Ordinal Optimization for Optimal Capacitor Placement and Network ...

2012 IEEE International Conference on Systems, Man, and Cybernetics October 14-17, 2012, COEX, Seoul, Korea

Ordinal Optimization for Optimal Capacitor Placement and Network Reconfiguration in Radial Distribution Networks R. EL Ramli+, M.Awad*, R.A. Jabr^ Department of Electrical & Computer Engineering, American University of Beirut, Beirut, Lebanon [email protected], *[email protected], ^[email protected] Abstract—Capacitor Placement (CP) is a complex non linear mixed integer optimization problem that includes both integer variables that correspond to the locations at which capacitors should be placed and discrete variables that represent the number of capacitor banks that should be installed. Motivated to efficiently reduce power losses in distribution networks by CP and network reconfiguration (NR) using some of the intelligent soft computing approaches, we propose to use in this study ordinal optimization (OO) which relies on order comparison and goal softening to make the computation more efficient. We tested OO on 33 and 69 bus systems and compared experimental results with other recently published methods. The reduction in losses obtained using OO for CP and NR motivates further studies. Keywords-ordinal optimization, distribution networks, soft computing.

I.

capacitor

placement,

INTRODUCTION

Capacitor placement (CP) is one of the practical methods to reduce power losses in distribution networks and improve nodal voltages. CP addresses the optimal placement, the type, size and number of capacitors required to be installed in the system in such a way to achieve a maximum power loss reduction. CP is a complex non linear mixed integer optimization problem: it includes both integer variables that correspond to the locations at which capacitors should be placed and discrete variables that represent the number of capacitor banks that should be installed. It is also a computationally exhaustive problem whose dimension increases enormously with network size. Because of the computational difficulty of CP, intelligent soft computing and more specifically the ordinal optimization (OO) technique seems to be a good candidate for that task. OO is based on the idea that the relative order (instead of the cardinal value) of the performance of different alternatives in a decision problem is robust with respect to estimation noise. OO narrows the search for optimum performance to a good enough subset in the design space instead of estimating the accurate values of the system performance. This implies that if a set of alternative designs is approximately evaluated and ordered according to a crude model, then there is high probability that the actual good alternatives can be found in the top-s estimated choices.

978-1-4673-1714-6/12/$31.00 ©2012 IEEE

In this paper we propose to solve CP problem by minimizing the system losses subject to the power flow balance equations, security and investment constraints of the fixed capacitor banks using OO and network reconfiguration (NR). The remainder of this paper is organized as follows. Section II presents the literature review. In section III we introduce the methodology for CP while the experimental results are compared with previously published ones in section IV. Finally section V concludes the study. II. LITERATURE REVIEW The different algorithms used for CP can be classified into four categories: analytical, numerical programming, heuristics, and artificial intelligence (AI) based. Due to the unavailability of powerful computing resources, analytical solutions such as in [1-3] were the first to be introduced. The objective of these studies was to develop a simple mathematical model with the following assumptions: no voltage regulation, uniformly distributed loads along the feeders, equal size of capacitor banks at each location, and power losses only due to the reactive current component. Later researchers such as in [4] provided a more realistic formulation where the switched and fixed capacitors as well as the load duration curve to account for the time variation of the reactive load were modeled. However their algorithm assumed a predefined number of fixed and switched capacitors and their relative placement with respect to each other along the feeder. In general, the major disadvantage of all the analytical methods is the fact that the capacitor sizes and locations were assumed to take continuous values which implied that the final results had to be rounded to the nearest value thus impacting the calculated voltages and loss reductions. As computer power and memory became larger and less expensive, numerical methods gained more interestReference [5] was the first to propose a numerical approach for the capacitor allocation problem using an objective function with no capacitor cost but discrete capacitor sizes. Extending the work of [5,6] included the released kVA into the objective function. Reference [7] used the method of local variations and took into consideration the effects of load growth, and switched capacitors for varying load. Similarly, [8] formulated the CP problem using mixed integer programming and [9] employed integer quadratic programming to coordinate the optimal

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operation of capacitors and regulators in a distribution system. The main disadvantage of the numerical methods is the possibility of obtaining a local optimum instead of a global one. Researchers in [10] presented a heuristic technique for reactive loss reduction in distribution networks. Capacitors were allocated in a two stage manner at certain sensitive nodes. First nodes were selected by identifying a section in the distribution system which has the largest losses due to the reactive power and second the node therein which has the greatest effect on the system loss reduction was identified as the sensitive one. Improving on the previous work, [11] solved directly for the sensitive nodes that have the greatest impact on loss reduction for the entire distribution system. While the voltage constraints were met, the study was only limited to fixed capacitors. Another heuristic constructive algorithm (HCA) based on the sigmoid function sensitivities was proposed in [12]. The sigmoid function approximated the discrete nature of the capacitor placement decision, and to reduce the number of candidate buses, reactive support was considered on those bursars where the voltages are below nominal value and reactive power demand is high. With the growing popularity and availability of AI software packages, many studies investigated AI algorithms to solve the CP problem. In [13] an expert system using a two-stage artificial neural network (ANN) was proposed for the optimal control of switched capacitors installed on a small distribution system. One ANN predicted the load profile from a set of previous load values and a second one selected the optimal capacitor tap positions based on the load profiles provided by the first network. The genetic algorithm (GA) method was proposed in [14] to determine optimal capacitor sizes and locations. These sizes and locations were encoded into binary strings and then crossover was performed to generate new populations. Reference [15] also recommended GA for varying load conditions and added voltage constraints as a penalty term in the objective function. H. Ng et al. [16] modeled voltage and power loss indices of the distribution system nodes by membership functions while a fuzzy expert system containing a set of heuristic rules inferred the capacitor placement suitability index of each node. The OO technique was proposed in [17] for solving the CP problem in transmission systems subject to the investment constraint on reactive power sources, the satisfaction of the power flow balance equations, and the security constraints. For simplicity, only the resource of fixed capacitor banks was considered. The OO technique was tested on IEEE 118-bus system and when compared to GA and tabu search, results showed that OO consumed 200 times less CPU time and gave 10% more reduction in power losses. Differently from [17], in this paper we are investigating the effectiveness of OO for CP in radial distribution network and combining it to our earlier work on NR [18] to further reduce power losses. III. OO FOR CAPACITOR PLACEMENT The OO method, a relatively recent method capable of handling enormously large search space problems with uncertainties, narrows the search for optimum performance by constructing a selected set S that contains good enough designs

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with high probability [19]. OO relies on soft computing principles: it is based on two tenets stating that the optimization of complex problems can be made much easier by order comparison and goal softening. Order comparison means that "order" is much more robust against estimation noise as compared to "value". That is, it is much easier to estimate whether one design is better than another rather than to estimate the difference in performance of the two designs. Goal softening means that for many practical problems it is enough to settle for a "good enough" solution instead of insisting on getting the "best". The procedure for applying OO to complex optimization problems can be summarized as follows: 1) Uniformly or heuristically sample N designs from the entire space. 2) Estimate the performance of these designs using a crude and computationally fast model. 3) Estimate the error level in the crude model as large, moderate, or small. 4) Order and plot the OPC based on the crude model evaluation. 5) Choose the size of the good enough subset, the required alignment level k, and the corresponding alignment probability (AP). 6) Based on the choices in steps 3-5, use the universal alignment probability table [20] to determine the size of the selected subset s. 7) Select the estimated top-s designs from step 2 to form the selected subset, S. 8) Evaluate the designs in S using an accurate model to determine the good enough solution by picking the best solution from the set S. OO theory ensures that S contains at least k truly good enough designs with a probability level no less than AP. In our CP study, the objective function is to minimize the system losses while satisfying the total investment constraint on the capacitor banks to be installed as well as meeting the power flow balance equations, the voltage and power constraints. If x represents the vector of continuous variables consisting of voltage magnitudes, angles, generators’ real and reactive powers and c the vector of discrete variables showing the size of the capacitor bank and the bus at which it is installed, then the optimal solution would consist of solving for x and c that minimizes the objective function. Assuming that: x J is a set that contains the indices of the candidate buses for installing capacitors. x

E j is an integer variable which takes a value of one if a capacitor bank was installed at bus j and zero if capacitor bank was not installed at bus j.

x

cmax denotes the maximum size of the capacitor bank.

x

f(x) is the objective function which represents the summation of real and reactive power injections at each generator.

x

g(x) represents the power balance equations.

x

p0 represents the fixed cost of installing a capacitor bank.

x

p1 is the cost per the size of the bank.

x

I is the total available investment value.

We followed the same six general steps proposed by [17] for applying OO to the CP problem and which are: Step 1: Find candidate buses. Step 2: Reduce the number of candidate buses. Step 3: Select set N=1000. Step 4: Choose s good enough solutions. Step 5: Apply two-stage OO. Step 6: Perform exact optimal power flow (OPF).

C. Step 3 From step 2 the optimal continuous values of capacitors to be installed were obtained. However since in practice capacitor sizes are discrete, we gave for every continuous solution ܿ௝௖ two approximated values one corresponding to the neighboring lefthand side discrete value ሺܿ௝ି ) and one corresponding to the neighboring right-hand side discrete value ሺܿ௝ା ). Thus by taking two possible values for every continuous variable ܿ௝௖ there will |J new |

A. Step 1 We first assume that one capacitor bank is installed at every bus of the network and without considering the investment constraint, the optimization problem can be directly solved using the OPF function from MATPOWER [21]. Once the OPF is solved, we select the indices of the candidate buses at which the objective function f is mostly decreased. These indices are chosen according to the values of Lagrange multipliers. If we assume Ȝ to be the Lagrange multiplier corresponding to the equality constraints, then the sensitivity theorem [22] states that the deviation in the objective function 䌛c f ( x ) can be estimated as follows: 䌛c f ( x ) = 䌛c g ( x, c ). Ȝ

solution will return continuous values of ܿ௝ . Denoting these continuous capacitor values by ܿ௝௖ the empirical threshold value r that is determined by the system planner can be used in order to reduce the size of Jnew such that if the efficiency factor of the p1c cj resource utilization ˺r , we set ȕj=0 otherwise set ȕj=1. p0 The set Jnew is updated by removing all the indices at which ȕj p1c cj was set to zero. This process should be repeated until ˻r p0 for all values of ݆ in the most recent Jnew [17].

be a total of 2 possible combinations of discrete solutions; among this space of solutions we will have to choose N=1000 samples since this is a constraint imposed by OO tables [19]. Let ‫ ݔ‬௖ and ߣ௖ denote the optimal solution of the continuous variable ‫ ݔ‬and the optimal Lagrange multiplier for the equality constraints respectively. Based on the sensitivity theorem, the deviation of optimal objective value due to the deviation of 'ccj c j  ccj or 'ccj c j  ccj is given by:

ǻf (䌛c cj ) ˷䌛c g ( x c , c cj ). Ȝc .ǻc cj j

(2)

min('f (c j  ccj ), 'f (cj  ccj )) and ranking

Setting 'f (c cj )

the values of ǻf (c cj ) from least to greatest such that the first

(1)

Thus, the larger values of 䌛c f ( x ) in the negative sense imply that increasing the capacitor value from the initial value c at this bus will result with reduction in the objective function. This infers that installing capacitors at buses with high ’ c f ( x ) will more effectively reduce the system’s losses. We then ordered the bus indices according to decreasing order of sensitivity such that if Į1, Į2, Į3 ..., Įn are the ordered indices Į1 corresponds to the index of the first candidate bus at which a capacitor will be installed. Once the indices are ordered, the number of how many one bank capacitors should be installed given the investment value needs to be identified. If t is the total number of candidate buses at which capacitors can be installed while satisfying the budget constraint, then the new set of candidate buses becomes Jnew= {Į1,Į2,Į3,…, Įt} [17]. B. Step 2 Step 1 reduced the number of candidate buses from n to t. If the value of t is still large, step 2 can be implemented to further reduce the set Jnew by determining the most effective buses using the empirical threshold value. Here we will assume ܿ௝ to be a continuous variable and again solve the optimization problem using the OPF function from MATPOWER [21]. The

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value has the highest order, the first m values of c cj are set to one discrete value such that to obtain 1000 samples of the set Jnew, and the rest c cj s , which are in total | J new | m , can have two possible values ሺܿ௝ା ) and ሺܿ௝ି ). Thus starting from the highest order of ܿ௝௖ until reaching the mth one we set:

c cj

c j

if

( 'f ( c j  c cj ) d 'f ( c j  c cj ))

c cj

c j

if

( 'f ( c j  c cj ) t 'f (c j  c cj ))

All the chosen 2|Jnew|-m discrete solutions are verified to satisfy the budget constraint. The set of discrete capacitor values that does not meet the budget is excluded, and if the set of feasible solutions become less than 1000 we would have to pick a lower value of m in order to allow two discrete values of ܿ௝௖ . The process of checking the budget is repeated again until 1000 possible sets of solutions which are feasible and satisfy the constraints are found. A set of one possible discrete solution of capacitor values at all the candidate buses is denoted by C(Ji) where the size of set C is equal to the number

of candidate buses determined in step 2 and i=1, 2,…, 1000. Hence this step reduces the huge search space down to 1000 feasible solutions [17].

their corresponding objective function value from the least to the greatest. Then we pick out the top s’=3 solutions that constitutes the new estimated good enough solutions [17].

D. Step 4 Now that we obtained N=1000 samples: C(Ji), i=1, 2… 1000, in this step we have to select s good enough solutions out of 1000. This is performed by using a crude model which estimates the deviation of the optimal objective function due to the deviation of the discrete solution form the optimal continuous solution. The crude model that will be implemented is an extension to the sensitivity model that was used before, but here instead of finding the deviation due to change in one capacitor value from continuous to discrete, we will find the deviation due to the change of the vector of capacitor values C(Ji) from the optimal continuous to discrete values as shown in (3).

F. Step 6 From the previous step we obtained s’ estimated good enough solutions, that we denote by C(Jl), l=1,2,…,s’. Now these s’ discrete solutions will be exactly evaluated by solving the OPF problem function from MATPOWER [21] s’ times and the solution corresponding to the minimum objective function is chosen as the good enough solution for the CP problem [17].

ǻf (ǻCc (Ji )) ˷[䌛C g(xc , Cc (J )).Ȝc ]T [C(Ji ) - Cc (J )]

(3)

IV.

COMPARISON AND DISCUSSION OF RESULTS

We ran OO in MATLAB 7.1 on Intel (R) core i5 CPU with 8 GB RAM and tested on the systems of 33-node [24] and 69node [25]. A. 33-Node System This test system is a 12.66 kV network with 33 nodes, 37 lines and only one energy input at node 33 which is considered to be the slack node. The system consists of 5 tie lines and sectionalizing switches on every branch of the system. The complete system data is given in [24]. The total system loads are 3715 kW and 2300 kVAr. The real power loss for the initial reconfiguration is 204.14 kW.

Similarly as before, we order the C(Ji), i=1, 2… 1000 in such a way that the sample with the lower value of | ǻf |, that is the one having least sensitivity to the optimal objective value will have a higher rank. Finally set S is selected form the top s ranked samples. By taking AP=0.95, k=1 and g=50, and a steep OPC with uniform large noise distribution [-2.5, 2.5], we found s=50 from the table given in [19]. E. Step 5 Although the huge size of the search space was reduced to s=50, but for large networks it is still computationally exhausting and time consuming to exactly evaluate the objective function of these s discrete solutions, since it requires solving the constrained non-linear optimization problem s times. Thus, a two-stage OO is applied in order to further reduce the search space. The selected subset s will be treated as the sampled set N and again by using a crude model, s’ good enough solutions from the set s are to be picked. Now it is important to determine the size of s’ solutions that should be chosen to ensure that s’ contains good enough solutions with very high probability. In this stage, as mentioned before s is considered to be the sampled set N, but since the size of s here is only around 50, it is far too small to apply the formula provided in [20]. Therefore in order to estimate the value of s’ a similar Monte Carlo study as the one in [20] was performed in [23], and for and AP= 0.99, it was found that the number of estimated good enough solutions s’ should be equal to 3 in order to include at least one actual good enough solution (in the top 5%) with a probability of 0.99.

OO method for capacitor allocation is applied to this 33node network. Practical sizes available for the switched capacitors are 150, 300, 450, 600 and 900 kVAr. For comparison purposes, three different configurations, including the initial configuration, were considered and the CP problem is solved for each of these configurations. The fixed installation cost of a capacitor bank p0 $2.5 u 105 and the cost per kVAr: p1=21 $/ kVAr [17]. The total installation cost is considered to be $500000. The results obtained are summarized in Table I. It can be observed that by installing capacitors the power losses are greatly reduced. In Table II, OO is compared with the Ant Colony Algorithm (ACA) [26], and for comparison purpose we limit the number of possible capacitors to three. It is obvious that OO gave better results with a fast execution time of an average of five seconds. B. 69-Node System This test system is also a 12.66- kV network with 69 nodes, and 7 tie lines and sectionalizing switches on every branch of the system. The complete system data is given in [25]. The total system loads are 1107.90 kW and 897.93 kVAr. Similarly to the 33-node system, capacitor allocation using OO method is performed for the 69-node system. The capacitors’ sizes that were considered are 50, 100, 200, 300, 400 kVAr and the total installation cost available is $ 300000 with the same capacitor costs as before. Since we are comparing results with [27], we consider the configuration having the following switches opened: 15-58-62-70 and 71. Obtained results are summarized in Table III and show that the reduction in power losses using OO for CP is superior. It

The crude model proposed for solving the power flow for these s samples is the fast decoupled method available in MATPOWER [21]. The budget constraint is not considered in the crude model since all the picked s solutions were checked to satisfy this constraint in the previous step. Let the set of the top s discrete solution be denoted C(Jr), where r=1, 2,…, 50. Once all the s samples are solved using the crude model, these C(Jr) should be ranked according to

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should be noted that OO method gave better results even with a smaller number of capacitors as compared with [27].

TABLE I.

RESULTS FOR THE 33-NODE SYSTEM Case 1

Case 2

x

kVAr

Node

kVAr

Node

kVAr

9

600

20

600

9

600

28

450

28

450

28

450

29

600

29

600

29

600

Power losses in kW

136.14

95.79

98.29

Node

kVAr

Node

kVAr

Node

kVAr

23

600

7

450

7

450

24

450

29

600

29

600

29

600

31

300

31

300

Capacitors added using OO (in kVAr) Power losses in kW

132.61

TABLE II.

85.21

85.59

COMPARISON OF OO WITH ACA[26] Case 1

Case 2

Case 3

Configuration

33-34-35-36-37

7-9-14-32-37

7-10-14-32-37

Power losses without capacitors

204.14

139.6

140.3

Capacitors added using OO (in kVAr)

Node

kVAr

Node

kVAr

6

600

7

450

7

450

23

300

23

300

23

300

24

150

24

300

24

300

29

600

29

600

29

600

31

300

31

300

31

300

Power losses using OO in kW

117.6

TABLE III.

Capacitors added in kVAr 8.88

Node

81.81

Network reconfiguration and capacitor placement was also applied on the 69-node system. The results are summarized in Table V, and the power losses are compared to the different cases. From the results we conclude that considering NR and CP jointly greatly reduces the power loss.

kVAr

V. CONCLUSION In this paper, we proposed OO technique for optimal CP and NR. The implemented algorithm selects the most effective candidate buses using the sensitivity theorem, then by solving for continuous capacitors values, every continuous solution is given two approximated discrete values and the values which cause the smallest deviation in the objective function are chosen in such a way to obtain a total of 1000 samples. Then a two-stage OO is performed and the three samples from the final selected subset are evaluated using an exact model. The results obtained showed superior power loss reduction as compared to other published results thus encouraging further investigation of OO method for CP. Results for integrating NR with CP appeared as a very powerful since the power loss reduction was greater than what could be achieved using CP or NR approaches alone.

82.91

RESULTS FOR THE 69- NODE SYSTEM

Without Cap

Power losses kW

Generating all the possible spanning trees that can be obtained from the given system. x Uniformly sample N=1000 configurations from the entire search space. x Evaluate these N configurations using the crude Bmatrix loss formula. x Order these N configurations based on the power losses obtained from least to greatest. x Pick the top s =53 configurations out of the ordered N samples. x Solve the capacitor allocation problem for the s configurations using the OO approach. x Evaluate these s configurations with the capacitors installed using an exact power flow analysis. x Pick the optimal solution for NR and CP to be the one that gave the minimum power losses out of the selected set s. The results of the proposed algorithm on the 33-node system are summarized in Table IV, where the power losses are compared before reconfiguration, with reconfiguration only, with capacitors only and with the combined problem.

Case 3

Node

Capacitors added using ACA [26] (in kVAr)

OO

[27]

Node

kVAr

Node

kVAr

9 and 49 48 60 63

200 50 300 100

7,11 and 15 48 and 49 60 63

100 100 300 100

5.81

picked using the crude model based on the B-matrix as discussed in [18]. Then the capacitor allocation problem is solved for the s samples using the OO technique discussed in section 3. The complete procedure of the joint problem is as listed below:

TABLE IV.

POWER LOSSES FOR THE COMBINED PROBLEM OF THE 33NODE SYSTEM

5.95 Initial Configuration Initial configuration and capacitor addition Reconfiguration without capacitor [18]

C. NR and CP We implemented both NR and CP jointly such that first the space of all configurations is generated, and the selected set s is

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Opened Switches

Power Losses

33-34-35-36-37

204.14

33-34-35-36-37

117.6

7-11-14-28-36

126.0

Reconfiguration with capacitor placement TABLE V.

Opened Switches

Power Losses

7-11-14-28-36

80.35

POWER LOSSES FOR THE COMBINED PROBLEM OF THE 69NODE SYSTEM Opened Switches

Initial Configuration Initial configuration and capacitor addition Reconfiguration without capacitor [18] Reconfiguration with capacitor placement

Power Losses

70-71-72-73-74

20.88

70-71-72-73-75

12.01

13-54-63-70-71

8.72

9-18-27-56-70

5.9

ACKNOWLEDGMENT The work was supported by the grant titled “Ordinal Optimization Formulation for Distribution Network Reconfiguration” that was provided by the Lebanese National Center for Scientific Research (LNCSR). REFERENCES [1]

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[9]

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