 0.3848 0.3813 0.1542 0.1204 0.0972 0.0820 0.0669 0.0416 0.0159 0.0000
Close degree Tj
Constraints met?
Scheme rating
0.1013 0.1205 0.3071 0.5423 0.5423 0.6627 0.6992 0.6992 0.7522 0.7685 0.7329 0.7618 0.8588 0.9263 0.9263 0.9343 0.8771 0.2480 0.1805 0.1465 0.1237 0.1009 0.0617 0.0231 0.0000
No No No No No No No No No No Yes Yes Yes Yes Yes Yes Yes No No No No No No No No
0 0 0 0 0 0 0 0 0 0 6 5 4 2 2 1 3 0 0 0 0 0 0 0 0
Note: ‘0’ represents the node voltage constraints and branch current constraints cannot be satisfied at the same time; ‘1’ represents the optimal scheme, ‘2’ represents the suboptimal scheme, and ‘3’ represents the 3rd-grade scheme, and so on.
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J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038
Table 9 Comparison of fault service restoration results. Methods
Breaking off switch set
Power loss (kW)
Lowest node voltage (kV)
Flow calculation times
AG Proposed method
11–43, 13–14, 20–21, 58–59, 27–65 11–43, 13–14, 20–21, 50–59, 61–62
118 125
11.95 12.04
783 25
operable switch set. The proposed method is realized in Matlab2012a on a computer with 2.00 GHz frequency, 2 GB memory, and Inter(R) Core(TM) 2 Duo CPU processor. Simulation results show that, the average time needed for each flow calculation of 33-bus and 69-bus system are 0.04 s and 0.056 s respectively. In 33-bus system, when fault occurs on line 8–9 and line 13–14 of at the same time, the power restoration time using the proposed method is 1.08 s. In 69-bus system, when fault occurs on line 13–14 and line 20–21 of 69-bus system at the same time, the power restoration time using the proposed method is 1.93 s. Besides, compared with the genetic algorithm, the flow calculation times of the proposed method are obviously reduced. Therefore, the proposed method could contribute to fast power restoration in the distribution network.
Conclusion A power restoration strategy for the distribution network based on the weighted ideal point method is proposed in this paper. First, with the power loss, voltage quality and load balancing as the reconfiguration goal, the comprehensive evaluation function is established. And then, the rough set theory is used to determine the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. Simulation results verify that the proposed method has the following advantages: (1) The weight coefficients of the objective functions obtained by the rough set theory are flexibly adaptive to the network structure. Thus, the uncertainty and immutability problem in traditional weight setting due to dependence on subjective experience can be avoided. (2) By applying multi-objective optimization to the virtual reconfiguration network, the breaking scheme is required to reach the optimal in multiple indexes. Therefore, the non-selectivity problem in single objective optimization due to more than one scheme reaching the optimal can be avoided.
Acknowledgements This work was supported by the National Basic Research Program of China (973 program) (2012CB215200), the National Natural Science Foundation of China (50907021, 51277193, 50837002), the Chinese University Scientific Fund Project (2014ZZD02), the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry ([2011] No. 1139), Hebei Natural Science Foundation (E2012502034), Electric Power Youth Science and Technology Creativity Foundation of CSEE ([2012] No. 46), the New-Star of Science and Technology supported by Beijing Metropolis Beijing Nova program (Z141101001814012),
the Excellent talents in Beijing City (2013B009005000001), the Fund of Fok Ying Tung Education Foundation (141057). References [1] Huang CM, Hsieh CT, Wang YS. Evolution of radial basic function neural network for fast restoration of distribution systems with load variations. Int J Electr Power Energy Syst 2011;33(4):916–68. [2] Sathish KK, Jayabarathi T. Power system reconfiguration and loss minimization for an distribution systems using bacterial foraging optimization algorithm. Int J Electr Power Energy Syst 2012;36(1):13–7. [3] González A, Echavarren F, Rouco L, Gómez T, Cabetas J. Reconfiguration of large-scale distribution networks for planning studies. Int J Electr Power Energy Syst 2012;37(1):86–94. [4] Stankovic AM, Calovic MS. Graph oriented algorithm for the steady-state security enhancement in distribution networks. IEEE Trans Power Del 1989;4(1):539–44. [5] Morelato AL, Monticelli A. Heuristic search approach distribution system restoration. IEEE Trans Power Del 1989;4(4):2235–41. [6] Nahman J, Strbac G. A new algorithm for service restoration in large-scale urban distribution systems. Electr Power Syst Res 1994;29(3):181–92. [7] Perez GR, Heydt GT, Jack NJ, Keel, Brian K, Castelhano Jr AR. Optimal restoration of distribution systems using dynamic programming. IEEE Trans Power Del 2008;23(2):1589–96. [8] Popovic DS, Ciric RM. A multi-objective algorithm for distribution networks restoration. IEEE Trans Power Del 1999;14(3):1134–41. [9] Miu KN, Chiang HD, Yuan B, Darling G. Fast service restoration for large-scale distribution systems with priority customers and constraints. IEEE Trans Power Syst 1998;13(3):789–95. [10] Li WX, Wang P, Li ZM, Liu YC. Reliability evaluation of complex radial distribution systems considering restoration sequence and network constraints. IEEE Trans Power Del 2004;19(2):753–8. [11] Shin DJ, Kim JO, Kin TK, Choo JB, Singh C. Optimal service restoration and reconfiguration of network using genetic-tabu algorithm. Electr Power Syst Res 2004;71(2):145–52. [12] Gomes FS, Carneiro Jr S, Pereira JLR, Vinagre MP, Garcia PAN, Araujo LR. A new heuristic reconfiguration algorithm for large distribution systems. IEEE Trans Power Syst 2005;20(3). [13] Ahuja A, Das S, Pahwa A. An AIS-ACO hybrid approach for multi-objective distribution system reconfiguration. IEEE Trans Power Syst 2007;22(3): 1101–11. [14] Kumar Y, Das B, Sharma J. Multiobjective, multiconstraints service restoration of electric power distribution system with priority customer. IEEE Trans Power Del 2008;23(1):261–70. [15] Singh SP, Raju GS, Rao GK, Afsari M. A heuristic method for feeder reconfiguration and service restoration in distribution networks. Int J Electr Power Energy Syst 2009;31(7-8):309–14. [16] EI-Zonkoly AM. Power system single step restoration incorporating cold load pickup aided by distributed generation. Int J Electr Power Energy Syst 2012;35(1):186–93. [17] Hor CL, Crossley PA. Substation event analysis using information from intelligent electronic devices. Int J Electr Power Energy Syst 2006;28(6): 374–86. [18] Othman ML, Aris I, Othman MR, Osman H. Rough-set-and-genetic-algorithm based data mining and rule quality measure to hypothesize distance protective relay operation characteristics from relay event report. Int J Electr Power Energy Syst 2011;33(8):1437–56. [19] Biswas S, Dey D, Chatterjee B, Chakravorti S. An approach based on rough set theroy for identification of single and multiple partial discharge source. Int J Electr Power Energy Syst 2013;46(1):163–74. [20] Aman MM, Jasmon GB, Bakar AHA, Mokhlis H. Optimum network reconfiguration based on maximization of system loadability using continuation power flow theorem. Int J Electr Power Energy Syst 2014; 54:123–33. [21] Lim IH, Sidhu TS, Choi MS, Lee SJ, Hong S, Lim SI, et al. Design and implementation of multiagent-based distributed restoration system in DAS. IEEE Trans Power Del 2013;28(2):585–93.
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A power restoration strategy for the distribution network based on the weighted ideal point method Jing Ma ⇑, Wei Ma, Dong Xu, Yang Qiu, Zengping Wang State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China
a r t i c l e
i n f o
Article history: Received 17 August 2013 Received in revised form 13 June 2014 Accepted 6 July 2014 Available online 30 July 2014 Keywords: Power recovery Multi-objective optimization Rough set theory Ideal point method
a b s t r a c t A power restoration strategy for the distribution network based on the weighted ideal point method is proposed in this paper. First, with the power loss, voltage quality and load balancing as the reconfiguration goal, the comprehensive evaluation function is established. And then, the rough set theory is used to calculate the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. Simulation tests on the 33-bus and 69-bus radial distribution networks verify that, the proposed method of using multi-objective function instead of single objective function to gain the optimal solution is more comprehensive in different considerations and more in line with the requirement of the actual distribution network power restoration. Furthermore, the uncertainty problem in traditional weight setting due to dependence on experience can be avoided. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Power restoration in the distribution network refers to the restoring of power supply to the lost load in the non-fault area by means of network reconfiguration after a fault occurs. Power restoration should meet the constraints of the node voltage and line current, as well as taking into account the economic and reliable operation of the distribution network after the fault recovery. Therefore, power restoration is a multi-objective optimization problem with multiple constraints [1–3]. Currently there are many methods to establish and solve the power restoration objective function and these methods can be sorted into three major categories. (1) Mathematical optimization methods [4–7]. This category of methods is based on a complete mathematical theory, and is not dependent on the initial structure of the distribution network. However, these methods have the ‘dimension calamity’ problem, thus are only applicable to power restoration of systems with relatively small scale and complexity. (2) Heuristic search methods [8–10]. This category of methods is widely used for fault restoration in the distribution network, and is mainly based on switch operation, according to the search mode guided by the heuristic rules. These methods are real-time, and ⇑ Corresponding author. Address: No. 52 Mailbox, North China Electric Power University, No. 2 Beinong Road, Changping District, Beijing 102206, China. Tel.: +86 10 80794899 (work), mobile: +86 15801659769. E-mail address: [email protected] (J. Ma). http://dx.doi.org/10.1016/j.ijepes.2014.07.017 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
have good versatility and practicability. However, the search result is easily affected by the initial state, thus the algorithm stability is not favorable. (3) Artificial intelligence methods [11–16]. This category of methods is effective in solving specific power restoration problems. However, parameters such as the penalty coefficient, the initial particle swarm, and the mutation probability are needed. These parameters will directly affect the calculation speed and convergence of the algorithm. And currently there is no clear theoretical basis as to what values the parameters should apply. Continuous grope is needed concerning the specific problem. The advantage of the above methods is that the power restoration problem can be expressed accurately in the form of objective function with constraints However, since power restoration is a complicated optimization problem with multiple objectives, and the sub-objectives are conflictingly inter-connected, it is difficult to describe the comprehensive objective function. Besides, the weight setting is often reliant on subjective experience, which makes the solving process complicated. In view of the above problems, a power restoration strategy for the distribution network based on the weighted ideal point method is proposed in this paper. First, with the power loss, voltage quality and load balancing as the reconfiguration goal, the comprehensive evaluation function is established. And then, the rough set theory is used to determine the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. Simulation tests on the 33-bus and 69-bus radial distribution networks verify
J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038
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Nomenclature U min j U max j Imax Imax i Ii D1 D2 C1 C2 C3 m Pi Qi Ui ri ki Pimax Um Ue S U A V
lower bound of the RMS value of the voltage at node j. upper bound of the RMS value of the voltage at node j. line current constraint matrix the maximum current allowed in line i the current on line i node voltage constraint line current constraint power loss function voltage quality function load balancing function number of lines active power on the end of line i reactive power on the end of line i voltage of the end node of line i resistance of line i state variable of line i maximum capacity of line i measured voltage vector of the system rated voltage vector of the system knowledge representation system universe of discourse finite set of properties value range of A
f C D posQ(P)
u(Ci) cC(D)
cCCi ðDÞ Y wi Hi(+) Hi() Ci(x) gj(+) gj(+) Li(+) Li() Ti
decision function condition property set decision property set the set of elements in U that can be accurately divided to U/P through Q importance degree of property Ci dependence degree of the decision property set D on the condition property set C dependence degree of the decision property set D on the condition property set C–Ci comprehensive evaluation function weight coefficient of the sub-objective function ideal point vector of the ith index of object H anti-ideal point vector of the ith index of object H value of the ith index positive evaluation function negative evaluation function the distance from the evaluation function to the ideal point the distance from the evaluation function to the anti-ideal point the weighed ideal point close degree
that, the proposed method of using multi-objective function instead of single objective function to gain the optimal solution is more comprehensive in different considerations and more in line with the requirement of the actual distribution network power restoration. Furthermore, the uncertainty problem in traditional weight setting due to dependence on experience can be avoided.
where Imax represents the maximum current allowed in line I; m is i the number of lines. The line current constraint is as follows:
Constraints and objective functions
Power loss function
After the fault is located and separated accurately, there will be lost load in the downstream of the fault area. First, all the contact switches in the original network that may help to restore power to the lost load are virtually closed. It should be noted that, this operation may cause branches to interconnect and form loops (the route between two different sources is also equivalent to a loop). On this basis, all the breakable branches in the loops constitute of an operable switch set. And then, virtual breaking of the switches in the operable switch set is done separately, and power flow of the resulting network is calculated to determine whether each virtual breaking meets the constraints of the node voltage and line current. With all the constraints met, the optimal breaking scheme can then be determined by the optimization of the multi-objective function of power loss, voltage quality and load balancing.
The node voltage amplitude constraint is as follows:
U min j
ð1Þ
U max j
where and are the lower bound and upper bound of the RMS value of the voltage at node j. In order to meet the thermal stability constraint of the distribution network lines, the line current constraint matrix Imax is defined according to the maximum current allowed in each line: max Imax ¼ ½Imax ; ; Imax 1 ; ; Ii m
T
ð3Þ
where Ii is the current of line i.
The objective function of the distributed system power loss is:
C1 ¼
m X P2i þ Q 2i i¼1
U 2i
! r i ki
ð2Þ
ð4Þ
where m is the number of lines; Pi and Qi are the active and reactive power of line i; Ui is the voltage of line i; ri is the resistance of line i; ki is the state variable of line i, with ki = 0 representing that line i is open and ki = 1 representing that line i is closed. The smaller the power loss function C1 is, the higher the power utilization rate is, and the more energy saving and economic the distribution network proves to be. Voltage quality function The objective function of the power quality is:
C 2 ¼ kU m U e k
Constraints of node voltage and line current
U min U j U max j j
Ii Imax i
ð5Þ
where Um and Ue are the measured voltage vector and rated voltage vector of the system. The smaller the voltage quality function C2 is, the closer the bus voltage is to the rated value, and the better the voltage quality is. Load balancing function Load balancing refers to the ability of the distribution system to match the load distribution with the power supply capacity of each line. It aims to improve operational efficiency of the lines and power supply capability of the distributed system, at the same time reducing power loss and lowering the risk of overloading.
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The function with a balanced load distribution as the optimization goal C3 can be expressed as:
C3 ¼
2 n 1X Pi n i¼1 Pmax i
ð6Þ
where n is the number of closed lines. Pmax is the maximum i capacity of line i. The smaller the load balancing function C3 is, the more balanced the load rate is for each line, and the higher the equipment utilization rate is, the bigger the overall power supply capacity is.
where the lower approximation set R-(X) represents the set of elements under knowledge R that definitely belong to X; the upper approximation set R(X) represents the set of elements under knowledge R that possibly belong to X; [x]R represents the equivalence class divided according to equivalence relation R that includes x. Suppose knowledge P and knowledge Q are both equivalence relations in theory domain U, and the Q-positive domain of P is defined as:
posQ ðPÞ ¼
[
Q ðXÞ
ð8Þ
X2U=P
Power restoration strategy for the distribution network based on the weighted ideal point method First, the comprehensive evaluation function is established. And then, the rough set theory is used to calculate the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. The rough set theory The rough set theory is a mathematical tool to deal with fuzzy and indefinite information, as well as qualitative and quantitative factors, which is widely applied in the power system [17–19]. Compared with other methods to deal with uncertainty problem (for example, data mining methods based on the probability theory/the fuzzy theory/the evidence theory), the rough set theory has a biggest advantage that no previous experience is needed other than the data set to be dealt with. In this paper, the rough set theory is used to determine the weight coefficients of the objective functions. By transforming the weight solving problem to the property importance evaluation problem in the rough set, the weights of the objective functions can be calculated using the dependence degree and property importance evaluation method in the rough set theory. Thus, the uncertainty problem in traditional weight setting due to dependence on previous experience can be avoided.
posQ(P) represents the set of elements in U that can be accurately divided to U/P through Q. (1) Knowledge dependence degree In order to illustrate the dependence between knowledge P and knowledge Q, the dependence degree of P on Q is defined as:
cQ ðPÞ ¼
cardðposQ ðPÞÞ cardðUÞ
ð9Þ
where card() is the set cardinality. Obviously, 0 6 cQ(P) 6 1, and the closer cQ(P) is to 1, the more dependent P is on Q. (1) Property importance degree In the decision table, the importance of each condition property Ci to the decision property set D differs. If Ci is deleted, then the dependence degree of D on the condition property C–Ci is defined as:
cCCi ðDÞ ¼
cardðposCC i ðDÞÞ cardðUÞ
ð10Þ
From (9) and (10), it can be seen that the importance degree of the condition property Ci to D is:
uðC i Þ ¼ cC ðDÞ cCCi ðDÞ
ð11Þ
The bigger u(Ci) is, the more important Ci is in the condition property set C.
(1) Basic concepts The knowledge representation system is S = (U, A, V, f), where U is a non-zero finite set of objects, which is called universe of discourse; A is a non-zero finite set of properties; V is the value range of A; f:U A ? V is an decision function which assigns an information value to each property of each object. Data in the knowledge representation system are expressed in relationships. When A = C[D and C\D = U, C is called the condition property set and D is called the decision property set. The knowledge representation system with both the condition property and decision property is called the decision table, which is a special and important kind of knowledge representation system. The theoretical foundation of RS is the classification mechanism, which is also interpreted as the equivalence relation (also called knowledge). The equivalence relations are used to divide a particular space. Suppose R is an equivalence relation in theory domain U, and U is divided by R into dis-intersected sub sets Ei, denoted as U/R = {E1, E2,. . ., Ek}, where Ei and the empty set are called the basic sets. For X # U, if X cannot be expressed accurately by the union of basic sets, then X is called a rough set. The upper and lower approximation sets of X are defined as follows:
R ðXÞ ¼ [fx 2 Uj½xR # Xg R ðXÞ ¼ [fx 2 Uj½xR \ X–Ug
ð7Þ
Power restoration strategy comprehensive evaluation function and calculation of the weight coefficients Comprehensive evaluation function According to ‘Constraints and objective functions’, there are three sub-objective functions of the power restoration strategy in this paper. Thus, the weight vector of the evaluation functions is W = [w1, w2, w3], which meets the following conditions:
8 3 > < Xw ¼ 1 i > i¼1 : 0 wi 1
ð12Þ
The comprehensive evaluation function Y is then established:
Y ¼ w1 C 1 þ w2 C 2 þ w3 C 3
ð13Þ
Weight coefficients of the sub-objective functions In order to calculate the weights of the sub-objective functions, the relational data model is first established. Taking the sub-objective functions as the condition properties, the condition property set C = {C1, C2,C3} is formed. And taking the constraints as the decision properties, the decision property set D = {D1, D2} is formed. Supposing there are k evaluation objects, by taking the sub-objective function and constraints of the tth evaluation object as a piece of information, information ut = {ct1, ct2,. . .,ctp, dt1, dt2,. . .,dtq} is
J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038
defined and then theory domain U = {u1, u2,. . .,ut,. . .uk} is formed. The information table consisting of uk is the relational data model of the evaluation objects. The weight coefficients of the sub-objective functions can be calculated with the following steps:
8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 > h i > uX p ffiffiffiffiffiffiffiffiffiffiffiffi > C i;j Hi;j ðþÞ 2 > t > L w Y j ðþÞ ¼ j ðþÞ ¼ i > ðþÞ H > i;j < i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 > > h i pffiffiffiffiffiffiffiffiffiffiffiffi uX > C i;j Hi;j ðÞ 2 > t > L ðÞ ¼ ðÞ w Y ¼ > j j i > Hi;j ðÞ :
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ð20Þ
i¼1
(1) Calculate the dependence degree of the decision property set D on the condition property set C: cC(D); (2) Calculate the dependence degree of the decision property set D on the condition property set C–Ci: cCC i ðDÞ; (3) Calculate the importance degree of property Ci: u(Ci); (4) Calculate the weight coefficients of the sub-objective functions according to (14).
wi ¼
uðC i Þ 3 X
ði ¼ 1; 2; 3Þ
ð14Þ
uðC j Þ
j¼1
Thus, the weighed ideal point close degree can be calculated as follows:
Tj ¼
Lj ðÞ Lj ðþÞ þ Lj ðÞ
ð21Þ
where the bigger Tj is, i.e. the closer the evaluation function is to the ideal point and the farther it is from the anti-ideal point, the more reasonable the solution of the multi-objective function is. And the maximum Tj is the optimal solution of the comprehensive evaluation function.
Optimal solution of the comprehensive evaluation function Graph theory for maintaining the radially of the system The evaluation index basically takes two forms: the income type index and the loss type index. The bigger the income type index is, the better the evaluation result is; the bigger the loss type index is, the worse the evaluation result is. On this basis, the ideal point and anti-ideal point can be defined as following: For the income type index:
Hi ðþÞ ¼ max C i
ð15Þ
Hi ðÞ ¼ min C i For the loss type index:
Hi ðþÞ ¼ min C i
ð16Þ
Hi ðÞ ¼ max C i
where Hi(+) and Hi() are the ideal point vector and anti-ideal point vector of the ith index of object H; Ci is the actual value of this index, i = 1, 2, 3. The weighted ideal point evaluation function is the distance from the index to the ideal point and anti-ideal point, i.e.
kC i Hi ðþÞk ! min
ð17Þ
kC i Hi ðÞk ! max
The closer the index solution is to the ideal point and the farther it is from the anti-ideal point, the better the index solution is. According to (17), the positive evaluation function and negative evaluation function are defined:
8 h i C Hi;j ðþÞ 2 > < g j ðþÞ ¼ i;jH ðþÞ i;j h i2 > : g ðÞ ¼ C i;j Hi;j ðÞ j Hi;j ðÞ
ð18Þ
To ensure the radially of the system, the Graph Theory approach could be applied [20]. First, the Depth First Search method is used to search for the operable switch set corresponding to the loops in the distribution network. Second, by breaking any line in the operable switch set, the loop could be opened and the radially of the system could be guaranteed. Take the distribution network in Fig. 1 for example, the formation of the operable switch set is illustrated as follows. First, by the Depth First Search method, the two independent loops in Fig. 1 could be found: Lloop1 and Lloop2. According to the graph theory, an independent loop is formed by the closing of some link branches, and the breaking of any link branch will open the loop. Therefore, the set of link branches that form the independent loops are called the operable switch set. Obviously, the operable switch set corresponding to Lloop1 is {1–2, 2–3, 3–4, 4–5, 5–7, 7–8, 8–9, 1–9}, and the operable switch set corresponding to Lloop2 is {2–3, 3–4, 4–10, 10–11, 11–13, 13–14, 14–15, 2–15}. After the operable switch sets corresponding to the loops are found, break any line in the operable switch sets, and the radially of the system in Fig. 1 could be guaranteed.
Simulation verification To test the proposed algorithm, 12.66 kV standard 33-bus and 69-bus radial distribution systems are used. The upper bound and lower bound of the node voltage RMS value are 1.05p.u and 0.95p.u respectively. The line limits of the 33-bus and 69-bus radial distribution test systems can be found in [20].
where Ci,j represents the jth element of Ci. Hi,j(+) and Hi,j() represent the jth element of Hi(+) and Hi() respectively. Applying (14), (18) and (13), the weighted ideal point evaluation function can be obtained:
8 3 h i X > C Hi;j ðþÞ 2 > > Y j ðþÞ ¼ wi i;jH ðþÞ > i;j < i¼1 3 h i > X > C Hi;j ðÞ 2 > > wi i;jH ðÞ : Y j ðÞ ¼ i¼1
ð19Þ
i;j
Then the distance from the evaluation function to the ideal point and anti-ideal point can be defined using the Euclidean distance:
Fig. 1. A connected graph derived from one distribution network with all tie breakers closed.
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J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038 Table 2 Power flow calculation result for IEEE 33-bus system.
Fig. 2. 33-bus system.
Table 1 Information data table. Object
C1
C2
C3
D1
D2
u1 u2 .. . ut .. . uk
C11 C21 .. . Ct1 .. . Ck1
C12 C22 .. . Ct2 .. . Ck2
C13 C23 .. . Ct3 .. . Ck3
D11 D21 .. . Dt1 .. . Dk1
D12 D22 .. . Dt2 .. . Dk2
33-bus radial distribution network The electrical wiring of the 33-bus test system is shown in Fig. 2, which includes 33 nodes, 32 lines and 5 contact switches. The base load is 4369.35 kVA (see Table 1). (1) Power flow calculation Suppose the fault occurs on both line 9–10 and line 14–15, the fault location system [21] will identify the fault and relevant switches will be tripped. Thus the load in the downstream of the fault area will lose power supply. The lost load set is defined as:
K 1 ¼ f10; 11; 12; 13; 14g K 2 ¼ f15; 16; 17; 18g
ð22Þ
When the lost load is known, connection switches BRK12–22, BRK9–14 and BRK18–33 will close automatically to restore power supply to the lost load. According to the principle of restoring as much load as possible, if the line current constraints are met, by closing switch BRK12–22 all the lost load in K1 can be restored to power supply quickly. However, if the line current constraints are not met, some load on this power supply route need to be cut according to the importance degree of the load. As for the lost load in K2, by closing switches BRK9–14 and BRK18–33, they can all be restored to power supply. However, the following loop will result: Lloop ¼ f6—7; 7—8; 8—9; 9—15; 15—16; 16—17; 17—18; 18—33; 33 —32;32—31;31—30; 30—29; 29—28; 28—27; 27—26; 26—6g
ð23Þ
In order to maintain the radiation-shaped structure of the distribution system, the virtual switches in the operable switch set need to be virtually operated and flow calculation needs to be carried out on the network after the virtual breaking. And then, it needs to be decided if each virtual breaking meets the node voltage constraint D1 and line current constraint D2, at the same time the values of the power loss function C1, the voltage quality function C2 and the load balancing function C3 are calculated, shown in Table 2.
Scheme
Breaking switches
C1 (kW)
C2 (kV)
C3
D1
D2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6–7 7–8 8–9 9–15 15–16 16–17 17–18 18–33 33–32 32–31 31–30 30–29 29–28 28–27 27–26 26–6
213 198 186 186 193 186 202 193 199 186 186 198 205 213 213 231
6.2 5.3 4.9 4.7 4.7 4.4 4.2 4.3 4.4 4.4 4.4 5.3 5.8 6.2 6.2 6.9
0.21 0.19 0.18 0.18 0.18 0.17 0.17 0.16 0.16 0.17 0.17 0.19 0.20 0.22 0.22 0.23
0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0
0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0
Note: in the constraints of node voltage and line current, ‘0’ means the constraints are not met and ‘1’ means the constraints are met.
(2) Weight coefficient In the decision table, the decision property set and the condition property set are D = {D1 the node voltage constraint, D2 the line current constraint} and C = {C1 the power loss, C2 the voltage quality, C3 the load balancing} respectively. (1) The dependence degree of the decision property set D on the condition property set C It can be seen from Table 2 that, the division of theory domain U by D is U/D = {Y1, Y2, Y3, Y4}, where Y1 = {1, 15, 16}, Y2 = {2, 3, 12, 13, 14}, Y3 = {4}, Y4 = {5,6,7,8,9,10,11}. The division of theory domain U by C is U/C = {X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12}, where X1 = {1}, X2 = {2, 12}, X3 = {3}, X4 = {4}, X5 = {5}, X6 = {6, 10, 11}, X7 = {7}, X8 = {8}, X9 = {9}, X10 = {13}, X11 = {14, 15}, X12 = {16}. The sub set Y1 represents the set of breaking schemes that meet neither the node voltage constraint nor the line current constraint. The C positive domain of D is:
posC ðDÞ ¼ fX 1 ; X 2 ; X 3 ; X 4 ; X 5 ; X 6 ; X 7 ; X 8 ; X 9 ; X 10 ; X 12 g
ð24Þ
¼ f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 16g According to (9), in the decision table the dependence degree of D on C is:
cC ðDÞ ¼
cardðposC ðDÞÞ 14 ¼ ¼ 0:8750 cardðUÞ 16
ð25Þ
(2) The dependence degree of the decision property set D on the condition property set C–Ci Take the power loss as an example for analysis. The division of theory domain U by C–C1 is U/C–C1 = {X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11}, where X1 = {1}, X2 = {2, 12}, X3 = {3}, X4 = {4, 5}, X5 = {6, 10, 11}, X6 = {7}, X7 = {8}, X8 = {9}, X9 = {13}, X10 = {14, 15}, X11 = {16}.
posCC 1 ðDÞ ¼ fX 1 ; X 2 ; X 3 ; X 5 ; X 6 ; X 7 ; X 8 ; X 9 ; X 11 g ¼ f1; 2; 3; 6; 7; 8; 9; 10; 11; 12; 13; 16g
ð26Þ
According to (10), the dependence degree of D on the condition property set without the power loss C–C1 is:
cCC1 ðDÞ ¼
cardðposCC 1 ðDÞÞ 12 ¼ ¼ 0:7500 cardðUÞ 16
ð27Þ
Similarly, it can be calculated that the dependence degrees corresponding to the voltage quality and the load balancing are 0.7500 and 0.8125 respectively.
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(3) The importance degree of property Ci According to (11), the importance degrees of the power loss, the voltage quality and the load balancing are:
8 > < uðC 1 Þ ¼ cC ðDÞ cCC 1 ðDÞ ¼ 0:1250 uðC 2 Þ ¼ cC ðDÞ cCC2 ðDÞ ¼ 0:1250 > : uðC Þ ¼ c ðDÞ c ðDÞ ¼ 0:0625 3
C
69-bus radial distribution network
ð28Þ
CC 3
(4) The weight coefficients of the sub-objective functions According to (14), the weight coefficients of the power loss, the voltage quality and the load balancing can be calculated, being 0.4000, 0.4000 and 0.2000 respectively. The dependence degree, importance degree and weight coefficient of the evaluation indexes are listed in Table 3.
The evaluation functions of power loss, voltage quality and load balancing are all loss-type indexes. The smaller their value is, the more reasonable the breaking scheme and the resulting network prove to be. The schemes that meet both the node voltage and line current constraints constitute of the feasible scheme set. According to (16), the ideal point and anti-ideal point vectors of the evaluation indexes can be calculated:
The electrical wiring of the 69-bus test system is shown in Fig. 3, which includes 69 nodes, 68 lines and 5 contact switches. The total active load is 3802.2 kW, and total reactive load is 2694.6 kVar. Suppose that fault occurs on line 13–14 and line 20–21 at the same time. The fault locating system [21] identifies the faults and relevant switches are tripped immediately. Therefore, power supply to the downstream load of the fault location is lost. Define the lost load set as:
(3) Close degree and scheme rating
Lloop ¼
this index is considered, there will be two optimal schemes. The multi-index optimization in the proposed method, however, is quick to distinguish that scheme 8 is better than scheme 9.
K 1 ¼ f14; 16; 17; 18; 20g
When the lost load is determined, the contact switches BRK15–46, BRK13–21 and BRK27–65 will close automatically to restore power supply to the lost load. For the lost load in K1, fast power restoration could be achieved by closing switch BRK15–46. For the lost load in K2, fast power restoration could be achieved by closing switch BRK13–21 and switch BRK27–65. However, the following loop is formed:
9—10; 10—11; 11—12; 12—13; 13—21; 21—22; 22—23; 23—24; 24—25; 25—26; 26—27; 27—65; 65—54; 64—63; 63—62; 62—61; 61—60; 60—59; 59—58; 58—58; 57—56; 56—55; 55—54; 54—53; 53—9
8 H1 ðþÞ ¼ ½186; 186; . . . ; 186116 > > > > > H1 ðÞ ¼ ½213; 231; . . . ; 231116 > > > < H2 ðþÞ ¼ ½4:4; 4:4; :::; 4:4116 > > H2 ðÞ ¼ ½6:9; 6:9; . . . ; 6:9116 > > > > > H3 ðþÞ ¼ ½0:16; 0:16; . . . ; 0:16116 > : H3 ðÞ ¼ ½0:23; 0:23; . . . ; 0:23116
ð29Þ
And then by (17) and (20), the weighted ideal point evaluation function of the feasible schemes can be calculated. Finally, the weighted ideal point close degree can be gained according to (21). The schemes in the feasible scheme set are rated according to the value of their close degree, shown in Table 4. Comparison between Tables 4 and 2 is shown in Table 5, where ‘0’ means the node voltage and line current constraints cannot be both met; ‘1’ represents the optimal scheme; ‘2’ represents the sub-optimal scheme; ‘3’ represents the 3rd grade scheme, and so on. It can be seen from Table 5 that, the optimal scheme obtained by the proposed method (corresponding to the biggest close degree of the optimal evaluation function) is scheme 8, i.e. breaking line 18–33. In this scheme, the load balancing index is the optimal, while the power loss index and the voltage quality index both reach the sub-optimal, i.e. rating ‘2’. However, the sub-optimal values of the two indexes are both very close to the optimal values. It can also be seen that, concerning the load balancing index, both scheme 8 and scheme 9 reach the optimal. Therefore, if only Table 3 Dependence degree, importance degree and weight coefficient of the indexes. Evaluation function
Power loss
Voltage quality
Load balancing
Dependence degree Importance degree Weight coefficient
0.7500 0.1250 0.4000
0.7500 0.1250 0.4000
0.8125 0.0625 0.2000
ð30Þ
K 2 ¼ f21; 22; 24; 26; 27g
ð31Þ
In order to keep the network structure radial, virtual breaking of the lines in Lloop are conducted, and flow calculation is carried out in the post-breaking network. And then, decide whether each virtual breaking meets the node voltage constraint D1 and line current constraint D2. At the same time, the power loss function C1, voltage quality function C2 and the load balance function C3 are calculated, shown in Table 6. It can be seen from Table 6 that, the division of the decision attribute set D on domain U is U/D = {Y1, Y2, Y3}, where Y1 = {1, 2}, Y2 = {3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 22, 23, 24, 25}, Y3 = {11, 12, 13, 14, 15, 16, 17}. Y3 represents the set of power restoration schemes that meet both the node voltage constraint and the line current constraint. The division of the condition attribute set C on domain U is U/C = {X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22}, where X1 = {1}, X2 = {2}, X3 = {3}, X4 = {4, 5}, X5 = {6}, X6 = {7, 8}, X7 = {9}, X8 = {10}, X9 = {11}, X10 = {12}, X11 = {13}, X12 = {14, 15}, X13 = {16}, X14 = {17}, X15 = {18}, X16 = {19}, X17 = {20}, X8 = {21}, X19 = {22}, X20 = {23}, X21 = {24}, X22 = {25}. The dependence degree of the decision attribute set D on the condition attribute set C is C ðDÞÞ cC ðDÞ ¼ cardðpos ¼ 1, which means the decision attribute set D is cardðUÞ
totally dependent on the condition attribute set C. According to the rough set theory, the dependence degree, importance degree and weight coefficient of the power loss, voltage quality and the load balance can be obtained, shown in Table 7. According to (14) and (18) to (13), the weighted ideal point evaluation function of each scheme is calculated. Finally, the weighted ideal point fitness degree could be calculated according to (21). And then, the grade of each scheme in the feasible scheme set is evaluated according to the fitness degree. Relevant calculation results and evaluation results are shown in Table 8. It can be seen from that, the optimal scheme obtained with the proposed
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Table 4 Evaluation function, close degree and scheme rating for 33-bus system. Scheme
Breaking line
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6–7 7–8 8–9 9–15 15–16 16–17 17–18 18–33 33–32 32–31 31–30 30–29 29–28 28–27 27–26 26–6
Evaluation function Li(+)
Li()
0.3444 0.1901 0.1193 0.0937 0.0967 0.0411 0.0612 0.0282 0.0535 0.0411 0.0411 0.1901 0.2734 0.3567 0.3567 0.4764
0.0898 0.1889 0.2413 0.2555 0.2469 0.2851 0.2849 0.2935 0.2806 0.2851 0.2851 0.1889 0.1365 0.0832 0.0832 0
Close degree Tj
Constraints met?
Scheme rating
0.2067 0.4985 0.6691 0.7315 0.7184 0.8740 0.8232 0.9124 0.8398 0.8740 0.8740 0.4985 0.3330 0.1891 0.1891 0
No No No No Yes Yes Yes Yes Yes Yes Yes No No No No No
0 0 0 0 5 2 4 1 3 2 2 0 0 0 0 0
Note: ‘0’ means the node voltage and line current constraints cannot be both met; ‘1’ represents the optimal scheme; ‘2’ represents the sub-optimal scheme; ‘3’ represents the 3rd grade scheme, and so on.
Table 5 Calculation results and rating evaluation for 33-bus system. Scheme
Breaking switches
C1 (kW)
C1 rating
C2 (kV)
C2 rating
C3
C3 rating
Close degree Tj
Tj rating
Constraints met?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6–7 7–8 8–9 9–15 15–16 16–17 17–18 18–33 33–32 32–31 31–30 30–29 29–28 28–27 27–26 26–6
213 198 186 186 193 186 202 193 199 186 186 198 205 213 213 231
0 0 0 0 2 1 5 2 4 1 1 3 0 0 0 0
6.2 5.3 4.9 4.7 4.7 4.4 4.2 4.3 4.4 4.4 4.4 5.3 5.8 6.2 6.2 6.9
0 0 0 0 4 3 1 2 3 3 3 6 0 0 0 0
0.21 0.19 0.18 0.18 0.18 0.17 0.17 0.16 0.16 0.17 0.17 0.19 0.20 0.22 0.22 0.23
0 0 0 0 3 2 2 1 1 2 2 0 0 0 0 0
0.2067 0.4985 0.6691 0.7315 0.7184 0.8740 0.8232 0.9124 0.8398 0.8740 0.8740 0.4985 0.3330 0.1891 0.1891 0
0 0 0 0 5 2 4 1 3 2 2 0 0 0 0 0
No No No No Yes Yes Yes Yes Yes Yes Yes No No No No No
Fig. 3. 69-bus test system.
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J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038 Table 6 Power flow calculation result for 69-bus system. Scheme
Breaking line
C1 (kW)
C2 (kV)
C3
D1
D2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
9–10 10–11 11–12 12–13 13–21 21–22 22–23 23–24 24–25 25–26 26–27 27–65 65–64 64–63 63–62 62–61 61–60 60–59 59–58 58–57 57–56 56–55 55–54 54–53 53–9
216 212 184 158 158 146 146 146 143 143 146 143 136 125 125 125 124 213 230 230 230 230 234 239 240
8.2 8.0 7.0 6.1 6.1 5.8 5.6 5.6 5.4 5.3 5.4 5.4 5.0 5.0 5.0 4.9 4.9 6.5 6.8 7.1 7.3 7.5 7.8 8.1 8.3
0.24 0.24 0.23 0.22 0.22 0.21 0.21 0.21 0.21 0.21 0.21 0.20 0.21 0.21 0.21 0.21 0.22 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Note: for the node voltage and line current constraints, ‘0’ represents the constraints are not satisfied and ‘1’ represents the constraints are satisfied.
method is scheme 16, i.e. breaking line 61–62. In scheme 16, the indexes of voltage quality and load balance reach the optimal solution, and the power loss index reaches the sub-optimal. Comparison with power restoration strategy based on genetic algorithm Take the 69-bus system for example, comparison is made between the proposed method and power restoration strategy based on genetic algorithm. In the fault restoration scheme based on genetic algorithm, each switch is viewed as a gene on the chromosome. The binary encoding method is used, i.e. when the switch is ‘off’, the corresponding gene is 0, and when the switch is ‘on’, the corresponding gene is 1. The initial population is 50. First, it is determined whether all the power restoration schemes are radial in network structure, and those with non-radial network structure are deleted. And then, the fitness degrees of the fault restoration schemes with radial network structure are calculated. During the genetic operation process, the roulette wheel selection method is used, and the crossover algorithm applies single-point crossover. The crossover probability is 0.7, and the mutation probability is 0.5. After the genetic operation, the fault restoration effect is shown in Table 9. It can be seen that, when applied for power restoration, the proposed method is superior to the genetic algorithm in lowest node voltage and flow calculation times. Analysis of computation time
Table 7 Dependence degree, importance degree and weight coefficient of the indexes for 69bus system. Evaluation function
Power loss
Voltage quality
Load balancing
Dependence degree Importance degree Weight coefficient
0.9200 0.0800 0.2857
0.8800 0.1200 0.4286
0.9200 0.0800 0.2857
In terms of computation time, the computation time of the proposed power restoration scheme is from when the loop is formed to when the optimal scheme is obtained. During this process, the computation time is mainly the flow calculation time, which equals to the time needed for each flow calculation multiplied by the flow calculation times. The time needed for each flow calculation is dependent on the power network itself, and the flow calculation times are dependent on the number of elements in the
Table 8 Evaluation function, close degree and scheme rating for 69-bus system. Scheme
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Breaking line
9–10 10–11 11–12 12–13 13–21 21–22 22–23 23–24 24–25 25–26 26–27 27–65 65–64 64–63 63–62 62–61 61–60 60–59 59–58 58–57 57–56 56–55 55–54 54–53 53–9
Evaluation function Li( + )
Li()
0.6025 0.5717 0.3899 0.2237 0.2237 0.1555 0.1358 0.1358 0.1090 0.1014 0.1190 0.1057 0.0597 0.0302 0.0302 0.0271 0.0535 0.4675 0.5467 0.5665 0.5808 0.5959 0.6330 0.6740 0.6943
0.0679 0.0784 0.1728 0.2650 0.2650 0.3054 0.3158 0.3158 0.3310 0.3365 0.3267 0.3379 0.3633 0.3794 0.3794 0.3848 0.3813 0.1542 0.1204 0.0972 0.0820 0.0669 0.0416 0.0159 0.0000
Close degree Tj
Constraints met?
Scheme rating
0.1013 0.1205 0.3071 0.5423 0.5423 0.6627 0.6992 0.6992 0.7522 0.7685 0.7329 0.7618 0.8588 0.9263 0.9263 0.9343 0.8771 0.2480 0.1805 0.1465 0.1237 0.1009 0.0617 0.0231 0.0000
No No No No No No No No No No Yes Yes Yes Yes Yes Yes Yes No No No No No No No No
0 0 0 0 0 0 0 0 0 0 6 5 4 2 2 1 3 0 0 0 0 0 0 0 0
Note: ‘0’ represents the node voltage constraints and branch current constraints cannot be satisfied at the same time; ‘1’ represents the optimal scheme, ‘2’ represents the suboptimal scheme, and ‘3’ represents the 3rd-grade scheme, and so on.
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J. Ma et al. / Electrical Power and Energy Systems 63 (2014) 1030–1038
Table 9 Comparison of fault service restoration results. Methods
Breaking off switch set
Power loss (kW)
Lowest node voltage (kV)
Flow calculation times
AG Proposed method
11–43, 13–14, 20–21, 58–59, 27–65 11–43, 13–14, 20–21, 50–59, 61–62
118 125
11.95 12.04
783 25
operable switch set. The proposed method is realized in Matlab2012a on a computer with 2.00 GHz frequency, 2 GB memory, and Inter(R) Core(TM) 2 Duo CPU processor. Simulation results show that, the average time needed for each flow calculation of 33-bus and 69-bus system are 0.04 s and 0.056 s respectively. In 33-bus system, when fault occurs on line 8–9 and line 13–14 of at the same time, the power restoration time using the proposed method is 1.08 s. In 69-bus system, when fault occurs on line 13–14 and line 20–21 of 69-bus system at the same time, the power restoration time using the proposed method is 1.93 s. Besides, compared with the genetic algorithm, the flow calculation times of the proposed method are obviously reduced. Therefore, the proposed method could contribute to fast power restoration in the distribution network.
Conclusion A power restoration strategy for the distribution network based on the weighted ideal point method is proposed in this paper. First, with the power loss, voltage quality and load balancing as the reconfiguration goal, the comprehensive evaluation function is established. And then, the rough set theory is used to determine the weight coefficients of the sub-objective functions. Finally, the optimal solution of the comprehensive evaluation function is obtained by applying the ideal point method. Simulation results verify that the proposed method has the following advantages: (1) The weight coefficients of the objective functions obtained by the rough set theory are flexibly adaptive to the network structure. Thus, the uncertainty and immutability problem in traditional weight setting due to dependence on subjective experience can be avoided. (2) By applying multi-objective optimization to the virtual reconfiguration network, the breaking scheme is required to reach the optimal in multiple indexes. Therefore, the non-selectivity problem in single objective optimization due to more than one scheme reaching the optimal can be avoided.
Acknowledgements This work was supported by the National Basic Research Program of China (973 program) (2012CB215200), the National Natural Science Foundation of China (50907021, 51277193, 50837002), the Chinese University Scientific Fund Project (2014ZZD02), the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry ([2011] No. 1139), Hebei Natural Science Foundation (E2012502034), Electric Power Youth Science and Technology Creativity Foundation of CSEE ([2012] No. 46), the New-Star of Science and Technology supported by Beijing Metropolis Beijing Nova program (Z141101001814012),
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