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An ordinal optimization based method for power distribution system control
Available online at www.sciencedirect.com
Electric Power Systems Research 78 (2008) 694–702
An ordinal optimization based method for power distribution system control Erkan Atmaca ∗ ˙ Istanbul University, Faculty of Engineering, Electrical Engineering Department, 34850 Avcılar, Istanbul, Turkey Received 18 August 2006; received in revised form 19 January 2007; accepted 17 May 2007 Available online 28 June 2007
Abstract This work presents an Ordinal Optimization Theory based method for power distribution system control. The solution procedure bases on an evolutionary search scheme. The method aims at minimizing the total power losses on network feeders while keeping the node voltages at near rated values against changing loading conditions. By relaxing the definition of optimality and softening the goal of optimization, ordinal optimization makes the problem easier as well as the solution process faster. A solution can be defined as good enough in performance value if it is one of the best m-percent solutions on the search space with probability P%. A solution can be defined as good enough also if its performance value approximates to the optimal value to a pre-defined extent. This work estimates the performance value, the total power loss, of the optimum state of power distribution network at a given loading condition. Then, with the two relaxed definitions for optimality, it finds the optimal settings of on-load tap changers and capacitors in a 24-h time frame. Results verify our method is proper for on-line and off-line operation. © 2007 Elsevier B.V. All rights reserved. Keywords: Power distribution control; Ordinal optimization; Performance value; Good enough criterion
1. Introduction Due to recent rises in energy prices and already common use of sensitive electronic loads, power distribution system control is an important issue concerned with power loss minimization and electric power quality. To deal with this issue utilities make use of new technologies providing for the automatic control structure necessary to manage the network from a remote center. The primary aim of power distribution system control is at minimizing power and energy losses while keeping the node voltages at near rated values against changing loading conditions. This control is realized by adjusting the on-load tap changer (OLTC) settings, which are located at substations, and by switching the capacitors, which are located at substations and on network feeders. Reconfiguration of the network by switching the tie-switches at branches is another tool for distribution control. Automatic control of power distribution system bases on an optimally configured network in which the capacitors are optimally placed and sized at the design stage. During opera-
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tion, states of the controlling elements are changed continuously by switching operations and total switching operations number of each element for a specific time interval should be limited because of their life expectancy constraints. Therefore, power distribution system control is a constrained, multi-objective optimization problem with integer variables and solution to this problem requires rather new techniques than the conventional approaches, particularly for large systems. Lu and Hsu [1] propose a dynamic programming method for properly dispatching the shunt capacitors and on-load tap changers at the distribution substation based on the forecasted hourly loads. Liang and Cheng [2] improved this method so as to include the capacitors installed on network feeders. But dynamic programming method is inadequate in large systems due to the computational burden involved with it. Ramakrishna and Rao [3] introduce an adaptive neuro-fuzzy inference system for volt/var control, which involves complex calculations for decision process and considerable training times for artificial neural networks. Due to the drawbacks of preceding studies, recent studies used heuristic techniques like genetic algorithms (GAs). Hu et al. [4] develops a GA based method for minimizing the total energy loss through the 24-h time frame. They reduce the search space
E. Atmaca / Electric Power Systems Research 78 (2008) 694–702
by dividing the 24-h load pattern into a number of load levels and OLTC settings change only when the load level changes. But since this method encodes the states of variables through the entire 24-h time frame, it leads to a considerable computational burden despite the reduced search space. Augugliaro et al. [5] combines the network reconfiguration problem with dispatching the capacitors and tap changers. Further, this study unifies the power loss minimization and voltage profile flattening objectives into a single one by a fuzzy reasoning approach. This unification, however, introduces an additional computational burden. Mori and Tani [6] proposes a Parallel Tabu search method for online capacitor control, which uses ordinal optimization (OO) concepts, however, disregards the voltage regulation problem. Although these studies work quite well, their efficiencies can be improved by using the OO concepts. Ordinal Optimization Theory is relatively a new topic, and found limited use in power system studies. By relaxing the definition of optimality and softening the goal of optimization, ordinal optimization makes the problem easier as well as the solution process faster. This study presents an OO based method for volt/var control in power distribution systems. It aims at minimizing total power losses at changing loading conditions. It uses two definitions for optimality: It identifies a solution as optimum according to either its ‘performance order’ in performances of a feasible set of solutions, or its ‘performance value’ compared with performance value of the ‘truly optimum’. Although the truly optimum cannot be identified in advance, we can estimate its performance value by setting the reactive components of the loads to a small amount. Following the forecasted daily load pattern, the method searches for the OLTC settings and capacitor on/off statuses at each hour that minimize the total power loss at that time interval. The search technique is evolutionary search (ES) based. The method is proper for on-line control as well. Results obtained from the tests on a sample system are promising as far as total computation time and quality of solutions are concerned. 2. Method 2.1. Formulation of the problem The considered power distribution system is assumed to have the following properties: (a) There is an OLTC at each HV/MV distribution substation, whose tap ranges from −8 to +8 with voltage range of variation ± 10%. (b) There are shunt capacitor banks installed at substations and on network feeders. (c) Loading condition at each node changes according to the daily load pattern shown in Fig. 1 The objective is the minimizing of the total power loss on network elements at any loading condition in a 24-h time frame Ploss =
NB i=1
Ri Ii2
(1)
695
Fig. 1. Daily load pattern of the power distribution network.
satisfying the voltage constraints Vmin ≤ Vi ≤ Vmax ,
i = 1, 2, . . . , NL
(2)
where NB is the number of branches and NL is the number of MV/LV transformers. To satisfy the requirements of many utilities, we assumed Vmin is 0.95 p.u. and Vmax is 1.05 p.u. The objective function is subject to power balancing constraint in addition to the following constraints: 24 i=1 24
|tapmi − tapmi−1 | ≤ Tmax
(3)
|Cni ⊕ Cni−1 | ≤ Cn max
(4)
i=1
where tapmi is the tap position of the mth OLTC at hour i; Cni the on/off status of the capacitor Cn at hour i, Cni = 0, 1; Tmax the maximum number of the switching operations allowed for each OLTC; Cn max is the maximum number of the switching operations allowed for the capacitor Cni . Maximum number of the switching operations of an element depends on its life expectancy. These numbers are utility dependent, however, considering the common practice, we can assume total switching operations number of an OLTC should be below 30 a day, and of capacitors installed at substations and on feeders below 10 and 4, respectively. Control variables are the OLTC settings and the on/off statuses of shunt capacitors. In an evolutionary search scheme these variables can be coded as real numbers. Such a coding formulates the volt/var control as an optimization problem with discrete search space. Size of the search space is determined by the number of control variables and the number of states associated with them. Theoretically, search space of the volt/var control problem is very large for a practical power distribution system. 2.2. Ordinal optimization (OO) Ordinal optimization was first introduced by Lau and Ho in 1997 [7]. OO is not a method on its own but rather a supplement to existing optimization methods. OO is based on two ideas: (i) it is easier to determine order than value. That is to say, determin-
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ing whether A is greater or less than B is easier than determining the values of A and B exactly. (ii) Softening the goal of optimization makes the problem easier as well. Instead of asking the ‘best for sure’, the ‘good enough with high probability’ can be proposed. For example, considering a search on the search space Θ, we can define the ‘good enough’ subset, G/Θ, as the top-1% of the space based on objective function values. Consider a ‘selected’ subset, S/Θ, determined by a sampling procedure. This sampling can be random, or be done by rough calculations and measurements. By requiring high probability for matching of these two subsets, we can narrow down the search space from Θ to S. Relaxation of the optimization objective can reduce computation time significantly for a large-scale problem. Let us define the good enough subset G as the top m-percent of the whole set of the solutions, based on objective function values. Let NS be the total number of solutions in the selected subset S. Then, the probability for at least one of the selected solutions being a member of G is P. Then, P = 1 − (1 − m)NS
(5)
Let us define P as the confidence level of the sampling procedure, and the percentage m as the solution quality. Therefore, from Eq. (5), we can determine at least how many candidates should be sampled so that they include at least one solution from the best m-percent set with confidence level P. Having decided for P and m by engineering judgment, the least number of solutions to be sampled is NS =
ln(1 − P) ln(1 − m)
(6)
In this way, size of the good enough set is determined beforehand by the operator and it is independent of the problem. But this approach does not tell us how much difference in performance value can be between the solution we found and the truly optimum. Therefore, a slightly different ‘good enough’ definition can be made, which measures the performance value of a solution with respect to the performance value of the truly optimum [8]. Consider an optimization problem minJ(θ) θ∈Θ
(7)
where Θ is the search space, J(θ) is the performance function. Θ = {θ 1 , . . ., θ N } is a finite discrete set, where N is the size of the search space and J(θ i ) = J(θ j ) if i = j. We can define the good enough set as G ≡ {solution that satisfies a good enough criterion}. Then, it is possible to make two different definitions for a good enough subset: G1 ≡ the top-1% solution
(8.a)
G2 ≡ {θi |J(θi ) ≤ 1.01 × J(θopt )}
(8.b)
where θ opt represents an optimal solution, which minimizes the objective function on the search space. But whether all the solutions in G1 or G2 are desirable is problem dependent. Consider we know performance values of all the solutions in Θ. By ordering them according to their performance values we
Fig. 2. Two categories of the ordered performance curves (OPCs) of an optimization problem.
can obtain the ordered performance curve (OPC) of the problem. The shape and characteristics of this curve can fall in one of the categories shown in Fig. 2. A flat OPC represents a problem with lots of good solutions, although a steep OPC represents a problem with few good solutions. Therefore, although the good enough criterion G1 can produce desirable results for a problem with flat OPC, it may not be satisfactory for a problem with steep OPC, in which the good enough criterion G2 is more reliable. For many optimization problems, performance value of the optimal solution is not known beforehand, unless all the solutions are calculated. But use of the good enough criterion G2 requires it to be known. In a properly compensated power distribution system the reactive power flows through the lines are negligible compared with the active power flows [9]. A proper compensation level can be simulated by setting the reactive components of the loads to a small amount. To be realistic, we set the reactive loads at 5% of their original values. We regulate the system by changing only OLTC settings. We define the performance value, the total power loss, calculated at these conditions as the reference optimal value for measuring quality of the solutions. This value is a reference and underestimates the value of the real optimum. The real optimum cannot be identified unless the computation was completed. This is why the good enough criterion G2 deals with the reference optimal value. On the other hand, we use the good enough criterion G1 as the lower bound for the quality of the solutions. During the search process at each hour, tap perturbations are performed in the neighborhood of the OLTC settings that found when calculating the reference optimal value of that hour. This procedure is prone to satisfy the bus voltage constraints. Nevertheless, the solutions that provoke intolerable voltages are not rejected but assigned to very large power loss values so that they are subordinated during the search process. 2.3. Solution procedure The solution procedure is based on an evolutionary search scheme, where only the mutation operator is considered. New
E. Atmaca / Electric Power Systems Research 78 (2008) 694–702
solutions are created by perturbing the existing solutions but not by recombination. A similar scheme is used successfully by Augugliaro et al. [5]. It is reported Tabu Search (TS) outperforms Genetic Algorithm [11]. However, TS needs a serious effort to model the solution strategy. Besides, creating a neighborhood around a solution can be time consuming for a large-scale problem. On the other hand, classical GA needs to tune up the cross-over and mutation operators. A convergence criterion is also needed by TS and GA. However, generating new solutions in evolutionary strategies is simple, and selection of parents is more flexible compared with classical GA. ES also allows the real-coding of variables, hence simplifies the representation of individuals. In our algorithm, the search at each hour starts with slight modifications on the optimum of previous hour. The load profile changing smoothly allows using such a scheme. However, for a daily load profile changing sharply, the search process demands more diversification than our algorithm produces, in which some kind of cross-over can be used to improve the solution quality. In this application, we used (λ + μ) selection scheme, where the symbol μ stands for the number of parents in one generation, and λ for the number of created offsprings. At each generation parents and offspring’s are joined and the best μ individuals are selected from this combined set. To exploit the smoothly changing pattern of the loading conditions, we have chosen the first generation solutions for each loading condition as the same as the last generation solutions of the previous one. This approach leads the optimization process to converge fast. Each solution is coded into an integer string; Z, containing the information of taps positions of OLTCs, and shunt capacitor statuses: Z = Ztap , Zcap
where k=
|ZC | , |ZC | − |ZT |
In our scheme, determination of the starting tap positions and estimation of the reference optimal values are related to each other. If the reactive power flows through feeders are negligible compared with active power flows, the total power loss on feeders and the node voltages are function of only the OLTC positions. To create a pseudo-state of optimal compensation, we set the reactive loads at 5% of their original values. Hence, for each loading factor (LF) in the 24-h time frame, we determine the starting tap position of the mth OLTC, tm start,h , as the following: tm start,h ∴ min f (tm,i,h ); f (tm,i,h ) =
t = 1 + 0.0125 × TAP
i = −8, −7, . . . , 7, 8
1 |1.00 − Vi,j,h |; NL
j = 1, 2, . . . NL
(11.a)
Qj = 0.05 × Qj ; (11.b)
where Vm,i,h is the tap position of the mth OLTC at hour h; Vi,j,h the voltage of the jth node at the tap position i at hour h; NL the number of the load nodes; Qj the original value of the reactive component of the load at the jth node; Qj is the reactive component of the load at the jth node. To avoid the unnecessary calculations we searched for tm start,h in the neighborhood (±1) of the tap settings found by the formula in Eq. (10). Namely, the starting tap positions at hour h are those that make the voltage profile of the network most flat. We define the reference optimal value, Ploss ref,h or performance value of the truly optimum, as the total power loss resulted from the conditions described in Eq. (11). The reference optimal value will be calculated at each hour. This value will serve for defining the good enough criterion G2 in Eq. (8).
(9)
Theoretically, the size of Ztap is the number of OLTCs and the size of Zcap is the number of capacitors. Size of the search space depends on the number of control variables and the number of states associated with each variable. Limiting the range of variations of the variables reduces the computational burden. To restrict the range of tap variations of an OLTC, some studies perform the perturbations around a forecasted starting tap position that is obtained by the analytical formula in Eq. (10) [1,5]. The formula is based on the active and reactive power demands from transformer and the desired voltage at secondary bus. ⎧ ⎡ ⎤⎫1/2 2 ⎨ |Z |2 ⎬ 2 V1 T ⎣ 2⎦ t= Q (10.a) + + P L L ⎩ V02 V12 ⎭ k|ZT |
697
Ploss ref,h =
NB i=1
Ri Ii2 ;
∴ tm,i,h =tm start,h ;
i = 1, 2, . . . NB Qj =0.05 × Qj ;
j=1, 2, . . . NL (12)
The solution procedure is summarized in the flow chart in Fig. 3. Loading factor (LF) is a number between 0.4 (the lightest loading condition) and 1.01 (the heaviest loading condition). They are obtained from the load forecasting data of the given day. Updating LF means continuing to the next hour calculations. Proper sizes for the numbers of population, Npop , and generations, Ngen , depend on search space size in an evolutionary search scheme. They are generally determined by engineering judgment. These two parameters result in size of the selected subset: NS = |S| = Npop × Ngen
(13)
(10.b)
In these equations, TAP is the position of an OLTC, ZC is the impedance of the switched capacitor banks at the substation, V0 and V1 the transformers primary and secondary voltages, ZT the transformer impedance, PL and QL are active and reactive loads of the transformer, and t is the transformer voltage ratio.
Considering Eqs. (6) and (13) we choose the numbers of generations and population according to desired confidence level and acceptable solution quality. Hence, the good enough criterion G1 results in the proper sizes for Npop and Ngen , and the good enough criterion G2 serves as the termination criterion of the search process as well as the quality measure for solutions.
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commercial size of 300 kvars is assumed for each capacitor unit. Because we perturb the tap settings (±1) around starting positions, for any loading condition, there are three possible states for each OLTC. The state of each group of capacitors can be represented by a number between 0 and 3; there are four states for each group. For example, number ‘2’ means two of the capacitor units are on and number ‘0’ none of them. Hence, we sample the solutions from (34 × 428 ) = 5.8 × 1018 possible states for each hour. The evolutionary search parameters used in this sampling is shown in Table 1. Hence, for a given loading condition, the maximum number of traversed solutions, NS , is 25 × 10 = 250. Based on this selected-subset size, Table 2 shows how the solution quality changes with confidence level according to Eq. (6). Hence, the set of 250 random samples includes at least one solution from the top-1% subset with probability 92%. However, since the search is guided to more promising areas through evolutions we can expect confidence levels higher than those in Table 2. The good enough criterion G2 can terminate the search process of the current hour without traversing all the 250 solutions if the following condition is satisfied: Ploss,h ≤ 1.01 × Ploss ref,h
(14)
where Ploss,h is performance value of the best solution in current population, and Ploss ref,h is the reference optimal value for hour h. The average voltage variation at hour h, Vh was defined as the following: Vh = Fig. 3. Flow chart of the solution procedure.
1 |1.00 − Vj, h |; NL
j = 1, 2, . . . , NL
(15)
where NL is the total number of load nodes and Vj,h is the p.u. value of the jth node voltage at hour h.
2.4. Test system
3. Results
We applied our method to a commonly used 115/23 kV/kV test system [3,4,10]. The network and load data were taken from Ref. [10]. To complicate the problem, we repeated the test system four times as shown in Fig. 4; hence created a 121-node system with 120 lines. Line parameters are same but the loads of the nodes 1–30, 31–60, 61–90 and 91–120 were set to 100, 90, 75 and 60%, respectively, of their original values. Further, we installed additional units at each capacitor bus to enlarge the search space. Since distribution system loads are sensitive to voltage, using only P–Q models for loads can lead to inaccuracies. Accordingly, we modeled half the load at each bus as constant current load and the other half as constant impedance load. We assumed loads change according to the daily load profile shown in Fig. 1. Transformers with OLTCs are installed between the node 0 and nodes 1, 31, 61 and 91. We assumed primary voltage of each transformer is constant and 1.00 p.u. Totally 28 groups of capacitors are installed at 24 nodes. A group consists of three units. A
The program was written in MATLAB and run on a PC with 1.5 GHz microprocessor. Many runs result in 85 s computation time in average for a 24-h time frame. This computation time is proper for on-line control operation. To test the efficiency of the good enough criterions G1 and 2 G , and the method overall, we extended the search space and increased the number of samplings for comparison purposes. We set the width of mutation tap to (±2); Npop to 20 and Ngen to 50 hence traversed 1000 solutions for a single loading factor without a termination criterion. Table 3 compares the limited and extended searches referring to the reference values. In Table 3, Ploss (%) denotes to total power loss error, where Ploss (%) =
Ploss − Ploss ref × 100 Ploss ref
Table 3 includes total power losses of the uncontrolled system as well.
E. Atmaca / Electric Power Systems Research 78 (2008) 694–702
699
Fig. 4. One-line diagram of the test distribution system. Table 1 Evolutionary search parameters used in this work Number of generations
Parent population, μ
Offspring population, λ
Width of mutation tap
Width of mutation capacitor
Mutation probability
25
10
10
±1
±300 kvars
1.0
Resulting objective function values of the limited search, with good enough criterions G1 and G2 , are comparable to those of the extended search. On the other hand, the extended search improves total power loss results slightly at the expense of minor Table 2 Solution qualities versus confidence level with 250 and 1000 samples Solution quality (m, %) vs. confidence level with 250 and 1000 samples
250 1000
Confidence level P (%)
63
78
87
92
97
99
0.40 0.10
0.60 0.15
0.80 0.21
1.00 0.25
1.40 0.35
1.80 0.46
increases in voltage variations. This is expected due to large variations in tap settings, since the extended search perturbs the OLTC settings (±2) around starting positions. As far as the definition in Eq. (15) is concerned for voltage variation, the limited search results in more smooth voltage profiles than the extended search does at the expense of minor increases in power losses. Although the good enough criterion G2 is not fully satisfied by the limited search at some hours (1, 3, 4, 5, 21, 23, 24), errors are inconsiderable. Many runs have converged in on an average of 10 generations per hour in the 24-h frame. On the other hand, solution process continued up to 25 generations at six loading factors on average. However, at some loading factors such as the first
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Table 3 Comparison of actual total power losses with reference values Hour
Unregulated
Reference values
Extended search
Ploss (kW)
Ploss ref (kW)
V (%)
Ngen
Ploss (kW)
Ploss (%)
V (%)
Ngen
Ploss (kW)
Ploss (%)
V (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
221.5 256.1 346.4 466.5 641.9 760.8 935.2 1128.1 1423.6 1339.5 1103.0 1005.4 912.4 1078.1 1258.0 1153.5 1029.3 720.0 568.4 499.3 389.3 306.0 268.1 210.6
185.0 213.8 289.1 389.0 534.7 633.3 777.6 936.8 1180.0 1111.0 916.1 835.6 758.8 895.6 1043.9 957.8 855.4 599.5 473.7 416.3 324.8 255.4 223.8 175.9
0.74 0.78 0.91 1.07 1.22 1.33 1.52 1.64 1.82 1.77 1.62 1.56 1.46 1.61 1.72 1.65 1.58 1.29 1.16 1.10 0.97 0.86 0.80 0.72
50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
187.0 215.9 291.7 392.6 539.9 638.4 783.6 943.8 1188.3 1118.9 923.1 841.7 764.3 902.2 1051.5 964.7 861.9 605.5 478.0 420.0 327.7 257.8 226.0 178.0
1.07 0.98 0.90 0.93 0.96 0.81 0.76 0.75 0.71 0.71 0.76 0.73 0.73 0.73 0.73 0.73 0.76 0.99 0.92 0.89 0.90 0.93 0.96 1.23
1.24 1.04 1.53 1.69 1.76 1.35 1.68 1.77 2.02 1.86 1.72 1.94 1.76 2.08 1.84 1.74 1.73 1.67 1.71 1.37 1.22 1.56 1.05 1.80
25 1 25 25 25 12 5 1 3 1 1 3 6 1 3 1 1 13 7 5 25 6 25 25
187.0 215.9 292.1 393.2 540.4 639.5 785.3 945.3 1190.7 1121.3 925.2 843.4 766.3 904.0 1054.0 966.4 863.1 605.5 478.3 420.1 328.2 258.0 226.2 178.0
1.07 0.98 1.07 1.07 1.06 0.98 0.98 0.91 0.91 0.93 0.99 0.93 0.99 0.93 0.97 0.90 0.90 0.99 0.98 0.93 1.05 1.00 1.06 1.23
0.83 0.78 0.94 1.22 1.35 1.51 1.70 1.69 1.87 1.80 1.72 1.70 1.46 1.66 1.80 1.69 1.65 1.30 1.18 1.28 1.08 1.01 0.85 1.09
Mean
750.9
624.3
1.29
50
629.3
0.80
1.63
10
630.3
0.96
1.38
and last ones, the extended search was lack of satisfying the second good enough criterion, either. This occurs especially at low loading factors, where underestimation of the truly optimal leads to larger percentage errors. The cross-over operator, inserted at these hours for further improvement, resulted in inconsiderable rise in solution quality despite the increase in total computation time due to the extent of solution diversification it introduced. However, the good enough criterion G2 still serves as a quality measure for solutions. Table 4 shows the tap positions and capacitor status variations at substations and on feeders through 24 h. Total switching operations numbers a day satisfy the constraints. These constraints are on-line, not final checks. Total switching operations number of each unit has a certain daily limit beyond which further switching is prohibited. However, since our search at each hour proceeds with slight modifications on existing solutions of the previous hour we never observed such a violation. Consider that for the system at hand total number of powerflow runs in the 24-h time frame is proportional to the total number of generations. The good enough criterion G2 terminated the search process for each hour averagely in 10 generations. This leads to a considerable saving in computation time compared with the extended search, which is performed without a termination criterion through 50 generations for each hour. Voltage profile of the node 15, which is the lowest voltage node of the network, is shown through 24 h in Fig. 5.
Limited search
The extended search regulates the voltage of the node 15 better than the limited search does, since the former resulted in the average voltage as 0.98 p.u. through 24 h while the latter 0.97 p.u. Despite such a difference for a specific node, as far as the overall voltage regulation capability is concerned, the limited search slightly outperformed the extended search.
Fig. 5. Voltage profile of the node 15 through 24 h. (*) Results of the extended search.
Table 4 Tap positions and capacitor switching operations through 24 h Hour
Capacitor switching operations
T1
T2
T3
T4
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
C26
C27
C28
2 2 3 2 2 3 4 4 5 4 4 3 4 4 4 4 4 3 2 3 3 2 2 2
1 2 2 3 3 3 4 4 4 4 4 3 3 4 4 4 3 4 3 2 2 2 1 1
2 1 2 1 2 2 2 3 3 3 3 2 3 3 3 3 3 3 1 2 1 2 0 1
0 1 1 2 2 2 2 2 3 3 2 2 2 2 2 2 2 3 2 2 1 1 0 0
0 0 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 0 1 1 1 1 0 0
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 0 0 0 1 1
1 1 1 1 2 2 2 2 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1
1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 2 2 2 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 1
1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1
1 1 1 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 2 3 3 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1
1 1 1 1 1 1 2 3 3 3 2 2 2 2 2 2 2 2 2 2 2 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 0 0 1 0
1 1 2 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
0 0 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1
1 1 1 1 2 2 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 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 1
0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 0
1 1 2 1 1 1 2 2 2 2 2 2 2 2 2 2 2 0 1 1 1 1 1 1
0 0 0 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E. Atmaca / Electric Power Systems Research 78 (2008) 694–702
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
OLTC taps
701
702
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4. Conclusions
References
This work deals with power losses minimization of power distribution systems while keeping the node voltages at near rated values. It treats the optimization problem by an evolutionary search scheme supplemented by Ordinal Optimization Theory. Ordinal optimization approach relaxes the definition of optimality and makes the optimization problem easier. In this work, two definitions of the good enough criterion were used to identify the optimality of a solution. A solution may be optimal in terms of its performance value order among all the solutions on the search space, or in terms of its performance value with respect to that of a truly optimum. The first good enough criterion was used to determine the number of solutions to be sampled for at least one of them included by the best m-percent subset with a desired probability. For identifying the optimality of a solution in terms of the second good enough criterion, this work first calculated a reference optimal value for each loading factor then interpreted it as the estimated performance value of the truly optimum. Results verify validity of this estimation. The method finds the OLTC settings and capacitor on/off statuses at each hour by performing perturbations on solutions of the previous hour. This procedure, with the relaxed definitions of optimality, produces a fast solution to the power distribution control problem, making the method proper for on-line and off-line operation.
[1] F.C. Lu, Y.Y. Hsu, Reactive power/voltage control in a distribution substation using dynamic programming, IEE Proc. Gen. Transm. Distr. 142 (1995) 639–645. [2] R.H. Liang, C.K. Cheng, Dispatch of main transformer ULTC and capacitors in a distribution system, IEEE Trans. Power Deliv. 16 (2001) 625– 630. [3] G. Ramakrishna, N.D. Rao, Adaptive neuro-fuzzy inference system for volt/var control in distribution systems, Electr. Power Syst. Res. 49 (1999) 87–97. [4] Z. Hu, X. Wang, H. Chen, G.A. Taylor, Volt/var control in distribution systems using a time-interval based approach, IEE Proc. Gen. Transm. Distr. 150 (2003) 548–554. [5] A. Augugliaro, L. Dusonchet, S. Favuzza, E.R. Sanseverino, Voltage regulation and power losses minimization in automated distribution networks by an evolutionary multiobjective approach, IEEE Trans. Power Syst. 19 (2004) 1516–1527. [6] H. Mori, H. Tani, Fast hybrid method of parallel tabu search and ordinal optimization for on-line capacitor control in distribution systems, in: IEEE Proceeding PES General Meeting, 2003, pp. 1553–1558. [7] T.W.E. Lau, Y.C. Ho, Universal alignment probabilities and subset selection for ordinal optimization, J. Optimiz. Theory Appl. 93 (1997) 455–489. [8] X. Lin, A discussion on performance value versus performance order, IEEE Trans. Autom. Control 45 (2000) 2355–2358. [9] J.J. Grainger, S. Civanlar, et al., Optimal voltage dependent continous-time control of reactive power on primary feeders, IEEE Trans. Power Apparatus Syst. 103 (1984) 2714–2723. [10] S. Civanlar, J.J. Grainger, Volt/var control on distribution systems with lateral branches using shunt capacitors and voltage regulators. Part III. Numerical results, IEEE Trans. Power Apparatus Syst. 104 (1985) 3291–3297. [11] A. Augugliaro, L. Dusonchet, S. Favuzza, E.R. Sanseverino, Genetic, simulated annealing and Tabu Search algorithms: three heuristic methods for optimal reconfiguration and compensation of distribution networks, ETEP Eur. Trans. Electr. Power Eng. 9 (1999) 35–41.
Acknowledgement The author would like to acknowledge the Istanbul University Scientific Researches Unit for financial support of this research under grant 450/27122005.
Electric Power Systems Research 78 (2008) 694–702
An ordinal optimization based method for power distribution system control Erkan Atmaca ∗ ˙ Istanbul University, Faculty of Engineering, Electrical Engineering Department, 34850 Avcılar, Istanbul, Turkey Received 18 August 2006; received in revised form 19 January 2007; accepted 17 May 2007 Available online 28 June 2007
Abstract This work presents an Ordinal Optimization Theory based method for power distribution system control. The solution procedure bases on an evolutionary search scheme. The method aims at minimizing the total power losses on network feeders while keeping the node voltages at near rated values against changing loading conditions. By relaxing the definition of optimality and softening the goal of optimization, ordinal optimization makes the problem easier as well as the solution process faster. A solution can be defined as good enough in performance value if it is one of the best m-percent solutions on the search space with probability P%. A solution can be defined as good enough also if its performance value approximates to the optimal value to a pre-defined extent. This work estimates the performance value, the total power loss, of the optimum state of power distribution network at a given loading condition. Then, with the two relaxed definitions for optimality, it finds the optimal settings of on-load tap changers and capacitors in a 24-h time frame. Results verify our method is proper for on-line and off-line operation. © 2007 Elsevier B.V. All rights reserved. Keywords: Power distribution control; Ordinal optimization; Performance value; Good enough criterion
1. Introduction Due to recent rises in energy prices and already common use of sensitive electronic loads, power distribution system control is an important issue concerned with power loss minimization and electric power quality. To deal with this issue utilities make use of new technologies providing for the automatic control structure necessary to manage the network from a remote center. The primary aim of power distribution system control is at minimizing power and energy losses while keeping the node voltages at near rated values against changing loading conditions. This control is realized by adjusting the on-load tap changer (OLTC) settings, which are located at substations, and by switching the capacitors, which are located at substations and on network feeders. Reconfiguration of the network by switching the tie-switches at branches is another tool for distribution control. Automatic control of power distribution system bases on an optimally configured network in which the capacitors are optimally placed and sized at the design stage. During opera-
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tion, states of the controlling elements are changed continuously by switching operations and total switching operations number of each element for a specific time interval should be limited because of their life expectancy constraints. Therefore, power distribution system control is a constrained, multi-objective optimization problem with integer variables and solution to this problem requires rather new techniques than the conventional approaches, particularly for large systems. Lu and Hsu [1] propose a dynamic programming method for properly dispatching the shunt capacitors and on-load tap changers at the distribution substation based on the forecasted hourly loads. Liang and Cheng [2] improved this method so as to include the capacitors installed on network feeders. But dynamic programming method is inadequate in large systems due to the computational burden involved with it. Ramakrishna and Rao [3] introduce an adaptive neuro-fuzzy inference system for volt/var control, which involves complex calculations for decision process and considerable training times for artificial neural networks. Due to the drawbacks of preceding studies, recent studies used heuristic techniques like genetic algorithms (GAs). Hu et al. [4] develops a GA based method for minimizing the total energy loss through the 24-h time frame. They reduce the search space
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by dividing the 24-h load pattern into a number of load levels and OLTC settings change only when the load level changes. But since this method encodes the states of variables through the entire 24-h time frame, it leads to a considerable computational burden despite the reduced search space. Augugliaro et al. [5] combines the network reconfiguration problem with dispatching the capacitors and tap changers. Further, this study unifies the power loss minimization and voltage profile flattening objectives into a single one by a fuzzy reasoning approach. This unification, however, introduces an additional computational burden. Mori and Tani [6] proposes a Parallel Tabu search method for online capacitor control, which uses ordinal optimization (OO) concepts, however, disregards the voltage regulation problem. Although these studies work quite well, their efficiencies can be improved by using the OO concepts. Ordinal Optimization Theory is relatively a new topic, and found limited use in power system studies. By relaxing the definition of optimality and softening the goal of optimization, ordinal optimization makes the problem easier as well as the solution process faster. This study presents an OO based method for volt/var control in power distribution systems. It aims at minimizing total power losses at changing loading conditions. It uses two definitions for optimality: It identifies a solution as optimum according to either its ‘performance order’ in performances of a feasible set of solutions, or its ‘performance value’ compared with performance value of the ‘truly optimum’. Although the truly optimum cannot be identified in advance, we can estimate its performance value by setting the reactive components of the loads to a small amount. Following the forecasted daily load pattern, the method searches for the OLTC settings and capacitor on/off statuses at each hour that minimize the total power loss at that time interval. The search technique is evolutionary search (ES) based. The method is proper for on-line control as well. Results obtained from the tests on a sample system are promising as far as total computation time and quality of solutions are concerned. 2. Method 2.1. Formulation of the problem The considered power distribution system is assumed to have the following properties: (a) There is an OLTC at each HV/MV distribution substation, whose tap ranges from −8 to +8 with voltage range of variation ± 10%. (b) There are shunt capacitor banks installed at substations and on network feeders. (c) Loading condition at each node changes according to the daily load pattern shown in Fig. 1 The objective is the minimizing of the total power loss on network elements at any loading condition in a 24-h time frame Ploss =
NB i=1
Ri Ii2
(1)
695
Fig. 1. Daily load pattern of the power distribution network.
satisfying the voltage constraints Vmin ≤ Vi ≤ Vmax ,
i = 1, 2, . . . , NL
(2)
where NB is the number of branches and NL is the number of MV/LV transformers. To satisfy the requirements of many utilities, we assumed Vmin is 0.95 p.u. and Vmax is 1.05 p.u. The objective function is subject to power balancing constraint in addition to the following constraints: 24 i=1 24
|tapmi − tapmi−1 | ≤ Tmax
(3)
|Cni ⊕ Cni−1 | ≤ Cn max
(4)
i=1
where tapmi is the tap position of the mth OLTC at hour i; Cni the on/off status of the capacitor Cn at hour i, Cni = 0, 1; Tmax the maximum number of the switching operations allowed for each OLTC; Cn max is the maximum number of the switching operations allowed for the capacitor Cni . Maximum number of the switching operations of an element depends on its life expectancy. These numbers are utility dependent, however, considering the common practice, we can assume total switching operations number of an OLTC should be below 30 a day, and of capacitors installed at substations and on feeders below 10 and 4, respectively. Control variables are the OLTC settings and the on/off statuses of shunt capacitors. In an evolutionary search scheme these variables can be coded as real numbers. Such a coding formulates the volt/var control as an optimization problem with discrete search space. Size of the search space is determined by the number of control variables and the number of states associated with them. Theoretically, search space of the volt/var control problem is very large for a practical power distribution system. 2.2. Ordinal optimization (OO) Ordinal optimization was first introduced by Lau and Ho in 1997 [7]. OO is not a method on its own but rather a supplement to existing optimization methods. OO is based on two ideas: (i) it is easier to determine order than value. That is to say, determin-
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ing whether A is greater or less than B is easier than determining the values of A and B exactly. (ii) Softening the goal of optimization makes the problem easier as well. Instead of asking the ‘best for sure’, the ‘good enough with high probability’ can be proposed. For example, considering a search on the search space Θ, we can define the ‘good enough’ subset, G/Θ, as the top-1% of the space based on objective function values. Consider a ‘selected’ subset, S/Θ, determined by a sampling procedure. This sampling can be random, or be done by rough calculations and measurements. By requiring high probability for matching of these two subsets, we can narrow down the search space from Θ to S. Relaxation of the optimization objective can reduce computation time significantly for a large-scale problem. Let us define the good enough subset G as the top m-percent of the whole set of the solutions, based on objective function values. Let NS be the total number of solutions in the selected subset S. Then, the probability for at least one of the selected solutions being a member of G is P. Then, P = 1 − (1 − m)NS
(5)
Let us define P as the confidence level of the sampling procedure, and the percentage m as the solution quality. Therefore, from Eq. (5), we can determine at least how many candidates should be sampled so that they include at least one solution from the best m-percent set with confidence level P. Having decided for P and m by engineering judgment, the least number of solutions to be sampled is NS =
ln(1 − P) ln(1 − m)
(6)
In this way, size of the good enough set is determined beforehand by the operator and it is independent of the problem. But this approach does not tell us how much difference in performance value can be between the solution we found and the truly optimum. Therefore, a slightly different ‘good enough’ definition can be made, which measures the performance value of a solution with respect to the performance value of the truly optimum [8]. Consider an optimization problem minJ(θ) θ∈Θ
(7)
where Θ is the search space, J(θ) is the performance function. Θ = {θ 1 , . . ., θ N } is a finite discrete set, where N is the size of the search space and J(θ i ) = J(θ j ) if i = j. We can define the good enough set as G ≡ {solution that satisfies a good enough criterion}. Then, it is possible to make two different definitions for a good enough subset: G1 ≡ the top-1% solution
(8.a)
G2 ≡ {θi |J(θi ) ≤ 1.01 × J(θopt )}
(8.b)
where θ opt represents an optimal solution, which minimizes the objective function on the search space. But whether all the solutions in G1 or G2 are desirable is problem dependent. Consider we know performance values of all the solutions in Θ. By ordering them according to their performance values we
Fig. 2. Two categories of the ordered performance curves (OPCs) of an optimization problem.
can obtain the ordered performance curve (OPC) of the problem. The shape and characteristics of this curve can fall in one of the categories shown in Fig. 2. A flat OPC represents a problem with lots of good solutions, although a steep OPC represents a problem with few good solutions. Therefore, although the good enough criterion G1 can produce desirable results for a problem with flat OPC, it may not be satisfactory for a problem with steep OPC, in which the good enough criterion G2 is more reliable. For many optimization problems, performance value of the optimal solution is not known beforehand, unless all the solutions are calculated. But use of the good enough criterion G2 requires it to be known. In a properly compensated power distribution system the reactive power flows through the lines are negligible compared with the active power flows [9]. A proper compensation level can be simulated by setting the reactive components of the loads to a small amount. To be realistic, we set the reactive loads at 5% of their original values. We regulate the system by changing only OLTC settings. We define the performance value, the total power loss, calculated at these conditions as the reference optimal value for measuring quality of the solutions. This value is a reference and underestimates the value of the real optimum. The real optimum cannot be identified unless the computation was completed. This is why the good enough criterion G2 deals with the reference optimal value. On the other hand, we use the good enough criterion G1 as the lower bound for the quality of the solutions. During the search process at each hour, tap perturbations are performed in the neighborhood of the OLTC settings that found when calculating the reference optimal value of that hour. This procedure is prone to satisfy the bus voltage constraints. Nevertheless, the solutions that provoke intolerable voltages are not rejected but assigned to very large power loss values so that they are subordinated during the search process. 2.3. Solution procedure The solution procedure is based on an evolutionary search scheme, where only the mutation operator is considered. New
E. Atmaca / Electric Power Systems Research 78 (2008) 694–702
solutions are created by perturbing the existing solutions but not by recombination. A similar scheme is used successfully by Augugliaro et al. [5]. It is reported Tabu Search (TS) outperforms Genetic Algorithm [11]. However, TS needs a serious effort to model the solution strategy. Besides, creating a neighborhood around a solution can be time consuming for a large-scale problem. On the other hand, classical GA needs to tune up the cross-over and mutation operators. A convergence criterion is also needed by TS and GA. However, generating new solutions in evolutionary strategies is simple, and selection of parents is more flexible compared with classical GA. ES also allows the real-coding of variables, hence simplifies the representation of individuals. In our algorithm, the search at each hour starts with slight modifications on the optimum of previous hour. The load profile changing smoothly allows using such a scheme. However, for a daily load profile changing sharply, the search process demands more diversification than our algorithm produces, in which some kind of cross-over can be used to improve the solution quality. In this application, we used (λ + μ) selection scheme, where the symbol μ stands for the number of parents in one generation, and λ for the number of created offsprings. At each generation parents and offspring’s are joined and the best μ individuals are selected from this combined set. To exploit the smoothly changing pattern of the loading conditions, we have chosen the first generation solutions for each loading condition as the same as the last generation solutions of the previous one. This approach leads the optimization process to converge fast. Each solution is coded into an integer string; Z, containing the information of taps positions of OLTCs, and shunt capacitor statuses: Z = Ztap , Zcap
where k=
|ZC | , |ZC | − |ZT |
In our scheme, determination of the starting tap positions and estimation of the reference optimal values are related to each other. If the reactive power flows through feeders are negligible compared with active power flows, the total power loss on feeders and the node voltages are function of only the OLTC positions. To create a pseudo-state of optimal compensation, we set the reactive loads at 5% of their original values. Hence, for each loading factor (LF) in the 24-h time frame, we determine the starting tap position of the mth OLTC, tm start,h , as the following: tm start,h ∴ min f (tm,i,h ); f (tm,i,h ) =
t = 1 + 0.0125 × TAP
i = −8, −7, . . . , 7, 8
1 |1.00 − Vi,j,h |; NL
j = 1, 2, . . . NL
(11.a)
Qj = 0.05 × Qj ; (11.b)
where Vm,i,h is the tap position of the mth OLTC at hour h; Vi,j,h the voltage of the jth node at the tap position i at hour h; NL the number of the load nodes; Qj the original value of the reactive component of the load at the jth node; Qj is the reactive component of the load at the jth node. To avoid the unnecessary calculations we searched for tm start,h in the neighborhood (±1) of the tap settings found by the formula in Eq. (10). Namely, the starting tap positions at hour h are those that make the voltage profile of the network most flat. We define the reference optimal value, Ploss ref,h or performance value of the truly optimum, as the total power loss resulted from the conditions described in Eq. (11). The reference optimal value will be calculated at each hour. This value will serve for defining the good enough criterion G2 in Eq. (8).
(9)
Theoretically, the size of Ztap is the number of OLTCs and the size of Zcap is the number of capacitors. Size of the search space depends on the number of control variables and the number of states associated with each variable. Limiting the range of variations of the variables reduces the computational burden. To restrict the range of tap variations of an OLTC, some studies perform the perturbations around a forecasted starting tap position that is obtained by the analytical formula in Eq. (10) [1,5]. The formula is based on the active and reactive power demands from transformer and the desired voltage at secondary bus. ⎧ ⎡ ⎤⎫1/2 2 ⎨ |Z |2 ⎬ 2 V1 T ⎣ 2⎦ t= Q (10.a) + + P L L ⎩ V02 V12 ⎭ k|ZT |
697
Ploss ref,h =
NB i=1
Ri Ii2 ;
∴ tm,i,h =tm start,h ;
i = 1, 2, . . . NB Qj =0.05 × Qj ;
j=1, 2, . . . NL (12)
The solution procedure is summarized in the flow chart in Fig. 3. Loading factor (LF) is a number between 0.4 (the lightest loading condition) and 1.01 (the heaviest loading condition). They are obtained from the load forecasting data of the given day. Updating LF means continuing to the next hour calculations. Proper sizes for the numbers of population, Npop , and generations, Ngen , depend on search space size in an evolutionary search scheme. They are generally determined by engineering judgment. These two parameters result in size of the selected subset: NS = |S| = Npop × Ngen
(13)
(10.b)
In these equations, TAP is the position of an OLTC, ZC is the impedance of the switched capacitor banks at the substation, V0 and V1 the transformers primary and secondary voltages, ZT the transformer impedance, PL and QL are active and reactive loads of the transformer, and t is the transformer voltage ratio.
Considering Eqs. (6) and (13) we choose the numbers of generations and population according to desired confidence level and acceptable solution quality. Hence, the good enough criterion G1 results in the proper sizes for Npop and Ngen , and the good enough criterion G2 serves as the termination criterion of the search process as well as the quality measure for solutions.
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commercial size of 300 kvars is assumed for each capacitor unit. Because we perturb the tap settings (±1) around starting positions, for any loading condition, there are three possible states for each OLTC. The state of each group of capacitors can be represented by a number between 0 and 3; there are four states for each group. For example, number ‘2’ means two of the capacitor units are on and number ‘0’ none of them. Hence, we sample the solutions from (34 × 428 ) = 5.8 × 1018 possible states for each hour. The evolutionary search parameters used in this sampling is shown in Table 1. Hence, for a given loading condition, the maximum number of traversed solutions, NS , is 25 × 10 = 250. Based on this selected-subset size, Table 2 shows how the solution quality changes with confidence level according to Eq. (6). Hence, the set of 250 random samples includes at least one solution from the top-1% subset with probability 92%. However, since the search is guided to more promising areas through evolutions we can expect confidence levels higher than those in Table 2. The good enough criterion G2 can terminate the search process of the current hour without traversing all the 250 solutions if the following condition is satisfied: Ploss,h ≤ 1.01 × Ploss ref,h
(14)
where Ploss,h is performance value of the best solution in current population, and Ploss ref,h is the reference optimal value for hour h. The average voltage variation at hour h, Vh was defined as the following: Vh = Fig. 3. Flow chart of the solution procedure.
1 |1.00 − Vj, h |; NL
j = 1, 2, . . . , NL
(15)
where NL is the total number of load nodes and Vj,h is the p.u. value of the jth node voltage at hour h.
2.4. Test system
3. Results
We applied our method to a commonly used 115/23 kV/kV test system [3,4,10]. The network and load data were taken from Ref. [10]. To complicate the problem, we repeated the test system four times as shown in Fig. 4; hence created a 121-node system with 120 lines. Line parameters are same but the loads of the nodes 1–30, 31–60, 61–90 and 91–120 were set to 100, 90, 75 and 60%, respectively, of their original values. Further, we installed additional units at each capacitor bus to enlarge the search space. Since distribution system loads are sensitive to voltage, using only P–Q models for loads can lead to inaccuracies. Accordingly, we modeled half the load at each bus as constant current load and the other half as constant impedance load. We assumed loads change according to the daily load profile shown in Fig. 1. Transformers with OLTCs are installed between the node 0 and nodes 1, 31, 61 and 91. We assumed primary voltage of each transformer is constant and 1.00 p.u. Totally 28 groups of capacitors are installed at 24 nodes. A group consists of three units. A
The program was written in MATLAB and run on a PC with 1.5 GHz microprocessor. Many runs result in 85 s computation time in average for a 24-h time frame. This computation time is proper for on-line control operation. To test the efficiency of the good enough criterions G1 and 2 G , and the method overall, we extended the search space and increased the number of samplings for comparison purposes. We set the width of mutation tap to (±2); Npop to 20 and Ngen to 50 hence traversed 1000 solutions for a single loading factor without a termination criterion. Table 3 compares the limited and extended searches referring to the reference values. In Table 3, Ploss (%) denotes to total power loss error, where Ploss (%) =
Ploss − Ploss ref × 100 Ploss ref
Table 3 includes total power losses of the uncontrolled system as well.
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699
Fig. 4. One-line diagram of the test distribution system. Table 1 Evolutionary search parameters used in this work Number of generations
Parent population, μ
Offspring population, λ
Width of mutation tap
Width of mutation capacitor
Mutation probability
25
10
10
±1
±300 kvars
1.0
Resulting objective function values of the limited search, with good enough criterions G1 and G2 , are comparable to those of the extended search. On the other hand, the extended search improves total power loss results slightly at the expense of minor Table 2 Solution qualities versus confidence level with 250 and 1000 samples Solution quality (m, %) vs. confidence level with 250 and 1000 samples
250 1000
Confidence level P (%)
63
78
87
92
97
99
0.40 0.10
0.60 0.15
0.80 0.21
1.00 0.25
1.40 0.35
1.80 0.46
increases in voltage variations. This is expected due to large variations in tap settings, since the extended search perturbs the OLTC settings (±2) around starting positions. As far as the definition in Eq. (15) is concerned for voltage variation, the limited search results in more smooth voltage profiles than the extended search does at the expense of minor increases in power losses. Although the good enough criterion G2 is not fully satisfied by the limited search at some hours (1, 3, 4, 5, 21, 23, 24), errors are inconsiderable. Many runs have converged in on an average of 10 generations per hour in the 24-h frame. On the other hand, solution process continued up to 25 generations at six loading factors on average. However, at some loading factors such as the first
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Table 3 Comparison of actual total power losses with reference values Hour
Unregulated
Reference values
Extended search
Ploss (kW)
Ploss ref (kW)
V (%)
Ngen
Ploss (kW)
Ploss (%)
V (%)
Ngen
Ploss (kW)
Ploss (%)
V (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
221.5 256.1 346.4 466.5 641.9 760.8 935.2 1128.1 1423.6 1339.5 1103.0 1005.4 912.4 1078.1 1258.0 1153.5 1029.3 720.0 568.4 499.3 389.3 306.0 268.1 210.6
185.0 213.8 289.1 389.0 534.7 633.3 777.6 936.8 1180.0 1111.0 916.1 835.6 758.8 895.6 1043.9 957.8 855.4 599.5 473.7 416.3 324.8 255.4 223.8 175.9
0.74 0.78 0.91 1.07 1.22 1.33 1.52 1.64 1.82 1.77 1.62 1.56 1.46 1.61 1.72 1.65 1.58 1.29 1.16 1.10 0.97 0.86 0.80 0.72
50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
187.0 215.9 291.7 392.6 539.9 638.4 783.6 943.8 1188.3 1118.9 923.1 841.7 764.3 902.2 1051.5 964.7 861.9 605.5 478.0 420.0 327.7 257.8 226.0 178.0
1.07 0.98 0.90 0.93 0.96 0.81 0.76 0.75 0.71 0.71 0.76 0.73 0.73 0.73 0.73 0.73 0.76 0.99 0.92 0.89 0.90 0.93 0.96 1.23
1.24 1.04 1.53 1.69 1.76 1.35 1.68 1.77 2.02 1.86 1.72 1.94 1.76 2.08 1.84 1.74 1.73 1.67 1.71 1.37 1.22 1.56 1.05 1.80
25 1 25 25 25 12 5 1 3 1 1 3 6 1 3 1 1 13 7 5 25 6 25 25
187.0 215.9 292.1 393.2 540.4 639.5 785.3 945.3 1190.7 1121.3 925.2 843.4 766.3 904.0 1054.0 966.4 863.1 605.5 478.3 420.1 328.2 258.0 226.2 178.0
1.07 0.98 1.07 1.07 1.06 0.98 0.98 0.91 0.91 0.93 0.99 0.93 0.99 0.93 0.97 0.90 0.90 0.99 0.98 0.93 1.05 1.00 1.06 1.23
0.83 0.78 0.94 1.22 1.35 1.51 1.70 1.69 1.87 1.80 1.72 1.70 1.46 1.66 1.80 1.69 1.65 1.30 1.18 1.28 1.08 1.01 0.85 1.09
Mean
750.9
624.3
1.29
50
629.3
0.80
1.63
10
630.3
0.96
1.38
and last ones, the extended search was lack of satisfying the second good enough criterion, either. This occurs especially at low loading factors, where underestimation of the truly optimal leads to larger percentage errors. The cross-over operator, inserted at these hours for further improvement, resulted in inconsiderable rise in solution quality despite the increase in total computation time due to the extent of solution diversification it introduced. However, the good enough criterion G2 still serves as a quality measure for solutions. Table 4 shows the tap positions and capacitor status variations at substations and on feeders through 24 h. Total switching operations numbers a day satisfy the constraints. These constraints are on-line, not final checks. Total switching operations number of each unit has a certain daily limit beyond which further switching is prohibited. However, since our search at each hour proceeds with slight modifications on existing solutions of the previous hour we never observed such a violation. Consider that for the system at hand total number of powerflow runs in the 24-h time frame is proportional to the total number of generations. The good enough criterion G2 terminated the search process for each hour averagely in 10 generations. This leads to a considerable saving in computation time compared with the extended search, which is performed without a termination criterion through 50 generations for each hour. Voltage profile of the node 15, which is the lowest voltage node of the network, is shown through 24 h in Fig. 5.
Limited search
The extended search regulates the voltage of the node 15 better than the limited search does, since the former resulted in the average voltage as 0.98 p.u. through 24 h while the latter 0.97 p.u. Despite such a difference for a specific node, as far as the overall voltage regulation capability is concerned, the limited search slightly outperformed the extended search.
Fig. 5. Voltage profile of the node 15 through 24 h. (*) Results of the extended search.
Table 4 Tap positions and capacitor switching operations through 24 h Hour
Capacitor switching operations
T1
T2
T3
T4
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
C26
C27
C28
2 2 3 2 2 3 4 4 5 4 4 3 4 4 4 4 4 3 2 3 3 2 2 2
1 2 2 3 3 3 4 4 4 4 4 3 3 4 4 4 3 4 3 2 2 2 1 1
2 1 2 1 2 2 2 3 3 3 3 2 3 3 3 3 3 3 1 2 1 2 0 1
0 1 1 2 2 2 2 2 3 3 2 2 2 2 2 2 2 3 2 2 1 1 0 0
0 0 1 2 1 1 2 2 2 2 2 2 2 2 2 2 2 0 1 1 1 1 0 0
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 0 0 0 1 1
1 1 1 1 2 2 2 2 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1
1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 2 2 2 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 1
1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1
1 1 1 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 2 3 3 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1
1 1 1 1 1 1 2 3 3 3 2 2 2 2 2 2 2 2 2 2 2 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 0 0 1 0
1 1 2 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
0 0 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1
1 1 1 1 2 2 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 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 1
0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 0
1 1 2 1 1 1 2 2 2 2 2 2 2 2 2 2 2 0 1 1 1 1 1 1
0 0 0 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E. Atmaca / Electric Power Systems Research 78 (2008) 694–702
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
OLTC taps
701
702
E. Atmaca / Electric Power Systems Research 78 (2008) 694–702
4. Conclusions
References
This work deals with power losses minimization of power distribution systems while keeping the node voltages at near rated values. It treats the optimization problem by an evolutionary search scheme supplemented by Ordinal Optimization Theory. Ordinal optimization approach relaxes the definition of optimality and makes the optimization problem easier. In this work, two definitions of the good enough criterion were used to identify the optimality of a solution. A solution may be optimal in terms of its performance value order among all the solutions on the search space, or in terms of its performance value with respect to that of a truly optimum. The first good enough criterion was used to determine the number of solutions to be sampled for at least one of them included by the best m-percent subset with a desired probability. For identifying the optimality of a solution in terms of the second good enough criterion, this work first calculated a reference optimal value for each loading factor then interpreted it as the estimated performance value of the truly optimum. Results verify validity of this estimation. The method finds the OLTC settings and capacitor on/off statuses at each hour by performing perturbations on solutions of the previous hour. This procedure, with the relaxed definitions of optimality, produces a fast solution to the power distribution control problem, making the method proper for on-line and off-line operation.
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Acknowledgement The author would like to acknowledge the Istanbul University Scientific Researches Unit for financial support of this research under grant 450/27122005.