A Tabu Search Multiobjective Approach to ... - Semantic Scholar

MIC’2001 - 4th Metaheuristics International Conference

169

A Tabu Search Multiobjective Approach to Capacitor Allocation in Radial Distribution Systems Dulce F. Pires∗

C. Henggeler Antunes†

A. Gomes Martins†

∗ Escola Superior de Tecnologia de Set´ ubal Campus do IPS, 2914-508 Set´ ubal, Portugal Email: [email protected] † Departamento de Engenharia Electrotécnica Polo II da Universidade de Coimbra, 3000 Coimbra, Portugal Email: {ch, amartins}@dee.uc.pt

1

Introduction

The installation of shunt capacitors in primary distribution electrical networks can effectively reduce energy and peak power losses, while improving voltage profile. Economic and operational benefits depend mainly on the number, locations and sizes of the capacitors installed. Several approaches devoted to this problem have been reported in the literature. However, most of them adopt some unrealistic assumptions (such as uniform load distribution, uniform feeder size and constant loads), suffer from lack of generality, or are computationally too demanding when applied to distribution systems of realistic sizes. Initially the problem of capacitor allocation has been handled with analytical methods [1]-[2]. However, more recently other methodologies have proposed: mixed integer programming [3]-[4], linear programming models [5] and methods based on heuristic search techniques: Genetic Algorithms [6], Simulated Annealing [7]-[8], Tabu Search [9]-[10]-[11], etc. Heuristic rules produce fast and practical strategies, which allows a more effective exploration of the search space and can lead to a solution that is near optimal. In these problems multiple and conflicting evaluation aspects are generally at stake, such as minimizing capacitor installation cost and reducing system losses. Multiple objective models explicitly address these different concerns, and they entail analyzing trade-offs between the different objectives to select a satisfactory compromise solution from the set of nondominated solutions, which can be accepted as the output of the decision process. In this paper, a Tabu Search (TS) based approach to provide decision support in the problem of capacitor allocation in radial distribution networks is presented. The model explicitly considers two conflicting and incommensurate objective functions, related to cost and operations aspects of evaluation. This offers the decision maker (DM) the possibility both to expand the range of potential alternative solutions and to express his/her preferences to select a satisfactory compromise solution.

1.1

Preamble

In this paper, a multiobjective approach to the capacitor allocation problem in radial distribution systems is described. The aim of this work is to provide decision support in the placement and dimenPorto, Portugal, July 16-20, 2001

170


sioning of capacitors, taking into account two conflicting objectives: minimizing line losses (resistive) and minimizing capacitors costs (acquisition, installation, operation and maintenance). The proposed methodology uses a Tabu Search based algorithm to decide the locations to install capacitors (number and place) and the sizes of capacitors to be installed. The methodology is applied to a test case of an actual radial distribution system.

2

Mathematical Model

In general, the problem of VAr optimisation consists in determining the optimal number, location, sizes, and switching times for capacitors to be installed in a distribution feeder in order to obtain maximum cost savings while maintaining good operating conditions. The proposed approach formulates this problem using a multiobjective mathematical programming model, with two conflicting objectives: minimizing feeder resistive losses and minimizing capacitor costs. Considering a radial distribution feeder with t nodes; t − 1 branches; s laterals and l shunt capacitors placed at the network nodes. The power flow through each branch is described by a set of recursive equations (1-5):

k Pn(i+1)

=

0 P0(i+1)

=

Qkn(i+1)

=

Q00(i+1)

=

k Vn(i+1)

2 =

k 2 k 2 Pni + Qni − QkCni k − − PLn(i+1) , ∀ n = 0 and k = 0 (1) k 2 Vni 0 2 0 2 S + Q0i − Q0C0i (i+1) 0 0 P0i 0 P0i − r0i − P − Pn0 (2) L0(i+1) 2 (V0i0 ) n=1 k 2 k 2 P + Qni − QkCni Qkni − xkni ni − QkLn(i+1) + QkCn(i+1) , ∀ n = 0 and k = 0 (3) k 2 Vni 0 2 0 2 S + Q0i − Q0C0i (i+1) 0 0 P0i 0 Q0i − x0i − Q − Qn0 + Q0C0(i+1) (4) L0(i+1) 2 (V0i0 ) n=1 ( 2 2 ) k & k 2 % k 2 + Qkni − QkCni Pni k k k k k 2 Vni − 2 × (rni Pni + xni Qni ) + (rni ) + (xni ) × (5) k 2 Vni k Pni

k rni

Where: k PLnm - load (real power) at bus m, (lateral n, derived from bus k at the main feeder).

QkLnm - load (reactive power) at bus m, (lateral n, derived from bus k ). k Pnm - real power which flows in connecting branch m → m + 1,(lateral n, derived from bus k ).

Qknm - reactive power which flows in connecting branch m → m + 1, (lateral n, derived from bus k ). QkCnm - reactive power injection from capacitor at node m, (lateral n, derived from bus k ). k Pnm - bus voltage magnitude at node m (lateral n, derived from bus k ). k rnm - resistance of connecting branch m → m + 1, (lateral n, derived from bus k ).

xknm - inductance of connecting branch m → m + 1, (lateral n, derived from bus k ) The principal feeder has the index n = 0, i.e. it is considered the 0th lateral, and k = 0, i.e. it begins at SE (sub-station). The capacities of new capacitors are given associated with their total cost, thus including acquisition, installation, operation and maintenance costs. Porto, Portugal, July 16-20, 2001


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Besides power flow equations there are other conditions to be satisfied for each lateral (including the feeder): from the last bus of each branch, there is no power (real or reactive) flowing to other branches, k therefore Pnm = Qknm = 0; and upper and lower bounds of the node’s voltage magnitude must be k = Vmax . considered, Vmin ≤ Pnm In this approach two objective functions have been considered. One represents the system resistive loss (6), and the other represents the cost associated with capacitor placement (7).

T S V

min

k rnm

k 2 k 2 Pnm + Qnm − QkCnm

m=0n=0k=0 T S V Y

min

2

k ) (Vnm

j aknm cj

(6)

(7)

m=0n=0k=0 j=0

Where: j aknm

=

QkCnm

=

bknm

bknm

=

k 1 if the new capacitor, QF j , is installed in Bnm , 0 otherwise Y

(8)

j aknm QF j ∀ m, n, k,

j=1 k 1 if it is possible to install capacitors in Bnm 0 otherwise

(9)

QkCnm is the compensation variable at node m, lateral n, derived from the principal node k QF j is the capacity of the capacitor j cj is capacitor (QF j ) cost (j = 1, ..., y) The constraints are described by a set of load flow equations (1—5), placement constraints which Y j guarantee that only one capacitor can be placed in each node, aknm < 1 ∀ m, n, k, and the voltage j=1 k constraints regarding the upper and lower bounds of the node’s voltage magnitude, Vmin ≤ Pnm = Vmax .

3

A Multiobjective Approach

Tabu Search (TS) [12]-[13] is a neighborhood search heuristic designed to avoid being trapped in local optima. The search is constrained by classifying certain moves as forbidden, i.e. tabu, in order to prevent the reversal, or sometimes repetitions, of the moves. The basic features used in this work are briefly described in this section. In order to avoid returning to the solution just visited, the reverse move that is detrimental to achieving a better solution must be forbidden. This is done by storing this move in a data structure, such as a finite length last-in-last-out structure called tabu list, whose elements are called tabu moves. As soon as a trial solution is generated, this one is checked with the tabu list to see whether it exists in the list, allowing more effective exploration of the search space. The tabu list dimension is called tabu list size (#TL). The tabu search version herein proposed is devoted to the capacitor allocation problem. The first phase of the search only uses a list of tabu moves. A move leading to a neighbour solution is Porto, Portugal, July 16-20, 2001

172


defined by changing compensation value of one bus of the electric system. The possible compensation changes are: • Decrementing the capacity installed at bus i; • Removing the capacity installed at bus i; • Moving the capacity installed at bus i to bus i − 1; • Moving the capacity installed at bus i to bus i + 1; Short-term tabu memory is implemented by a tabu list (last-in-last-out), which records the last #TL variables changed (#TL= 5 or 7). The choice of #TL is critical: if the size is too large, appealing moves may be forbidden and higher quality solutions cannot be explored; if this value is too low, cycling may occur in the search process and the process often returns to the solution just visited. The tabu search in our approach includes two phases: the first one only uses a list of tabu moves (as described above), and the second one, which is a diversification phase. The diversification phase attempts to generate solutions that embody different features (variable values) from those previously found, driving the search into new regions. It uses a frequency memory whose data have been collected from the beginning of the first phase. This memory vector records the number of times each variable took a value different from 0 (no capacitor installed) in feasible solutions. At the diversification phase, a variable is allowed to change from 0 to a compensation value x (or the reversal) if this move is non-tabu and the variable frequency of x’s is smaller (larger) than a threshold. The threshold is initialized to be the average frequency of x’s of all the variables. The method used to compute nondominated solutions to the multiobjective problem [14] is based on the tabu search version just described. The basic idea underlying the method is the progressive search of nondominated or “good” approximations of nondominated (potentially nondominated) solutions. The use of reservation levels (worst values the decision maker/planning engineer is willing to accept) aims at bringing the search process closer to regions of greater interest. A computing phase consists in running a tabu search routine k times (k being the number of objectives). The ith run (i = 1, . . . , k) gives privilege to the objective function i by using standard selection and acceptance criteria. A step-by-step TS-based algorithm for the problem of capacitor placement is presented below. A computer program has been developed using this algorithm. • Step 1: Input feeder data (including nodal loads, resistance and reactance of each section of the feeder, substation voltage, standard capacitor sizes and costs, etc).

• Step 2: Run the load flow application to compute the system’s initial conditions. • Step 3: Randomly generate solutions and check their feasibility. • Step 4: From the set of potentially nondominated (p.n.d.) solutions generated, choose an initial solution (which represents a cost/losses compromise). Repeat (Li = lm × zm ) (m = 1, ..., s):

• Step 5: Choose reservation levels for all or some objective values: fi (x) ≤ Li ,i = 1, ..., k . For j := 1, ..., k :

• Step 6: Run the TS algorithm (with a specified number of iterations), adding fi (x) ≤ Li to the original set of constrains, considering fj (x) the cost function, and updating the set of p.n.d. solutions at each iteration.

• Step 7: Output the set of p.n.d. solutions. Porto, Portugal, July 16-20, 2001


160

60

140

173

55

55

50

120 50 Cost ($)

Cost ($)

Cost ($)

45

100

45

80

40 40

60

20 0.1

35

35

40

0.2 0.3 0.4 0.5 Resistive Losses (MW)

0.6

(a) All (feasible) solutions after applying steps 1,2,3.

30

30

0.19 0.195 Resistive Losses (MW)

(b) Set of p.n.d. feasible solutions generated.

0.19 0.195 Resistive Losses (MW)

(c) Set of p.n.d. feasible solutions after aplying steps 5, 6 and 7.

Figure 1: Feasible solutions

4

An Actual Case Study

The proposed methodology has been applied to an actual Portuguese radial distribution system. The main phases of the study are briefly described below. Fig. 1(a) shows all (feasible) solutions after applying steps 1, 2 and 3. Fig. 1(b) presents the set of p.n.d. feasible solutions generated. The results after applying steps 5, 6 and 7 are displayed in Fig. 1(c). Each solution is associated with a compensation scheme, which consists of the number and sizes of the capacitors, the nodes in the system where they will be installed, and the corresponding cost and resistive losses. The results, Fig. 1(c), show that the proposed methodology is able to optimize the allocation of capacitors in radial distribution systems, according to different goals.

5

Conclusions

In this paper, a multiobjective model and a TS-based approach to provide decision support in the capacitor allocation problem have been presented. This formulation takes into account two objective functions: minimizing line losses and minimizing capacitor costs. Nondominated solutions are computed by using a methodology based on Tabu Search. The proposed methodology has been applied to an actual radial distribution system. The obtained results show the effectiveness of the proposed method.

References [1] R. F.Cook. Calculating loss reduction afforded by shunt capacitor application. IEEE Trans. on Power Apparatus and Systems, PAS - 83: 1227-1230, 1964. [2] Y. G. Bae. Analytical method of capacitor allocation on distribution primary feeders. IEEE Trans. on Power Apparatus and Systems, PAS - 87(11): 1232-1238, 1978. Porto, Portugal, July 16-20, 2001

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[3] W. M. Lebow et al. A hieralchical approach to VAR optimization in system planning. IEEE Trans. on Power Apparatus and Systems, PAS - 104(8): 549-557, 1985. [4] S. Granville et al. An Integrated Methodology for VAR Sources Planning. IEEE Trans. on Power Systems, PWRS-3(2): 549-557, 1988. [5] G. Opoku. Optimal Power System VAR Planning. IEEE Trans. on Power Systems, PWRS-5(1):5359, 1990. [6] K. Y. Lee et al. Optimization Method for Reactive Power Planning by Using a Modified Simple Genetic Algorithm. IEEE Trans. on Power Systems, PWRS-10(4):1843-1850, 1995. [7] Y.-T. Hsiao et al. A New Approach for Optimal VAR Sources Planning in Large Scale Electric Power Systems. IEEE Trans. on Power Systems, PWRS-8(3):988-996, 1993. [8] W.-S. Jwo, et al. Hybrid Expert System and Simulated Annealing Approach to Optimal Reactive Power Planning. IEE Proc. - Generation, Transmission and Distribution, 142(4):381-385, 1995. [9] D. Gan et al. Large Scale Var Optimization and Planning by Tabu Search. Electric Power Systems Research, 39:195-204, 1996. [10] Y. C. Huang et al. Solving the Capacitor Placement Problem in a radial Distribution System Using Tabu Search Approach. IEEE Trans. on Power Delivery, 11(4):1868-1873, 1996. [11] H. Mori and Y. Ogita. Parallel Tabu Search for Capacitor Placement in Radial Distribution Systems. Proc. of IEEE Power Engineering Society Winter Meeting, 4:2334-2339, 2000. [12] F. Glover. A User’s Guide to Tabu Search, Annals of OR, 41:3-28, 1993. [13] F. Glover and M. Laguna. Tabu Search. Kluwer Academic Publishers, 1997. [14] M. Jo˜ ao Alves et al. An Interactive Method For 0-1 Multiobjective Problems Using Simulated Annealing and Tabu Search. Journal of Heuristics, 6 (3): 385-403, 2000.

Porto, Portugal, July 16-20, 2001