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Abstract-- This paper presents a new formulation including

robustness of solution of constrained multiobjective design or reactive power compensation. The algorithm used for optimization is the NSGA-II (Non dominated Sorting Genetic Algorithm II) with a special crowded comparison operator for constraints handling. The need for including the issue of robustness of solutions derives from the simple observation that loads are uncertain in distribution systems and their estimation is often affected by errors. In design problems it is desirable to consider the loads with a certain range of variation. In this paper the NSGA-II algorithm is applied to efficiently solve the issue and the solutions attained considering stochastic loads are compared to those considering precise loads.

Index Terms—Optimal reactive power design, Multiobjective optimization, robust optimization, distribution systems.

I. INTRODUCTION HE issue of the optimal design of the compensation

system for distribution networks has been variously formulated. Most approaches deal with it using combinatorial or mixed-integer optimization algorithms. Optimal capacitor sizing and placement is an interesting issue for utilities since it results in losses reduction, voltage profile and power quality improvement. Some technical constraints have also been considered in order to increase the lifetime of capacitor banks and to improve the quality of the energy, by the limitation of the number of maneuvers and the voltage profile flattening. The problem of optimal capacitor placement in the literature has been extensively treated. More recently, new formulations or more efficient methods have been proposed to solve it. Many papers present solution techniques based on sub-symbolic Artificial Intelligence approaches [1],[2],[3]. Most of the papers in literature formulate the optimal compensation design as a minimum cost problem (losses and installation) with technical constraints [4],[5],[6]. In [4], the Authors use a hybrid approach drawn upon the Tabu Search approach, extended with features taken from other heuristic approaches. In [5], Linear Programming is used to evaluate the objective function of the different solutions of the Genetic

A. Augugliaro, L. Dusonchet, E. Riva Sanseverino, S. Favuzza, M. G. Ippolito are with Dipartimento di Ingegneria Elettrica, Elettronica e delle Telecomunicazioni, Università di Palermo, PALERMO, ITALIA (e-mail: augugliaro,dusonchet,eriva,favuzza,ippolito@diepa.unipa.it).

Algorithm. In [6], the Author uses an Immune algorithm to solve the capacitor placement problem. Many papers in the literature dealing with distribution systems optimal planning and automation do not account for uncertainties in measurements and input data. This issue may cause the identification of optimal solutions that are no more feasible or even optimal when these parameters variations are effective and of a certain entity. In [12] the effects of uncertain input data on the performance evaluation of a distribution system are analyzed. A criterion based on fuzzy logic is introduced for assessing the grade of uncertainty of the results obtained in the calculation of some technical and quality indices of a network. High uncertainty of some results obtained indicates that a re-examination of relevant uncertain input data would be recommendable for a more precise quantification. Other authors [13] propose an algorithm for reactive power optimization with time-varying fuzzy load model (RPOTF). The methodology considers the uncertainties in load demands. Fuzzy Logic certainly offers a way to take into account uncertainty but requires a domain knowledge that allows the designer to draw the correct membership functions. Thus it is reasonable to examine or even formulate all the operational and planning problems in this view. In this paper, the formulation of the problem of optimal reactive design in distribution systems is multiobjective, since the first year maximum Return On Investment, ROI, is searched together with the objective of solutions robustness in terms of the stochastic loads variation. The second objective is considered as one of the main objectives in order to follow a formulation which is between those proposed in [9] and namely Multi-objective robust solution of type I and of type II. The solution methodology is a Multi Objective, MO, optimization Evolutionary approach, NSGA-II [10] suitably modified in order to treat constrained problems. The paper is organized as follows. First, the problem formulation is presented. Then, the multiobjective solution method using the Pareto optimality concept for handling constraints and robustness as optimization function is presented. Finally test results on the 15 nodes system in [11] are reported.

II. THE PROBLEM OF ROBUST MULTIOBJECTIVE COMPENSATION SYSTEM DESIGN

In this section the objectives considered for the optimal

Constrained Robust MultiObjective Optimization for Reactive Design in

Distribution Systems A. Augugliaro, L. Dusonchet, S. Favuzza, M.G. Ippolito, E. Riva Sanseverino, Member, IEEE

T

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compensation system design are described. The adopted formulation indeed considers together with an economic index of Return On Investment maximization and a technical index of optimal operation even an index considering solutions robustness with respect to possible loads variations. A. Objective 1- Return On Investment

The ROI index expressed the redditivity of a given investment. It must be calculated considering the difference between two different situations: the current one and the proposed one. The economic terms referred to one year and to be considered are: a) variable cost connected to the energy losses in the network

lines; b) variable cost connected to the energy losses in HV/MV

transformer; c) fixed cost connected to the peak power loss reflecting

released systems capacity. d) installation cost for all the capacitor banks.

For one year supplied loads variability must be considered as well as the capacitor banks insertion state.

The seasonal modulation has been considered as well as the daily variability. Therefore optimization is carried out on two typical days (winter and summer), hypothesising that loads have a stochastic variation of 10% on the considered seasonal and hourly value.

Therefore capacitor banks can change their status hourly during 24 hours per day. On the basis of these hypotheses, the terms a)-d) can be expressed as follows.

To express term a), calling ΔPt(h) power losses at hour h and considering the daily load diagram with an hourly modulation for each load node of the MV network, the power losses all over the network can be written as:

[ ]∑=

−+=Δnr

i

stepC

iCii

i

it QhkhQhP

VR

hP1

222

))()(())(()( (6)

where, k iC (h) is the integer value representing the insertion step of the capacitor bank at the ending node of the i-th branch at hour h, QC

step is the unity insertion step of each capacitor bank, Pi(h) and Qi(h) are the real and reactive power flows through the i-th branch, Ri is the resistance of the i-th branch and Vi is the voltage of the sending bus of the i-th branch; terms Pi(h) and Qi(h) are also made up of the loads and real and reactive losses below the i-th branch. The summation is extended to all the nr branches of the network. The energy lost in one year, considering that there is a seasonal modulation, is:

EtΔ = Hw* ∑=

Δ24,..1

)(h

hPt + Hs* ∑=

Δ24,..1

)(h

hPt

(7) Where Hw and Hs are the number of hours of winter and summer respectively.

The economic benefit deriving from the power losses reduction can be expressed as:

RΔEt=(CEt0 - CEt) = (ΔEt0- ΔEt)*CΔEt (8)

where ΔEt0 are the losses in absence of compensation, CEt0

and CEt respectively are the costs related to the energy losses

in the network without (k iC (h)=0, i=1,…nr) and with

compensation and CΔEt is the unity cost of the energy losses [€/kWh].

The coefficients of string kC (h) represent the working status (on/off) of all the steps of the capacitor banks at any hour of the day.

Term b) can be expressed using the expression of power losses in the transformer at hour h:

[ ]222 ))()(())(()( step

CTR

TR QhKhQhPVR

hPt −+=Δ (9)

where RTR is the series resistance of the transformer, V is the rated voltage of the MV network, K(h) the string of integer values expressing the insertion status of capacitor banks at the HV/MV transformer at hour h, P(h), Q(h) represent the real and reactive power required by the network (loads and losses), evaluated at the MV busbars of the transformer and QC

step is the unity insertion step of each capacitor bank. Both real and reactive power required by the MV network at the MV busbars keep into account the insertion state of all the capacitor banks in the network.

The power losses in the transformer in one year are:

TREtΔ = Hw* ∑=

Δ24,..1

)(h

TR hPt + Hs* ∑=

Δ24,..1

)(h

TR hPt (10)

Calling ΔEtTR0 the losses in absence of compensation

(setting the coefficients of the string K(h) and kC(h) to zero), the benefit deriving from the reduced value of energy losses in the HV/MV transformers is:

RΔEtTR=(CETR0 - CETR) = (ΔEtTR0- ΔEtTR)*CΔEt (11)

where CETR0 and CETR respectively are the costs related to the energy losses in the transformer without and with compensation.

Term c) can be considered using the following expressions for the peak power loss in absence of compensation,

0peakPtΔ , and in presence of compensation, peakPtΔ :

0peakPtΔ = .)(max,..2,1;0,..2,1;024...1,24..1 ====== Δ

iKikhwhs iC

ihPt. (12)

peakPtΔ = )(max 24...1,24...1 hPthwhs Δ== . (13)

The economic benefit deriving from the release of system

capacity can be computed using the following expression:

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Rpeak=(Cpeak0 - Cpeak) = ( 0peakPtΔ - peakPtΔ )*CΔpeak (14)

Where Cpeak0 and Cpeak respectively are the costs related to the peak power loss in absence and in presence of compensation and CΔpeak is the unity cost of peak power loss. Finally the term of proposition d) can be attained in the following way. Capacitors are grouped into banks of standard discrete capacities (e.g. 5 to 300 kVAR); therefore it is realistic to assume that capacitor sizes are regarded as integer multiples of the standard size of one bank. To describe the cost of capacitors placement, the cost function can be formulated as follows:

)*(1

stepC

jCinstalled

Sw

N

jInstInstT Q

QKKCC

+= ∑=

(15)

where KInst is a constant representing a fixed installation cost of capacitor banks at the j-th bus, Ksw is a cost per bank of switchable capacitors installation, the ratio step

Cj

Cinstalled QQ /

gives the number of modules of size stepCQ constituting the j-th

installation and NC is the number of compensated buses in the network.

The economic effects of the compensation executed over the distribution system can be expressed using the expressions written above.

The ROI index here considered expresses the ratio Operational Revenue over net invested capital. The Operational Revenue is given by the difference between sum of the revenues and the total costs per year, Cyear

InstT, before the interest rate; therefore, in the considered case it will be the (8)+(11)+(14) minus the (15) per year, InstT

yearC . The invested capital InstTC is instead expressed by (15).

B. Objective 2- Voltage drop

The second objective here considered is the limitation of voltage drop. This objective is actually not totally concurrent with the minimization of operating costs in the Return On Investment, even though the latter also considers installation costs.

C. Constraint. Number of manoeuvres

In this formulation the only considered constraint concerns the limitation of the number of manoeuvres, nman, on the capacitor banks along the considered timeframe.

Therefore the constraint is: g1(x) = nman(x) ≤ max_man (16)

where x ∈ X is a mixed-integer vector representing the optimization variables string.

III. ROBUSTNESS IN CONSTRAINED MULTI-CRITERIA OPTIMAL DESIGN OF COMPENSATION

In [8][9] Deb and Gupta propose an approach to treat

robust multicriteria optimization both for unconstrained and for constrained problems. The problem of the exact loads estimation in distribution systems is actually one of the main difficulties when treating planning or operational problems and this issue can significantly affect solutions and their validity. In this paper the problem of optimal reactive power planning has been solved using the definition of Multiobjective robust solution of type I that can be found in [9]. It must be observed that in this case robustness is referred to some parameters (real and reactive loads) influencing the objective function. §In this paper the notion of Multiobjective robust solution of type I and II have been extended to the case of parametric shifts. A solution is called a multi-objective robust solution of type I if it is global feasible Pareto-optimal solution to the following multi-objective minimization problem:

Minimize (f1eff(x,p),…fMeff(x,p)) Subject to x∈S, for all pc∈Bδ(p) (17) where Bδ(p) is a δ neighbourhood of a parameter p,

fjeff(x,p) is defined as follows:

fjeff(x,p) = ∫

∈ )(),(

)(1

pBpcj dppxf

pB δδ

(18)

In general, the robustness of the constrained solutions is

ensured by condition (17), for which it is required that all the solutions used to compute the efficient objective function that are in a δ neighbourhood of a solution x are also feasible. In this case, the feasibility of solutions is not affected by the parametric variations since constraints about the number of manoeuvres are not affected by the considered parameters.

Furthermore, in [10] the authors show that it is easily possible to calculate the approximate effective objective values (18). In this paper the same procedure has been adapted to parametric shifts. The procedure starts choosing and calculating the objective function values in H different points in the δ-neighbourhood of p. In practice the perturbation domain of each parameter (-δ, δ) is divided into exactly H equal grids thereby dividing the δ-neighbourhood into nH hyperboxes. Thereafter H hyperboxes are picked randomly from nH so that in each dimension H different distinct values are represented.

The constraint handling strategy is incorporated into the NSGA-II using the constraint domination principle [AA] in the following way.

The formulation here considered is: Objective 1: Max(ROIeff) = Max

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −++ ΔΔ

),(),(),(R),(R),(R peakEtTREt

)( pxCpxCpxpxpx

averageInstT

InstTyear

pBδ

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Objective 2:

Min(ΔVxeff) = Min [ ]⎭⎬⎫

⎩⎨⎧

Δ ),(Vx)(

pxaveragepBδ

Constraint 1: Min(α1(g(x)) = Min{ }manxnmanba max)((* −+ The latter constraint do not depend on p and is the effective values are the same as the actual values.

It must be observed however that in this case the value of constraints 1 and 2 are not affected by the considered parametric variations.

In general, of course constraints can be affected by parametric variations and thus the formulation should be completed by including the effective formulation of constraints and an additional constraints expressing condition (17). This paper thus takes into account the issue of robustness in constrained multiobjective optimization but with reference to parameters, and not optimization variables, shift affecting the identification of the Pareto optimal set of solutions. It is thus slightly different as compared to the numerical optimization examples proposed both in [8] and [9], where the

issue of robustness is considered only with reference to the optimization variables.

IV. APPLICATION A test system of 15 nodes [11] has been considered for the

application. Some runs have been carried out with different values of the perturbation domain and of the number of representative samples H. The loading factors have a seasonal modulation. The elementary step of switchable capacitor banks is of 10 kVAR.

For all the runs of the NSGAII the population size is 70 and the number of generations 2000.

The following runs have been executed: - with fixed δ and variable H - with fixed H and variable δ First, δ has been fixed to 0.1 (loads variation = 10%). The

results are shown in figure 1. As it can be noted there is a difference between H=70 and H=150, because in the latter case the approximation of the effective front is more precise.

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Dvxeff [%]

-ROIeff

H=70H=100H=150

Figure 1 – Executions with fixed δ=0.1 (in the figure d) and variable H

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-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Dvxeff [%]

-ROIeff

delta=0delta=0.05delta=0.1delta=0.2

Figure 2 – Effective value of ROI versus Effective value of maximum voltage drop. Executions with variable δ and fixed H=100

Then some runs with H=100 and different values of δ

(figure 2) have been carried out and it can be observed that some values of ROI that the utility could presume from the solution of a traditional design problem are actually not realistic since the loads may have an actually unpredictable variation.

The two design solutions, reported in Table I, provide the same maximum voltage drop (5.5%) and different ROI index, respectively 10% and 38% for δ=0 and δ=0.1. It can be observed that they are different both in terms of size and of locations of the capacitor banks (differences are evidenced in bold). If the planner considers a small uncertainty about the loads values, he might expect higher Returns On Investments (four times the value that can be actually obtained with larger dispersions) and thus experience financial problems in the middle term.

TABLE I – Installations for the two design solutions attained with H=100

and different loads perturbations ID δ=0 δ=0.1 1 0 0 2 160 50 3 0 0 4 0 0 5 0 0 6 0 0 7 0 3 8 0 0 9 30 0 10 0 30 11 0 0 12 0 0 13 0 0 14 0 0

V. CONCLUSIONS This paper shows the importance of considering uncertain loads in design problems for distribution systems. In particular, the problem of optimal reactive power design has been here studied. The problem is multiobjective and constrained and thus the Non Dominated Sorting II algorithm has been used for its solution. The item of robustness of solutions with respect to parametric variations has been considered using an efficient methodology proposed by Deb et al.. The results show the importance of considering the issue of robustness in multiobjective and constrained optimization through the analysis of the different attained reactive design solutions. Further research will be addressed towards the investigation of the problem of robustness in other real world applications also with reference with optimization parameters and the analysis of different approaches towards the same problem.

VI. REFERENCES [1] H. Mori, Y. Ogita, “Parallel Tabu Search for capacitor placement in

radial distribution systems”, Proc. 2000 IEEE Power Engineering Society Winter Meeting, vol.4 , 2000 January 23-27 pp.:2334 – 2339.

[2] Ching-Tzong Su, Chu-Sheng Lee, “Modified differential evolution method for capacitor placement of distribution systems”, Transmission and Distribution Conference and Exhibition 2002: Asia Pacific. IEEE/PES ,vol.: 1 , 2002, October 6-10, pp.:208 – 213.

[3] Xin-Mei Yu, Xin-Yin Xiong, Yao-Wu Wu, “A PSO based approach to optimal capacitor placement with harmonic distortion consideration”, Electric Power Systems Research, vol.71, pp.27-33, 2004.

[4] Delfanti, M.; Granelli, G.P.; Marannino, P.; Montagna, M.;"Optimal capacitor placement using deterministic and genetic algorithms" IEEE Trans. on Power Systems, Volume: 15 Issue: 3 , Aug. 2000 , pp. 1041 -1046.

[5] Shyh-Jier Huang; "An immune-based optimization method to capacitor placement in a radial distribution system" IEEE Trans. on Power Delivery, Volume: 15 Issue: 2 , April 2000 , pp. 744 -749.

[6] Optimal Capacitor placement in radial distribution networks; R. A. Gallego, A. J. Monticelli, R. Romero, IEEE Trans. On Power Systems, vol. 16, n.4 Nov. 2001

[7] G.B. Jasmon, L.H.C.C Lee, “Maximizing voltage stability in distribution networks via loss minimization”, International Journal on Electrical Power and Energy Systems, vol. 13, no. 3, pp. 148-152, June 1991.

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[8] K. Deb, H. Gupta Searching for robust pareto-optimal solutions in multiobjective optimization. In Evolutionary Multi-criteria optimization, 2005.

[9] K. Deb, H. Gupta (January, 2005). A constraint handling strategy for robust multi-criterion optimization. KanGAL Report No. 2005001.

[10] K. Deb, S. Agrawal, A. Pratap, T Meyarivan, “A Fast Elitist Non-Dominated Sorting Genetic Algorithm for Multi-Objective Optimization: NSGA-II”, Proc. 2000 Parallel Problem Solving from Nature PPSN VI Conference, September 18-20, Paris, France.

[11] D. Das, H. S. Nagi, A. Kalam - "Novel method for solving radial distribution networks", IEE Proc. - Gener. Transm. Distrib., vol. 141, no. 4, July 1994, pp. 291-298.

[12] Nahman, J.; Peric, D. “Distribution system performance evaluation accounting for data uncertainty”; Power Delivery, IEEE Transactions on Volume 18, Issue 3, July 2003 Page(s):694 – 700

[13] Changcheng Zhao; Youman Deng; Xiaojuan Ren Reactive power optimization with time-varying fuzzy load demand for distribution systems; Power System Technology, 2002. Proceedings. PowerCon 2002. International Conference, Volume 4, 13-17 Oct. 2002 Page(s):2326 - 2330 vol.4

VII. BIOGRAPHIES Antonino Augugliaro (1949) received the Doctor’s degree in Electrical Engineering from the University of Palermo, Italy, in 1975. Since 1978 to 1994 he has been Associate Professor and now he is Full Professor of Electrical Power Generation Plants at the Faculty of Engineering of the University of Palermo. His main research interests are in the following fields: simulation of electrical power system; transmission over long distances; mixed three-phase/six-phase power system analysis; optimisation methods in electrical distribution system’s design and operation; distribution automation. (Dipartimento di Ingegneria Elettrica, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italy, T +39 91/6615241, Fax +39 91/488452).

Luigi Dusonchet (1948) received the Doctor’s degree in Electrical Engineering from the University of Palermo, Italy, in 1975. Since 1978 to 1990 he has been Associate Professor and now he is Full Professor of Industrial Electrical Systems at the Faculty of Engineering of the University of Palermo. His main research interests are in the following fields: simulation of electrical power system; transmission over long distances; mixed three-phase/six-phase power system analysis; optimisation methods in electrical distribution system’s design and operation; distribution automation. (Dipartimento di Ingegneria Elettrica, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italy, T +39 91/6615242, Fax +39 91/488452)

Salvatore Favuzza (1969) received the Doctor’s degree in Electrical Engineering in 1996 from the University of Palermo (Italy) and the Ph.D. in Electrical Engineering in February 2000 from the same University. Since October 2002 he is a researcher at the Department of Electrical Engineering of University of Palermo. His main research interests are in the areas of applications of artificial intelligence techniques to the control of electrical power systems, power systems analysis, optimal planning, design and control of electrical distribution systems. (Dipartimento di Ingegneria Elettrica, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italia, T +390916615221, Fax +39091488452) Eleonora Riva Sanseverino (1971) received the Doctor’s degree in Electrical Engineering from the University of Palermo, Italy, in 1995. Since 1995 she’s been working in the Research Group of Electrical Power Systems. In February 2000 she’s successfully finished her Ph. D in Electrical Engineering at the same University. Afterwards she’s had a researcher position at the Dept. of Electrical Engineering in Palermo. From December 2001 till October 2002 she’s been working as a researcher at the National Council of Research in the field of Computer Systems and Computer Networks. Now she is Associate Professor in Power Systems at the University of Palermo. Her main research interest is in the field of optimization methods for electrical distribution systems design, operation and planning. (Dipartimento di Ingegneria Elettrica, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italia, T +390916615262, Fax +39091488452) Mariano Giuseppe Ippolito (1965) received the Doctor degree in Electrical Engineering in 1990 and the Ph.D. in 1994 from the University of Palermo (Italy). Since 1995 to 2001, he is employed as a Researcher at the Department

of Electrical Engineering in Palermo. Now he is Associate Professor of Power Systems at the Faculty of Engineering of the University of Palermo, Italy. His main research interests are in the areas of power systems analysis, optimal planning, design and control of electrical distribution system. He also made some contributions to the advancement of power quality study. (Dipartimento di Ingegneria Elettrica, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italia, T +39 0916615204, Fax+39091488452)

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