A novel multi-zone reactive power market settlement model: A pareto-optimization approach

Energy 51 (2013) 85e100

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A novel multi-zone reactive power market settlement model: A pareto-optimization approach Amit Saraswat, Ashish Saini*, Ajay Kumar Saxena Department of Electrical Engineering, Faculty of Engineering, Dayalbagh Educational Institute, Agra 282110, Uttar Pradesh, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 June 2012 Received in revised form 10 December 2012 Accepted 11 December 2012 Available online 29 January 2013

This paper presents a Pareto-optimization based zonal day-ahead reactive power market settlement model named as multi-zone DA-RPMS model. Three competing objective functions such as Total Payment Function (TPF) for reactive power support services from generators/synchronous condensers, Total Real Transmission Loss (TRTL) and Voltage Stability Enhancement Index (VSEI) are optimized simultaneously by satisfying various power system operating constraints while settling the day-ahead reactive power market. The proposed multi-zone DA-RPMS model is tested and compared with singlezone DA-RPMS model on standard IEEE 24 bus reliability test system. A Hybrid Fuzzy Multi-Objective Evolutionary Algorithm (HFMOEA) approach is applied and compared with NSGA-II for solving these DA-RPMS models in competitive electricity market environment. Further, both the single-zone and multi-zone DA-RPMS models are also analyzed on the basis of market power owned by any generator/any generating company. The simulation results obtained confirm the superiority of HFMOEA in finding the better Pareto-optimal fronts in order to take better day-ahead reactive power market settlement decisions.
2012 Elsevier Ltd. All rights reserved.

Keywords: Day-ahead competitive electricity market Hybrid fuzzy evolutionary algorithm Market power Multi-objective optimization Pareto-optimal front Zonal reactive power market settlement

1. Introduction Of late, an appropriate reactive power provision has been one of the major concerns by the Independent System Operator (ISO) in order to maintain the reliable, economical and secure power system operations in deregulated environment. Unlike the real power, reactive power does not accomplish useful work (e.g., runs motors and lights lamps) but it is necessary to improve the capability to transfer bulk Alternating Current (AC) power over transmission lines. Moreover, it is responsible to establish and maintain electric and magnetic fields in ac equipment. Therefore, reactive power is not only necessary to operate the transmission system reliably, but it can also substantially improve the efficiency with which real power is delivered to customers. Increasing reactive power production at certain locations (usually near a load center) can sometimes alleviate transmission constraints and allow cheaper real power to be delivered into a load pocket. The detailed analysis of characteristics, its urgent needs and pricing of reactive power issues are presented in the report submitted by Federal Energy Regulatory Commission (FERC) [1]. This report also summarizes many conceptual aspects and current practices, points out various deficiencies in the reactive

* Corresponding author. Tel.: þ91 562 2801224; fax: þ91 562 2801226. E-mail address: [email protected] (A. Saini). 0360-5442/$ e see front matter
2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.12.009

power procurement in the US markets and provides recommendations for, and lists a number of challenges in the reactive power supply and its usage area. Furthermore, reactive power is tightly coupled with bus voltages throughout the power system, and hence it has a significant effect on system security. In fact, inadequate reactive power led to voltage collapses and has been a main cause of major power outages across the world in the past. Now, it has been a well established fact that there is a need of proper management of reactive power as one of the six ancillary services which must be provided through the competition in electricity markets [1]. The strongly local nature of reactive power restricts its ability to be transmitted over electrically large distances. More importantly, such characteristics imply that reactive power cannot be treated as a commodity of the same type as active power or active energy. Therefore, it renders the economics of reactive power and voltage support to be challenging and makes highly questionable the feasibility of setting up a workable market structure for reactive power provision. Transparent market processes and efficient market clearing mechanisms are needed for achieving optimal reactive power management in competitive electricity market. Hence, the literature review presented in subsequent paragraphs is focused on recent developments of reactive power market models and market clearing schemes along with their solution techniques. Initially, the research efforts were made in order to develop an optimal pricing scheme for reactive power provision using

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Nomenclature

Abbreviation TPF total power payment function ($) TRTL total real transmission loss (MW) VSEI voltage stability enhancement index (L-index) r0 uniform availability price ($/MVAr-h) r1 uniform cost of loss prices for absorbing reactive power ($/MVAr-h) r2 uniform cost of loss prices for producing reactive power ($/MVAr-h) r3 Uniform opportunity price ($/MVAr-h)/MVAr-h cost of availability price offer (in $) a0,i cost of loss component price offer for operating in m1,i under excited mode (absorb reactive power), Q Gmin,i  Q G,i  0 (in $/MVAr-h), cost of loss component price offer for operating in the m2,i region Q Gbase,i  Q G,i  Q GA,i (in $/MVAr-h) cost of lost opportunity price offer for operating in the m3,i region Q GA,i  Q G,i  Q GB,iin ($/MVAr-h)/MVAr-h W0,i, W1,i, W2,i and W3,i binary variables associated with ith generator Q G1,i, Q G2,i Q G3,i and Q A,i reactive power output of ith generator in the region (Q Gmin,0),(Q Gbase,QGA),(Q GA,Q GB) and(0,Q Gbase), respectively real power loss in kth transmission line Pk,loss PG,i,PD,i real power generation and demand at ith bus Q G,i,Q D,i reactive power generation and demand at ith bus Q Gmin,i,Q Gmax,i minimum, maximum limits of reactive power generation and demand

conventional marginal price theory [2e4] with having an assumption that all consumers should pay and all producers are remunerated for reactive power services. A market model process to manage reactive services by independent transmission operators is presented in [5]. It used a piece-wise linear representation of the capability curve of each generator for computing reactive power cost curves. In reference [6], a two-step approach for reactive power procurement is proposed. In first step, the marginal benefit of each reactive power bid with respect to total system losses is determined, and in second step, an OPF-based model maximizing a social welfare function is solved to determine the optimal reactive power procurement. This work is further extended in reference [7], where a uniform price auction model was proposed to competitively determine the prices for different components of reactive power services namely: availability, operation and opportunity. Market clearing was achieved by simultaneously minimizing of payment, total system losses, and deviations from contracted transactions using compromise programming approach. Mainly, a problem of market power (some of the reactive power producers misusing the situation by giving by extraordinary high prices of their services) may arise while establishing reactive power market for voltage control ancillary services [8]. This problem is caused due to local nature of reactive power and voltage phenomenon in electrical networks. Therefore, a need of effective design of localized/zonal reactive power market considering Voltage Control Areas (VCAs) is realized to overcome the same problem. In [8], a localized or zonal reactive power market is proposed using the concept of VCAs/zones in which the reactive power market is settled by calculating the zonal uniform market clearing prices.

shunt capacitor/indictor at ith bus Q C,i Q Cmin,i,Q Cmax,i minimum, maximum values of shunt capacitor/ inductor at ith bus power flow and its maximum value at lth transmission Sl,Smax l line conductance of kth transmission line gk transfer conductance between ith and jth bus (p.u.) Gij transfer susceptance between ith and jth bus (p.u.) Bij qij voltage angle difference between buses and (radian) voltage at ith bus (p.u.) Vi Vimin ; Vimax minimum, maximum limits of voltage at ith bus (p.u.) transformer tap setting at kth transmission line (p.u.) Tk Tkmin ; Tkmax minimum, maximum limits of transformer tap setting at kth transmission line (p.u.) total numbers of buses NB total of numbers of buses adjacent to ith bus, including Ni ith bus total number of generator buses NPV total number of load buses NPQ total number of transmission lines NL total number of transformer taps NT total number of shunt capacitors/inductors NC total number of VCAs/zones formed in the power NZ system terminal voltage of ith generator Vt,i steady state armature current of ith generator Ia,i armature e.m.f. generated of ith generator Eaf,i synchronous reactance of ith generator Xs,i rated real power output of ith generator PGR,i

However, seasonal market for reactive power encounters couple of problems [9]. Firstly, the reactive power consumption of system is volatile that its forecasting over a season becomes very hard. Secondly, the reactive power requirement of system strongly depends on the loading condition of network. Some of the recent publications [9e12] advocated a day-ahead reactive power market instead of long-term reactive market. In reference [10], a pay-as-bid based reactive power market clearing scheme is presented which implicitly considers the local nature of reactive power during the clearing of reactive power market. The uncertainty of generating units in the form of system contingencies is considered in the market clearing procedure by the stochastic model [11,12]. In all these reactive power market models, the Reactive Power Market Clearing (RPMC) problem is formulated as single objective optimization problems. In recent years, all real world optimization problems are being tried to be formulated in multi-objective optimization framework, in which multiple objective functions are optimized simultaneously. In fact, these objective functions are non-commensurable and often conflicting objectives. Multi-objective Optimization Problems (MOPs) with such conflicting objective functions give rise to a set of optimal solutions, instead of one optimal solution, called as Pareto-optimal solutions [13]. Therefore, in a multi-objective optimization framework, the main aim is to find out a set of feasible and non-dominating solutions which forms a Paretooptimal front within the entire search space. Many multiobjective reactive power optimization problems such as Optimal Reactive Power Dispatch (ORPD) [14e18] and RPMC [19,20] are formulated as MOPs and several Multi-Objective Evolutionary

A. Saraswat et al. / Energy 51 (2013) 85e100

87

Energy Market Clearing (EMC) Generators Real power output schedules (PG,i) Synchronous generator capability curve information

Generator Companies

TPF (Objective 1) Output MultiObjective RPMC Mechanism

TRTL (Objective 2) Reactive power offers (Bids) (a0 , m1, m2 and m3)

Best • Uniform market clearing compromised prices ( 0, 1, 2 and 3) solution • Generators reactive power output schedules

VSEI (Objective 3)

ISO Reactive Power Market Clearing (RPMC)

Fig. 1. Day-ahead reactive power market settlement model.

Algorithms (MOEAs) have been developed for solving these MOPs in last decade. These MOEAs such as Strength Pareto Evolutionary Algorithm (SPEA) [14,15], Fuzzy Adaptive Particle Swarm Optimization (FAPSO) [16], a Modified Non-dominated Sorting Genetic Algorithm (MNSGA-II) [19] are applied to reactive power optimization problems. In the light of above literature survey, this paper presents a new multi-objective zonal Day-Ahead Reactive Power Market Settlement (DA-RPMS) model based on Pareo-optimization approach. A zonal DA-RPMS problem is formulated as a mixed integer nonlinear multi-objective optimization problem which simultaneously minimizes three objectives such as Total Payment Function (TPF) for reactive power support from generators/syncronus condensers, Total Real Transmission Loss (TRTL) and Voltage Stability Enhancement Index (VSEI) subjected to various power system and zonal market operating conditions. Some of the significant contributions of present work may be briefly summarized as follows: (a) The proposed multi-zone DA-RPMS model (which minimizes three objectives TPF, TRTL and VSEI simultaneously) is tested and compared with a single-zone DA-RPMS model (which minimizes the same objectives) to explore its superiority. In multi-zone DA-RPMS model, the whole power sytem is devided in to various VCAs/zones and the reactive power market is settled on zonal basis and a set of uniform prices is determined separately for each VCA/zone. In contrast to the prosposed model, the single-zone DA-RPMS model settles the reactive power market by considering the whole power system as one control area, and obtains one set of uniform market price for the whole system. (b) A hybrid fuzzy multi-objective evolutionary algorithm (HFMOEA) approach as developed by the authors in [22] is applied for solving both the above complex multi-objective DARPMS models. In HFMOEA, crossover and mutation probabilities (i.e. PC and PM) are adjusted, dynamically during the execution of the program according to a fuzzy rule base which is developed based on heuristics. (c) Two case studies are conducted: In first case study, the performance of HFMOEA is compared with a well known Nondominated Sorting Genetic Algorithm (NSGA-II) [23] to solve above mentioned DA-RPMS models. In second case study, the market power exercised by any generator or any generation company is investigated in different gaming scenarios to demonstrate the superiority of multi-zone DA-RPMS model.

The rest of the paper is organized as follows: Section 2 of present paper describes the structure of reactive power market and various critical issues involved in reactive power market settlement model. The detailed problem formulation of multi-objective zonal DARPMS model is given in Section 3. In Section 4, proposed HFMOEA for solving zonal multi-objective DA-RPMS model is described. The simulation results of HFMOEA based multi-objective DA-RPMS models are presented in Section 5. Finally, the conclusion is drawn in Section 6. 2. Reactive power market structure The proposed day-ahead reactive power market settlement model is illustrated in Fig. 1. In this model, two separate markets, i.e. reactive power market and energy markets are assumed to be settled in different time frames. Therefore, the output of Energy Market Settlement (EMS) becomes one of the inputs for Reactive Power Market Settlement (RPMS). In reactive power market, all providers are required to submit their reactive power offers (bids) in terms of four components such as a0,i, m1,i, m2,i and m3,i (as defined in the Nomenclature section) along with their capability curve details (see Fig. 1). The structure of the reactive power offers and the capability curve for any generator are explained in subsequent subsections. ISO will also have the information regarding the real power output schedules of all the generators from day-ahead energy market settlement. Thus, on the basis of related information, ISO runs the RPMS program to evaluate the required reactive output schedules for generators along with Uniform Market Clearing Prices (UMCP) and finally, ISO settles the reactive power market. In proposed DA-RPMS model, three objective functions such as TPF, TRTL and VSEI are optimized simultaneously subjected to various system equality and inequality constraints in multiobjective optimization framework using proposed HFMOEA. The best compromise solution obtained by HFMOEA encompasses the UMCP along with the reactive power output obligations for the providers. The following assumptions are made regarding the design of a reactive power market.  The Reactive Power Market Settlement (RPMS) is taken place after the Energy Market Settlement (EMS) as in references [6,7].  Only the ISO has monopsony [1] power in the reactive power market which means that only the ISO is a sole buyer of

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reactive power ancillary services. Therefore, the ISO calls for reactive power offers (bids) from all the reactive power providers.  Only synchronous generators and synchronous condensers are considered as reactive power market participants recognized for receiving the payment for providing the reactive power compensations. The reactive power market is settled on first price, uniform auction at the level of individual voltage control areas/zone. This means that all selected reactive power providers receive a uniform price (UMCP) for their ancillary services, which is the highest priced offer accepted in the particular VCA/zone. As discussed in Refs. [24,25], this provides the players enough incentives to bid their true costs. It is worth mentioning here that some efforts are being made in the area of renewable energy sources especially wind power as in [26] for providing reactive power services. More work is expected in coming years.

however, result in the form of increased losses in the windings and, hence, increase the Cost of Loss Component (CLC). If the generator operates on the limiting curve, any increase in Q G will require a decrease in PG to adhere to the winding heating limits. Consider the operating point “A” on the curve defined by (PGA, Q GA). If more reactive power is required from the unit, say Q B at another operating point “B”, the operating point requires shifting back along the curve to point (PGB, Q GB), where PGB < PGA. This signifies that the unit has to reduce its real power output to adhere to field heating limits when higher reactive power is demanded. The loss in revenue to the generator due to the reduced production of real power is termed as Cost of Lost Opportunity (CLO) and is a significant issue. Once any generator is selected to provide reactive power service, that generator is entitled to get payment component called as Cost of Availability (CA). The availability offer typically represents a small part of the generator’s capital cost that goes toward providing reactive power [7]. All three cost components of a reactive power supply from a generator are depicted in Fig. 3.

2.1. Synchronous generator capability curve

2.2. Cost of reactive power production from a generator

The reactive power output ability of a synchronous generator is limited by its capability curve [6e8] as shown in Fig. 2. The synchronous generator capability curve provides the relation/coupling between its real and reactive power outputs. When real power and terminal voltage are fixed, the armature and field winding heating limits determine the reactive power capability of the generator. The synchronous generator’s MVA rating is the point of intersection of the two curves as represented by a point “R” in Fig. 2, and therefore its MW rating is given by PGR. At an operating point “A”, with real power output PGA such that PGA < PGR, the limit on QG is imposed by the generator’s field heating limit; whereas, when PGA > PGR, the limit on Q G is imposed by the generator’s armature heating limit. Q Gbase is the reactive power required by the generator for its auxiliary equipment. If the operating point lies inside the limiting curves, say, at (PGA, Q Gbase), then the unit can increase its reactive generation from Q Gbase up to Q GA without requiring readjustment of PGA. This will,

The reactive power market structure is designed on the basis of providers Expected Payment Function (EPF) for their reactive power support services. A realistic representation of reactive power offer price structure is given as shown in Fig. 3. Based on the information related to synchronous generator’s capability curve presented before, three operating regions for ith generator on the reactive power coordinate are identified as follows. Region I (Q Gmin,i  QG,i ¼ QG1,i  0) refers to the under excitation region, in which ith generator is required to absorb reactive power form the system and it would incur cost of loss and, hence, can expect to be paid for its services. Its EPF would consist of two components in this region such as Cost of Availability (CA) and Cost of Loss Component (CLC). Thus, the EPF may be expressed as (EPF¼CAþCLC). Region II (0  QG,i ¼ QG2,i  QGA,i) refers to the over excitation region, in which ith generator is required to supply reactive power within its reactive power capability limits. Thus in similar manner, as in the Region I, EPF may be expressed as (EPF ¼ CA þ CLC). Region III: (QGA,i  QG,i ¼ QG3,i  QGB,i): refers to the loss of opportunity region, in which ith generator is asked to reduce its active power production in order to meet the system reactive power requirements. Therefore, generator is entitled to receive a payment commensurate with its Cost of Lost Opportunity (CLO) of reduced real power production, along with the other components. The EPF may be expressed as (EPF ¼ CA þ CLC þ CLO). However, the ISO will not be in a position to estimate the EPF for a generator in deregulated markets. An appropriate option for the ISO is to call for reactive offers from all generators based on the EPF structure. Therefore, a possible structure of such reactive offers is discussed next.

Generator reactive power output (MVAr)

QG Field heating limits B

QGB QGA

A R

Armature heating limits

QGbase 0

Mandatory PGB

PGA

PGR

PG PGmax Generator real power output (MW)

QGmin

2.3. Structure of reactive power offer bids Based on the classification of reactive power production costs discussed in previous subsection, a generalized EPF and, hence, an offer structure (see Fig. 3) [7] can be formulated mathematically as follows:

Z0 EPFi ¼ a0;i þ

Q ZGA;i

m1;i $dQ G;i þ Q Gmin;i

Under excitation limits

Q ZGB;i

þ Fig. 2. Capability curve of synchronous generator [7].

Q GA;i

  m3;i $Q G;i $dQ G;i

m2;i $dQ G;i Q Gbase;i

(1)

A. Saraswat et al. / Energy 51 (2013) 85e100

89

Fig. 3. Structure of reactive power offers from providers [7].

The coefficients in Eq. (1) represent the various components of reactive power cost incurred by ith provider that need to be offered in the market. These are explained as: a0,i is cost of availability price offer (in $), m1,i is the cost of loss component price offer for operating in under excited mode (absorb reactive power), Q Gmin,i  Q G,i  (in $/MVAr-h), m2,i is the cost of loss component price offer for operating in the region QGbase,i  Q G,i  Q GA,i (in $/MVAr-h) and m3,i is the cost of lost opportunity price offer for operating in the region Q GA,i  Q G,i  QGB,iin ($/MVAr-h)/MVAr-h. Above discussion also holds true for synchronous condensers, except for the opportunity cost component. Synchronous condensers will be assumed to offer all components, except for the opportunity price.

market price in all NZ number of VCAs/zones namely Zone-z (i.e. Zone-1, Zone-2, Zone-3 up to Zone-NZ). The principle of highest priced offer selected within particular zone determines the zonal market clearing price (r0, r1z, r2z and r3z). Therefore, the TPF for reactive power supports based on zonal uniform prices may be formulated as follows:

F1 ¼ TPF ¼

X

r0 $W0;i

i˛gen

þ

X

0 @

z˛NZ

X  i˛gen;z

2.4. Formation of voltage control areas/zones In order to settle a reactive power market on zonal/localized basis, a given power system is to be separated into few nonoverlapping voltage control areas/zones comprising coherent bus groups. These groups are the sets of such buses forming voltage control areas/zones if they are sufficiently uncoupled electrically from their neighboring areas. Each VCA consists of those buses which have significant electrical couplings (dependencies) among them. In reference [8], a two-step method is proposed to identify the voltage control areas/zones. The first step involves a calculation of electrical distances between all nodes in the power system and second step classifies the areas to decide the borders of each areas using hierarchical classification algorithm. An alternative approach is suggested by authors in Ref. [27] to identify the voltage control areas/zones using K-means clustering algorithm for reactive power management and voltage control. 3. Problem formulation for multi-objective DA-RPMS model 3.1. Objectives in multi-objective DA-RPMS model In this proposed multi-objective DA-RPMS model, the first objective is to minimize the Total Payment Function (TPF) for reactive power support services provided by synchronous generators and synchronous condensers in order to settle the reactive power market on zonal basis. The total payment will depend on the market price of the four components of reactive power being offered to the providers that seeks the uniform reactive power



 r1z $W1;iz $Q G1;iz þ r2z $W2;iz $Q G2;iz 1  1 2 þ r2z $W3;iz $Q GA;iz þ r3z $W3;iz $Q G3;iz A 2 (2)

In Eq. (2), the subscript ‘gen,z’ denotes all the generators in zth VCA/zone (i.e.cz ¼ 1; 2; .; NZ ); r0 is the uniform availability price for whole system; r1z and r2z are the uniform operating prices for reactive power absorption and production respectively within the zth VCA/zone; whereas r3z is the uniform opportunity price within the zth VCA/zone. Here, the reactive power output from ith provider is classified into three components Q G1,i, Q G2,ior Q G3,i that represent the regions (Q Gmin,i,0), (QGbase,i,Q GA,i) and(Q GA,i,Q GB,i) respectively. The binary variables W1, W2 andW3 are associated with Q G1,i, Q G2,ior Q G3,i, respectively to determine if the generator is selected to provide reactive power in the three regions discussed above, accordingly only one of the binary variables W1, W2 andW3 can be selected. If any provider is selected, W0will be one, and it will receive the availability price, irrespective of its reactive power output. In this multi-objective DA-RPMS model, the second objective is to minimize the Total Real Transmission Losses (TRTL) which may be expressed as follows:

F2 ¼ TRTL ¼

P

Pk;loss ¼

k˛NL

P k˛NL

h  i gk Vi2 þ Vj2  2Vi Vj cos di  dj

where k ¼ ði; jÞ; i˛NB ; j˛Ni (3) where Vi :di and Vj :dj are bus voltages at the end buses i.e. ith and jth of the kth transmission line, respectively. The symbols of the above equation and in the following context are given in the Nomenclature section.

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A. Saraswat et al. / Energy 51 (2013) 85e100

Start

Input Power System Data (Bus data, Generator data, Line data), information about VCAs/zones and initialize HFMOEA Parameters (popsize, maxgen, no. of control variables, constraints limits, k, PC, PM) Generate Random Initial Population and run load flow analysis to evaluate normalized fitness values for each valid individual Non Domination Sorting of initial population Set generation counter (t = 0) Tournament Selection for obtaining parent population

Normalized fitness values of parent population

(BLX-α) Crossover Fuzzy Logic Controller (FLC_MOEA) PCA based Mutation Run load flow analysis and evaluate normalized fitness values for all individuals in offspring population

Normalized fitness values of offspring population

Combine the parent and offspring populations to obtain the intermediate population ensures elitism

Evaluate fuzzy input variables BCF

Non Domination Sorting of intermediate population

UN

Fuzzy Rule Base

Remove worse individuals based on crowding distance criteria to maintain the size of new population constant Check Whether k generations completed ?

VF

PC

Yes

PM

Update HFMOEA parameters (PC and PM)

No t=t+1 Check current generation no. = = maxgen

No

Yes Select the best compromise solution using fuzzy set theory

End Fig. 4. Flowchart for HFMOEA process for solving multi-objective DA-RPMS problem [22].

The third objective is to minimize a Voltage Stability Enhancement Index (VSEI) also known as L-index [28] in order to incorporate the voltage stability improvement in multi-objective DA-RPMS model. It is a static voltage stability measure of power system, computed based on normal load flow solution as presented in Refs. [29,30] solution. Its value may be defined as follows:

  ) ( NPV  X V   F3 ¼ VSEI ¼ L  index ¼ max Lj ¼ 1  Fji i ; j˛NPQ  Vj 



(4) All the terms within the sigma of Eq. (4) are complex quantities. The values Fjiare the elements of sub-matrix FLG as obtained from the Y-bus matrix as follows.



 ¼

YGG YLG

YGL YLL



VG VL

 (5)

where [IG], [IL] and [VG], [VL] represent the complex currents and bus voltages, respectively; whereas [YGG], [YGL], [YLG] and [YLL] are corresponding portions of network Y-bus matrix. Rearranging (5), we obtain



i¼1

IG IL

VL IG



 ¼

ZLL RGL

FLG YGG



IL VG

 (6)

where

FLG ¼ ½YLL 1 ½YLG 

(7)

A. Saraswat et al. / Energy 51 (2013) 85e100

The value of L-index lies between 0 and 1 [29]. The limiting values such as 0 represents that there is no load available in power system and as the system load is increases, its value increases until reaches to unity at the point of voltage collapse or voltage instability. Therefore, any value of L-index which is less than 1 and close to 0 indicates a system stable state i.e. system voltage stability margin. Thus L-index gives an indication of how far the system is from voltage collapse. This feature of this indicator is exploited by minimizing its value along with other two objective functions in proposed model to improve the systems voltage stability margin. 3.2. System constraints in multi-objective DA-RPMS model The minimization of all three objective functions (2)e(4) are subjected to various system equality and inequality constraints as described below. 3.2.1. Load flow equality constraints

PG;i  PD;i  Vi

X

  Vj Gij cos qij þ Bij sin qij ¼ 0;

91

3.2.4. Reactive power provision limits

Q Gmin;i  Q G;i  Q Gmax;i ;

i˛NPV

(20)

Q Cmin;i  Q C;i  Q Cmax;i ;

i˛NC

(21)

3.2.5. Reactive power capability limits of generators The constraint given in Eq. (22) is applicable to ensure that the armature heating limits (i.e. forPG,i  PGR,i) and field heating limits (i.e. forPG,i  PGR,i) are followed by any operating point (Q G,i, PG,i) over the capability curve for ith generator [6].

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  > > > < Vt;i Ia;i PG;i if PG;i  PGR;i

ðArmature heating limitÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Q G;i  u 2 u Vt;i Eaf ;i 2 Vt;i > t 2 > PG;i if PG;i  PGR;i ðField heating limitÞ  > : Xs;i Xs;i (22)

i˛NB

(8)

3.2.6. Bus voltage limits

(9)

Vimin  Vi  Vimax ;

j˛Ni

Q G;i  Q D;i  Vi

X

  Vj Gij sin qij  Bij cos qij ¼ 0;

i˛NB

ci˛NB

(23)

j˛Ni

3.2.7. Security constraints 3.2.2. Reactive power relational constraints and limits The reactive power offer regions as explained in Section 2, a set of governing algebraic relations is required to ensure appropriate allocation for each i ðci˛genÞ as given in Eqs. (10)e(14).

Q Gi ¼ Q G1;i þ Q G2;i þ Q G3;i

(10)

W1;i $Q Gmin;i  Q G1;i  W1;i $Q Gbase;i

(11)

W2;i $Q Gbase;i  Q G2;i  W2i $Q GA;i

(12)

W3;i $Q GA;i  Q G3;i  W3;i $Q GB;i

Sl  Smax ; l

l˛NL

(24)

PGmin;Slack  PG;Slack  PGmax;Slack (25) where PGmin,SlackandPGmax,Slack are the minimum and maximum limits of real power output of slack bus. 3.2.8. Transformer taps setting constraints

Tkmin  Tk  Tkmax ;

k˛NT

(26)

(13) 3.3. Generalized augmented objective function

W1;i þ W2;i þ W3;i  1

(14)

3.2.3. Constraints determining the multi-zone market clearing prices The multi-zone market clearing prices (r0, r1z, r2z and r3z) are determined separately for each component of reactive power. The constraints as mentioned in Eqs. (15)e(19) ensure that the market price, for a given set of offers, is the highest priced offer accepted:

W0;i ¼ W1;i þ W2;i þ W3;i ; W0;i $a0;i  r0 ; W1;i $m1;i  r1z ;

ci˛gen

(15) (16)

ci˛gen; z

  W2;i þ W3;i $m2;i  r2z ; W3;i $m3;i  r3z ;

ci˛gen

ci˛gen; z

ci˛gen; z

(17) (18) (19)

In this paper, a static penalty function approach [31] is used to handle the inequality constraint violations. Infeasible solutions are penalized, by applying a constant penalty to those solutions, which violate feasibility in any way. Thus, the penalty functions corresponding to voltage violations at all load busses, reactive power violations at all generator busses, real power violations at slack bus and power flow violations at all transmission lines (lVL,i, lQG,j, lPG,Slack and lS,l) are included in objective function as follows:

Faug;n ¼ Fn þ

X i˛NPQ

þ

X

k˛NG;Slack



lVL;i Vi  Vilim

2

þ

X j˛NG

lim lPG;Slack PG;k  PG;k

2



lQG;j Q G;j  Q lim G;j

þ

X



2

lS;l Sl  Slim l

2

;

l˛NL

cn˛Nobj (27) where Nobj is the total number of objectives (in this DA-RPMS model, Nobj ¼ 3); Fn is nth objective function value and the dependent variables limiting values may be considered as:

92

A. Saraswat et al. / Energy 51 (2013) 85e100

Fig. 5. Input and output membership functions for fuzzy logic controller [22].



if Vi > Vimax ; if Vi < Vimin

Vilim ¼ Q lim G;j ¼ Slim ¼ l lim PG;k ¼

ci˛NPQ

if Q G;j > Q Gmax;j ; if Q G;j < Q Gmin;j

if Sl > Smax l ; if Sl < Smin l

(28)

cj˛NPV

(29)

cl˛NL

PGmax;k ; if PG;k > PGmax;k ; PGmin;k ; if PG;k < PGmin;k

(30)

ck˛NG;Slack

(31)

maximum numbers of generations (maxgen), number of control variables, system variables limits, initial values of crossover and mutation probabilities (PC and PM). Implementation of HFMOEA for a DA-RPMS problem involves the parameter encoding (i.e., the representation of the problem). Each individual in the genetic population represents a candidate solution. The elements of that solution consist of all the control variables in the system. The system variables include all control variables as well as dependent variables. For the proposed DA-RPMS model, the control variables are the voltage magnitudes of all generators, transformer tap settings and shunt capacitors/inductors settings. The dependent variables are reactive power output of all generators, load bus voltage magnitudes and line flows, uniform market clearing prices (r0, r1, r2 and r3) for reactive power market, reactive power payment of all the providers.

4. HFMOEA for solving multi-objective DA-RPMS problem

4.2. Generation of initial population

HFMOEA approach is developed for solving complex mixed integer nonlinear multi-objective DA-RPMS problem. A flowchart of proposed algorithm for solution of multi-objective DA-RPMS problem is shown in Fig. 4. The details of proposed algorithm are discussed as below.

Initial population is generated randomly according to following procedural steps.

4.1. Initialization The very first step in proposed approach is to read the power system data (i.e. bus data, generator data and transmission line data), the information about the VCAs/zones and initialize various parameters of HFMOEA such as population size (popsize),

Step 1: Generate a string of real valued random numbers for system control variables within their specified limits to form a single individual; Step 2: Run NewtoneRaphson based load flow analysis to check load flow equality constraints given by Eqs. (8) and (9); Step 3: Check whether the load flow analysis is converged or not. If it is not converged (as the equality constraints are not satisfied), go to step 1; Step 4: Place the individual as valid individual in initial population; Step 5: Determine normalized fitness values defined by Eq. (35) for valid individual by evaluating augmented functions expressed by Eq. (27); Step 6: Check if the initial population is not completed then go to step 1. 4.3. Non-domination sorting The generated initial population is sorted on the basis of non-domination sorting algorithm as suggested in reference [13] and [23]. Table 1 Specifications of multi-objective evolutionary algorithms. Algorithm parameters

NSGA-II

HFMOEA

Number of control variables Population size (Popsize) Selection operator

17 200 Tournament selection SBX crossover Polynomial mutation 0.95 0.015 500

17 200 Tournament selection BLX-a crossover PCA mutation

20

20

Crossover operator Mutation operator

Fig. 6. IEEE 24 bus Reliability Test System (IEEE 24 bus RTS) showing three voltage control areas/zones.

Crossover probability (PC) Mutation probability (PM) Maximum number of generations Number of optimization runs

Varying based on FLC output Varying based on FLC output 500

A. Saraswat et al. / Energy 51 (2013) 85e100

93

Table 2 Comparison of best compromise solutions and uniform market clearing prices obtained from both multi-objective DA-RPMS model using NSGA-II and HFMOEA. Best compromised solutions Objective function

Case 1: Single-zone DA-RPMC model a

TPF TRTL VSEI

Case 2: Multi-zone DA-RPMC model

GAMS [21]

NSGA-II

HFMOEA

NSGA-II

HFMOEA

496.55 -

588.99 40.85 0.1724

488.60 39.25 0.1699

506.12 39.876 0.1825

465.74 39.33 0.1678

Comparison of uniform market clearing prices UMCPs

r0 r1 r2 r3 a

Case 1: Single-zone DA-RPMC model

Case 2: Multi-zone DA-RPMC model

GAMSa [21]

NSGA-II

NSGA-II

0.96 0 0.86 0.46

HFMOEA

0.96 0.89 0.86 0.46

0.96 0.48 0.86 0.38

HFMOEA

Zone#1

Zone#2

Zone#3

Zone#1

Zone#2

Zone#3

0.96 0.82 0.86 0

0.96 0 0.81 0

0.96 0.48 0.85 0

0.96 0 0.86 0

0.96 0.89 0.81 0

0.96 0.48 0.73 0.38

Multi-objective model i.e. Case II [21].

4.4. Evolutionary operators

generation by going through following procedural steps, till the termination condition is not satisfied:

For producing the new population for next generation, the following evolutionary operators are applied to parent population:  Selection: tournament selection operator based on the crowding comparison procedure [23] is applied for reproducing the mating pool of parent individuals for crossover and mutation operations.  Crossover: the blend crossover operator (BLX-a) [13] based on the theory of interval schemata is employed in proposed ð1;tÞ

algorithm. For two randomly selected parent solutions xi ð2;tÞ

and xi

ð1;tÞ

(assuming xi

ð2;tÞ

< xi

), the BLX-a randomly picks ð1;tÞ

a new individual in the range ½xi ð2;tÞ

ð1;tÞ

ð2;tÞ

 aðxi

ð1;tÞ

 xi

Þ;

ð2;tÞ

xi

þaðxi  xi Þ. If, ui is a random number between 0 and 1, new individual is generated as follows: ð1;tþ1Þ

xi

ð1;tÞ

¼ ð1  gi Þxi

ð2;tÞ

þ gi x i

ð1;tÞ

; where gi ¼ ð1 þ 2aÞui  a

(32)

ð2;tÞ

In Eq. (32), xi and xi represent ith control variable of individual 1 and 2, respectively, in the parent population before ð1;tþ1Þ

BLX-a crossover; xi is ith control variable of an individual in a new population after BLX-a crossover. If a is zero, this crossover creates a random individual in the

½

ð1;tÞ

ð2;tÞ



range xi ; xi . In this work, BLX-a (witha ¼ 0.5) is applied with a varying crossover probability which performs better than BLX operators with any other avalue as suggested in reference [13].  Mutation: the new individuals thus obtained by BLX-a crossover replace their parents in the new population which is then subjected to a Principal Component Analysis (PCA) based mutation operator. The PCA based mutation as proposed by authors in Ref. [32] is used with varying mutation probability in HFMOEA to generate the offspring population. In PCA mutation, a well known principal component analysis [33] is applied in order to achieve higher diversity in offspring population. This mutation operation is performed by adding a nonnegative random number in principal components extracted from new population obtained after crossover. 4.5. Criterion to prepare population for next generation After the execution of above evolutionary operators, offspring population is checked to prepare new population for next

Step 1: Run the NewtoneRaphson based load flow analysis on each individual in offspring population to check load flow equality constraints and also evaluate normalized fitness values corresponding to all objective functions (i.e. TPF, TRTL and VSEI); Step 2: Combine parent and offspring populations to obtain intermediate population which ensures elitism; Step 3: Perform the non-domination sorting algorithm on intermediate population; Step 4: Remove the worse individuals based on their crowding distance in the ascending order until the population size is equal to popsize to maintain the new population size constant. Here the new population for next generation is prepared; Step 5: Check if k generations are completed go to next step 6 otherwise go to step 7; Step 6: Update HFMOEA parameters (i.e. PC and PM) by using fuzzy logic controller (FLC_MOEA); Step 7: Check the termination condition of HFMOEA. i.e. if the current generation number is equal to maxgen, terminate the iterative process otherwise go to next generation; Step 8: Select the best compromise solution using fuzzy set theory;

4.6. Best compromise solution Upon having the Pareto-optimal set of non-dominated solutions using the proposed HFMOEA approach, a fuzzy approach proposed in Ref. [34] selects one solution to the decision maker as the best compromise solution as used in [14]. This approach suggests that due to imprecise nature of the decision maker’s judgment wherein a linear membership function (mi) is defined for ith objective function Fi as follows:

8 > > > 1 > > < F max  F i i mi ¼ > Fimax  Fimin > > > > :0

Fi  Fimin Fimin < Fi < Fimax Fi 

(33)

Fimax

where Fimin and Fimax are the minimum and maximum values of the ith objective function among all non-dominated solutions respectively. The defined membership function mi indicates the degree of optimality for the ith objective function. For each jth non-

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A. Saraswat et al. / Energy 51 (2013) 85e100

dominated solution, the normalized membership function mjis calculated based on its individual membership functions as:

P

Nobj j m mj ¼ PN i ¼P1 N i obj dom j¼1

i¼1

mji

(34)

whereNdomis the number of non-dominated solutions. The best compromise solution is that having the maximum value ofmj. 4.7. Normalized fitness function The normalized fitness values corresponding to each individual in the population are assigned based on their respective generalized augmented functions as determined in Eq. (27). Thus the normalized fitness function (Hn) for nth objective is evaluated as:

Hn ¼

Kn ; 1 þ Faug;n

cn ¼ 1 : Nobj

1. 2. 3. 4.

(35)

whereNobjis the total number of objectives and Knis the appropriate constant corresponding to nth objective. In this work, the values of Kn are selected after conducting several initial independent runs of HFMOEA with various settings of K1, K2 and K3. These values are chosen in such a manner that normalized fitness values of individuals must remain within range [0 1.0] as clear from Fig. 5 to fire correct fuzzy rule. 4.8. Fuzzy logic controller In most of the MOEAs reported in literature, the various algorithm parameters such as crossover and mutation probabilities are initialized at the beginning of MOEA’s execution and kept constant throughout the optimization process. These MOEA parameters greatly influence the convergence of optimization process and often become very critical by limiting its performance to reach near a global Pareto-optimal front. Moreover, it has been experienced that it may lead to slow convergence and seldom trapped into local Pareto-optimal fronts. In this situation, a large change in the decision vector (i.e. control variables vector) is needed to get out of a local optimum. Unless mutation or crossover operators are capable of creating solutions in the basin of another better attractor, the improvement in the convergence toward the global Paretooptimal front is not possible [23]. Therefore, based on this knowledge, a fuzzy logic controller FLC_MOEA is designed for adjusting PC and PM dynamically during the optimization process. The block diagram of FLC_MOEA is shown as gray shaded area of Fig. 4. The following important considerations are taken into account to form the fuzzy rule base.  The best compromised fitness (BCF) for each iteration is expected to change over a number of iterations, but if it does not change significantly over a number of iterations (UN) then this information is considered to cause changes in both PC and PM.  The diversity of a population is one of the factors, which influences the search for a global Pareto-optimal front. The variance of the fitness values of objective function (VF) of a population is a measure of its diversity. Hence, it is considered as another factor on which both PC and PM may be changed. The Mamdani-type fuzzy rule is used to formulate the conditional statements that comprise fuzzy logic. Total 15 Mamdani-type rules are framed as conditional statements, which are listed as below.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

If (BCF is Low) then (PC is High) (PM is Low) (1) If (BCF is Medium) and (UN is Low) then (PC is High) (PM is Low) (1) If (BCF is High) and (UN is Low) then (PC is High) (PM is Low) (1) If (BCF is Medium) and (UN is Medium) then (PC is Medium) (PM is Medium) (1) If (BCF is High) and (UN is Medium) then (PC is Medium) (1) If (UN is High) and (VF is Low) then (PC is Low) (PM is High) (1) If (UN is High) and (VF is Medium) then (PC is Low) (1) If (UN is High) and (VF is High) then (PC is Medium) (1) If (BCF is High) and (VF is Medium) then (PM is Low) (1) If (BCF is High) and (VF is High) then (PM is Low) (1) If (VF is High) then (PC is High) (PM is Low) (1) If (VF is Medium) then (PC is High) (PM is Low) (1) If (BCF is High) and (VF is Low) then (PC is High) (PM is Low) (1) If (BCF is Medium) and (VF is Medium) then (PC is Low) (PM is High) (1) If (BCF is Low) and (UN is Low) and (VF is Low) then (PC is High) (PM is Low) (1)

Three input fuzzy parameters (i.e. BCF, UN and VF) and also two output fuzzy parameters (i.e. PC and PM) are represented by three linguistic terms as LOW, MEDIUM and HIGH as shown in Fig. 5. 5. Simulation results and discussion The effectiveness of proposed HFMOEA based multi-objective zonal DA-RPMS model is demonstrated on the IEEE 24 bus Reliability Test System (IEEE 24 RTS) [35]. As shown in Fig. 6, the power system consists of 32 synchronous generators,1 synchronous condenser (located at bus 14), and 17 constant-power type loads. The system total active and reactive loads are 2850 MW and 580 MVAr, respectively. The simulations are carried out in MATLAB 7.0 programming environment on Pentium IV 2.27 GHz, 2.0 GB RAM computer system. In IEEE 24 RTS, system control variables are eleven generator bus voltage magnitudes, five transformer tap settings and one bus shunt inductor. Therefore, the search space has 17 dimensions. The lower and upper permissible limits of all bus voltages are 0.95 p.u. and 1.05 p.u., respectively. The lower and upper limits of all transformer tap settings are 0.9 p.u. and 1.1 p.u, respectively. In order to carry out the simulations for DA-RPMS in competitive electricity market environment, the ISO needs the following information from the reactive power providers. Voltage control areas/zones: the IEEE 24 RTS is divided into three voltage control areas/zones using method of K-means clustering algorithm [27] as shown in Fig. 6. Three zones are as follows: Zone#1 consists of three generators at bus number 1, 2 and 7 along with five load buses (bus number 3, 4, 5, 6 and 8); Zone#2 is having two generator buses (bus number 13 and 23), one synchronous condenser at bus number 14 and five load buses (bus number 9, 10, 11, 12 and 20); and Zone#3 includes five generator buses (bus number 15, 16, 18, 21 and 22) and three load buses (bus number 17, 19 and 24). Offer prices: the ISO is supposed to receive four components of the reactive power offer prices (a0,i,m1,i,m2,iand m3,i), directly from the participants of the reactive power market. In this simulation, the reactive power offer prices (bids) submitted by all generators and synchronous condensers are taken as given in Ref. [9]. A synchronous condenser at bus 14 also participates in the reactive power market with its opportunity cost (mi;u 3 ) equal to zero. Generator’s reactive power capability data: each participant of the reactive power market (i.e. each generating unit and synchronous condenser) is also required to submit the information regarding its reactive power capability diagram i.e. QBase,QA and QB (see Fig. 1). In the present case study, the assumptions are followed as in references [7,9], i.e. QBase¼0.10Qmax, QA is limited either by the field or the armature heating limit, as per operating condition, and QB¼1.5QA.

6000

3000

5000

2500

F1 : TPF ($)

F1 : TPF ($)

A. Saraswat et al. / Energy 51 (2013) 85e100

4000 3000 2000

2000 1500 1000 500

1000

0 42

0 43 42 41 0.18

40 39

F2 : TRTL (MW)

0.16

0.19

0.2

41

0.21

40 39

0.17

38

F2 : TRTL (MW)

F3 : VSEI (L-index)

0.15

0.16

0.17

0.18

0.2

0.19

F3 : VSEI (L-index)

(b) Case 1: Single-Zone DA-RPMS Model: HFMOEA

(a) Case 1: Single-Zone DA-RPMS Model: NSGA-II 5000

2500

4000

2000

F1 : TPF ($)

F1 : TPF ($)

95

3000 2000 1000

1500 1000 500 0 42

0 43 42 41 40 39

F2 : TRTL (MW)

0.16

0.18

0.17

0.19

0.2

41

0.21

40 39

F2 : TRTL (MW)

F3 : VSEI (L-index)

(c) Case 2: Multi-Zone DA-RPMS Model: NSGA-II

38

0.155

0.16

0.165

0.17

0.175

0.18

F3 : VSEI (L-index)

(d) Case 2: Multi-Zone DA-RPMS Model: HFMOEA

Fig. 7. Best obtained pareto-optimal fronts using NSGA-II and HFMOEA for both multi-objective DA-RPMS Models.

5.1. Case study A: base case scenario In this case study, two test cases are performed on base operating conditions, namely Case 1: Single-Zone DA-RPMS model and Case 2: Multi-Zone DA-RPMS model. Both the test cases use same offer prices from generators. In case 1, the simulations are carried out for the reactive power market considering the whole system as one voltage control area/single-zone and one set of uniform market clearing price for the whole system is obtained. Case 2 simulates the market considering three voltage control areas/multi-zones as obtained by K-means clustering approach [27] and a set of uniform prices is determined separately for each area. By comparing the results of Case 1 and Case 2, the advantages of considering voltage

Table 3 Statistical results: mean and standard deviation for MOEAs performance metrics in 20 optimization runs. MOEA

Case 1: Single-zone DA-RPMS model Mean

St. Dev.

Mean

St. Dev.

S

NSGA-II HFMOEA NSGA-II HFMOEA

13.1794 6.5502 0.2206 0.7198

2.7066 1.2083 0.1391 0.0025

10.3902 2.4840 0.2824 0.8011

4.7556 0.7537 0.1605 0.0030

Case 2: Multi-zone DA-RPMS model

Crossover Probability (Pc)

0.95 0.9 0.85 0.8 0.75

0

100

200

300

400

500

0.95 0.9 0.85 0.8 0.75

0

100

Generation Number 0.08

Mutation Probability (Pm)

Mutation Probability (Pm)

HV

Crossover Probability (Pc)

Performance metric

0.06 0.04 0.02

0

0

100

200

300

400

500

200 300 400 Generation Number

500

0.08 0.06 0.04 0.02

0

0

100

200

300

400

Generation Number

Generation Number

Case 1: Single-Zone DA-RPMS Model

Case 2: Multi-Zone DA-RPMS Model

500

Fig. 8. Variations in PC and PM during HFMOEA based optimization in both multi-objective DA-RPMS Models corresponding to best pareto-optimal solutions.

96

A. Saraswat et al. / Energy 51 (2013) 85e100

Table 4 Output control variables obtained from both the multi-objective DA-RPMS-VS models. Market model

Case 1: Single-zone DARPMS model

Case 2: Multi-zone DARPMS model

Bus No.

NSGA-II

NSGA-II

Generator 1 2 7 13 14 15 16 18 21 22 23

Bus ID

HFMOEA

bus voltages (p.u.) 1.0307 VG1 1.0278 VG2 1.0500 VG7 1.0384 VG13 1.0164 VG14 VG15 1.0049 1.0090 VG16 1.0128 VG18 1.0226 VG21 1.0382 VG22 1.0361 VG23

HFMOEA

1.0496 1.0500 1.0500 1.0487 1.0258 1.0204 1.0252 1.0340 1.0352 1.0489 1.0496

1.0122 1.0165 1.0480 1.0449 1.0277 1.0110 1.0188 1.0264 1.0280 1.0372 1.0481

1.0495 1.0500 1.0498 1.0497 1.0395 1.0254 1.0317 1.0315 1.0356 1.0446 1.0500

(p.u.) 1.07 1.08 1.01 0.99 0.97

1.08 1.06 1.03 1.01 0.99

1.06 1.03 1.06 0.99 0.97

1.04 1.05 1.03 0.99 0.98

Shunt capacitances (MVA-r) 137.05 6 QC6

L124.33

96.90

L130.47

Transformer tap settings 3e24 T3e24 9e11 T9e11 9e12 T9e12 10e11 T10e11 10e12 T10e12

control areas for reactive power services are evaluated. In both the cases, the performance of proposed HFMOEA is also compared with NSGA-II. The detailed specifications of NSGA-II and HFMOEA are summarized in Table 1. In both the test cases, 20 independent optimization runs are performed for NSGA-II and HFMOEA in order to restrict the randomness in the optimization results. The detailed comparison of best compromise solutions and output uniform market clearing prices obtained from NSGA-II and HFMOEA in both the cases is presented in Table 2. The bold values are representing the results obtained corresponding to the proposed HFMOEA based method. In case 1 for Single-zone DA-RPMS model, the best Pareto-optimal solutions are obtained as (588.99$, 40.85 MW and 0.1724) and (488.60$, 39.25 MW and 0.1699) from NSGA-II and HFMOEA, respectively. Similarly in Case 2 for Multi-zone DA-RPMS model, the best Pareto-optimal solutions are obtained as (506.12$, 39.876 MW and 0.1825) and (465.74$, 39.33 MW and 0.1678) from NSGA-II and HFMOEA, respectively. The best pareto-optimal fronts obtained from NSGA-II and HFMOEA for both the cases of Single and Multi-Zone DA-RPMS models are compared as shown in Fig. 7. It is cleared that the Pareto-optimal front and best compromise solution obtained using HFMOEA are superior compared to the same obtained using NSGA-II in their respective cases. Moreover, the results of a different multi-objective reactive power market clearing model as presented in reference [21] are also listed in Table 2 to make a comparison with the result obtained

Table 5 Generators reactive power output schedule and payments of all generating units obtained from both multi-objective DA-RPMS models. Market model Bus No.

1

2

7

13

14a 15

16 18 21 22

23

Total a

Case 1: Single-zone DA-RPMS model Unit No.

1 2 3 4 1 2 3 4 1 2 3 1 2 3 1 1 2 3 4 5 6 1 1 1 1 2 3 4 5 6 1 2 3

NSGA-II

Case 2: Multi-zone DA-RPMS model

HFMOEA

NSGA-II

HFMOEA

QG (MVAr)

Payment ($)

QG (MVAr)

Payment ($)

QG (MVAr)

Payment ($)

QG (MVAr)

Payment ($)

7.61 7.61 16.86 16.86 4.11 4.11 2.42 2.42 21.37 21.37 21.37 46.09 46.09 46.09 120.15 4.47 4.47 4.47 4.47 4.47 46.84 2.10 21.74 83.22 1.37 1.37 1.37 1.37 1.37 1.37 0.73 0.73 41.32

20.01 20.01 15.46 15.46 4.49 4.49 3.11 3.11 19.34 19.34 19.34 40.60 40.60 40.60 104.29 8.99 8.99 8.99 8.99 8.99 41.24 0.00 20.31 72.53 0.00 0.00 0.00 0.00 0.00 0.00 1.61 1.61 36.50

5.23 5.23 3.74 3.74 5.65 5.65 6.07 6.07 15.92 15.92 15.92 44.52 44.52 44.52 90.57 4.74 4.74 4.74 4.74 4.74 52.72 16.22 62.95 4.54 L0.49 L0.49 L0.49 L0.49 L0.49 L0.49 0.52 0.52 43.00

5.45 5.45 4.18 4.18 5.82 5.82 6.18 6.18 14.65 14.65 14.65 39.25 39.25 39.25 78.85 8.67 8.67 8.67 8.67 8.67 46.30 14.91 55.10 0.00 1.20 1.20 1.20 1.20 1.20 1.20 0.00 0.00 37.94

1.83 1.83 14.93 14.93 6.40 6.40 10.22 10.22 19.66 19.66 19.66 43.56 43.56 43.56 113.87 3.80 3.80 3.80 3.80 3.80 32.38 8.85 59.13 25.49 2.26 2.26 2.26 2.26 2.26 2.26 6.78 6.78 51.44

2.53 2.53 13.21 13.21 6.47 6.47 9.75 9.75 17.87 17.87 17.87 36.25 36.25 36.25 93.19 4.19 4.19 4.19 4.19 4.19 28.48 8.48 51.22 22.63 2.05 2.05 2.05 2.05 2.05 2.05 0.00 0.00 42.63

5.93 5.93 7.60 7.60 6.23 6.23 9.25 9.25 17.34 17.34 17.34 39.70 39.70 39.70 121.61 4.66 4.66 4.66 4.66 4.66 50.89 41.91 19.39 16.35 L2.42 L2.42 L2.42 L2.42 L2.42 L2.42 L3.17 L3.17 38.04

6.06 6.06 7.50 7.50 6.32 6.32 8.91 8.91 15.87 15.87 15.87 33.12 33.12 33.12 99.46 8.00 8.00 8.00 8.00 8.00 38.11 31.56 0.00 0.00 2.12 2.12 2.12 2.12 2.12 2.12 3.78 3.78 31.77

555.65

589.00

514.49

488.60

506.89

506.12

519.72

465.74

Synchronous condenser (SC).

A. Saraswat et al. / Energy 51 (2013) 85e100

97

Reactive Power output at Generator Bus (MVAr)

200 QGout (Case 1: Single-Zone DA-RPMS Model: HFMOEA) QGout (Case 2: Multi-Zone DA-RPMSModel: HFMOEA) QGmax QGmin

150

100

50

0

-50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Generator Bus Number Fig. 9. Generators reactive power output schedules from both multi-objective DA-RPMS models using HFMOEA.

power system as only one VCA/zone. Therefore, it reduces the payment burden of the ISO for reactive power services in the system moreover the problem of “market power” owned by any reactive power provider is limited as compared to other model. The issue related to market power is tested in subsequent subsection.

from the proposed DA-RPMS model. Reference [21] proposes a multi-objective day-ahead reactive power market model which is solved by a deterministic approach using Generalized Algebraic Modeling Systems (GAMS) solvers such as CPLEX and CONOPT. The proposed HFMOEA based multi-zone DA-RPMS model has following advantages in comparison to the model presented in Ref. [21]:

Furthermore, the average total CPU time taken by HFMOEA based multi-zone and single-zone DA-RPMS models are (23 min and 51.38 s) and (22 min and 38.01 s), respectively. Although, these CPU times are slightly higher as compared to the computation time (19 min and 55.26 s) as taken by a deterministic optimization approach like GAMS as used in Ref. [21] but are feasible for a dayahead reactive power market model. In order to further explore the superiority of HFMOEA, the statistical results in terms of mean and standard deviation of two different performance metrics such as spacing (S) suggested by Schott [36] and hypervolume (HV) [37] are determined from 20 independent optimization run for both MOEA as listed in Table 3. The bold values represent the results obtained corresponding to the proposed HFMOEA based

(a) It is evident that the values of TPF as obtained from both the HFMOEA based DA-RPMS models (i.e. single-zone and multi-zone market models) are smaller as compared to the same obtained from the multiobjective model i.e. Case II presented in Ref. [21]. (b) The better values of uniform market clearing prices are obtained from HFMOEA based DA-RPMS model as compare to other model. One of the obvious reasons is that HFMOEA is able to obtain better Pareto-optimal solutions (front) as compared to a deterministic optimization approach like GAMS as used in Ref. [21]. (c) In HFMOEA based multi-zone DA-RPMS model, the reactive power market is settled on localized/zonal basis whereas the model proposed in Ref. [21] is settled by considering the whole 1.1

Maximum permissible bus voltage limit

Bus Voltage (p.u.)

1.05

Case 1: Single-Zone DA-RPMS Model: NSGA-II Case 1: Single-Zone DA-RPMS Model: HFMOEA Case 2: Multi-Zone DA-RPMS Model: NSGA-II Case 2: Multi-Zone DA-RPMS Model: HFMOEA 0.95

Minimum permissible bus voltage limit 0.9 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Bus Number Fig. 10. Output bus voltage profiles from both multi-objective DA-RPMS models.

20

21

22

23

24

98

A. Saraswat et al. / Energy 51 (2013) 85e100

Table 6 Comparison of reactive power output schedules and offer prices in different gaming scenarios to study the effect of market power in day-ahead reactive power market. Voltage control areas (zones)

Zone#1

Bus No.

Unit No.

1

1 2 3 4 1 2 3 1 2 3 1 1 2 3

Case 1, 2 (0.961,2, 0.86, 0.861,2, 0.46) Case 3, 5 (0.963,5, 1.118, 1.1183,5, 0.598) Case 4, 6 (0.964,6, 1.376, 1.3764,6, 0.736) Case 1, 2 (0.94, 0.82, 0.82, 0.45) Case 3, 5 (0.94, 1.066, 1.066, 0.585) Case 4, 6 (0.94, 1.312, 1.312, 0.72) Case 1, 2 (0.85, 0.79, 0.79, 0.39) Case 3, 5 (0.85, 1.027, 1.027, 0.507) Case 4, 6 (0.85, 1.264, 1.264, 0.624) Case 1, 2 (0.83, 0.82, 0.82, 0.4) Case 3, 5 (0.83, 1.066, 1.066, 0.52) Case 4, 6 (0.83, 1.312, 1.312, 0.64) (0.5, 0.54, 0.54, 0.28) (0.42, 0.42, 0.42, 0.356) (0.69, 0.68, 0.68, 0.39) (0.65, 0.62, 0.62, 0.37) (0.75, 0.61, 0.61, 0.43) (0.8, 0.753, 0.75, 0.36) (0.7, 0.65, 0.65, 0.32) (0.68, 0.5, 0.5, 0.31) (0.7, 0.54, 0.54, 0.39) (0.75, 0.6, 0.6, 0.5) (0.94, 0.81, 0.812,5,6, 0) (0.9, 0.85, 0.85, 0.48) (0.95, 0.892,6, 0.89, 0.55) (0.86, 0.8, 0.8, 0.45)

1 2 3 4 5 6 1 1 1 1 2 3 4 5 6

(0.65, 0.6, 0.6, 0.3) (0.5, 0.58, 0.58, 0.25) (0.6, 0.73, 0.732,5, 0.381,2,3,4,5) (0.55, 0.61, 0.61, 0.27) (0.52, 0.5, 0.5, 0.26) (0.51, 0.51, 0.51, 0.27) (0.5, 0.5, 0.5, 0.3) (0.9, 0.85, 0.85, 0.48) (0.8, 0.75, 0.756, 0.41) (0.42, 0.42, 0.42, 0.17) (0.5, 0.481,2,4,6, 0.48, 0.2) (0.45, 0.42, 0.42, 0.38) (0.48, 0.44, 0.44, 0.35) (0.49, 0.45, 0.45, 0.33) (0.55, 0.46, 0.46, 0.32)

1

3

4

2

7

13

14a 23

Zone#3

Reactive power output (in MVAr) Base case scenario

2

Zone#2

Offer prices (a0, m1, m2, m3)

15

16 18 21 22

Total reactive power generation (MVAr) Total payment for reactive power support Services ($)

Gaming scenario (by generating units 1, 2, 3 and 4 at bus no. 1)

Case 1

Case 2

5.23

5.93

e e

e e 5.23

e e e e e e

Case 3 (30%)

Case 4 (60%)

Case 5 (30%)

Case 6 (60%)

e

e

6.06

e e

6.10

e e

6.06

e e

6.10

e e

8.57

e e

8.57

e e

e e

7.60

e e

7.60

e e

e e 3.74

Multi-zone DA-RPMS model

5.93 e e

3.74

Single-zone DA-RPMS model

4.70

4.70 8.34

e e 0.88 e e

e e

e

5.65 5.65 6.07 6.07 15.92 15.92 15.92 44.52 44.52 44.52 90.57 0.52 0.52 43.00

6.23 6.23 9.25 9.25 17.34 17.34 17.34 39.70 39.70 39.70 121.61 L3.17 L3.17 38.04

6.70 6.70 11.86 11.86 18.34 18.34 18.34 42.21 42.21 42.21 120.63 4.72 4.72 48.66

0.88 5.36 5.36 4.45 4.45 16.24 16.24 16.24 48.00 48.00 48.00 109.20 6.15 6.15 50.58

4.74 4.74 4.74 4.74 4.74 52.72 16.22 62.95 4.54 L0.49 L0.49 L0.49 L0.49 L0.49 L0.49

4.66 4.66 4.66 4.66 4.66 50.89 41.91 19.39 16.35 L2.42 L2.42 L2.42 L2.42 L2.42 L2.42

4.23 4.23 4.23 4.23 4.23 41.64 46.18 19.50 1.14 1.02 1.02 1.02 1.02 1.02 1.02

514.49 488.60

519.72 465.74

547.54 654.87

8.34

e e

5.12

e e

5.12

e e

3.18

e 6.37 6.37 10.01 10.01 17.82 17.82 17.82 50.46 50.46 50.46 95.79 6.86 6.86 51.55

3.18 7.04 7.04 13.72 13.72 19.17 19.17 19.17 41.42 41.42 41.42 105.99 3.21 3.21 37.99

4.77 4.77 4.77 4.77 4.77 53.30 34.69 31.19 3.97 0.39 0.39 0.39 0.39 0.39 0.39

4.39 4.39 4.39 4.39 4.39 45.02 23.53 13.16 15.78 0.18 0.18 0.18 0.18 0.18 0.18

3.69 3.69 3.69 3.69 3.69 29.95 50.16 11.89 53.70 1.99 1.99 1.99 1.99 1.99 1.99

540.23 766.12

548.49 476.15

529.69 549.10

Zone#1: {1, 2, 7}, Zone#2: {13, 14, 23}, and Zone#3: {15, 16, 18, 21, 22}. a Synchronous condenser (SC).

method. The metric S is used to evaluate the spacing of the obtained non-dominated solutions, respectively. An algorithm finding smaller values of S is able to find a better diverse set of non-dominated solutions. HV metric provides a qualitative measure of convergence as well as diversity for obtained Pareto-fronts in a combined sense. An algorithm with high value of HV is desirable. Nevertheless, it can be used along with the above metric to get a better overall picture of algorithm performance [13]. From Table 3, it is clear that the proposed HFMOEA has smaller values of S metric as compared to NSGA-II in both the cases. It suggests that the best Pareto-optimal front obtained using HFMOEA is more uniformly distributed as compared to the same obtained by NSGA-II. Whereas, the higher values of HV metric are obtained from HFMOEA in comparison to NSGA-II, which illustrates the overall superiority of HFMOEA. The variations in HFMOEA parameters such as PC and PM for in both the cases corresponding to best Pareto-optimal solutions are

shown in Fig. 8. It is observed that the variations in crossover and mutation probabilities are such that if PC is going to reduce, PM will increase (see Fig. 8). These variations in PC and PM would improve the stochastic search to arrive at near to global Pareto-optimal front as cleared from Fig. 7. Further, the system output control variables such as generator bus voltages, transformer tap settings and shunt inductor obtained after optimization in both the multi-objective DA-RPMS models are given in Table 4. Furthermore, the generators reactive power output schedules and their respective payments obtained after optimization in both the cases of multiobjective DA-RPMS models are listed in Table 5. The bold values in Tables 4 and 5 represent the results obtained corresponding to the proposed HFMOEA based method. The reactive power generation output schedules obtained after optimization using HFMOEA (best compromise solutions) from both multi-objective DA-RPMS Models are graphically compared

A. Saraswat et al. / Energy 51 (2013) 85e100

99

Table 7 Comparison of uniform market clearing prices in different gaming scenario for evaluating market power. Market model

Test cases scenario

Single-zone DA-RPMS model

Case 1 (Base) Case 3 (30%) Case 4 (60%)

Multi-zone DA-RPMS model

Case 2 (Base)

Case 5 (30%)

Case 6 (60%)

Uniform market clearing prices

Zone#1 Zone#2 Zone#3 Zone#1 Zone#2 Zone#3 Zone#1 Zone#2 Zone#3

as shown as in Fig. 9. It is clear that there is no violation in generators reactive power outputs in both the cases. In other words, all the generators reactive power output values are within their corresponding ranges of minimum and maximum permissible limits in both the cases. The bus voltage profiles obtained after optimization using NSGA-II and HFMOEA in both multiobjective DA-RPMS models are also compared as shown in Fig. 10. It is noticed that the bus voltage profiles obtained in case 2 of multi-zone DA-RPMS model is more flat as compared to the same obtained in case 1 of single-zone DA-RPMS model. Hence, on the bases of this analysis, it is clear that multi-zone DA-RPMS model may provides superior optimal solutions as compared to single-zone DA-RPMS model to take better decisions by ISO for clearing the reactive power market. 5.2. Case study B: generators gaming scenario The purpose of this case study is to simulate different gaming scenarios arise when any reactive power service provider (i.e. any generator or any generation company) tries to exercise market power by increasing its offer prices, intentionally. In this case study, four additional test cases along with two base cases (i.e. Case 1 and Case 2) are considered as: In Case 3 and Case 4, the offer prices (i.e. m1, m2 and m3) submitted by four generating units at bus no. 1 are intentionally increased by 30%, and 60% respectively over Case 1 (single-zone DA-RPMS) offer prices. Similarly, in Case 5 and Case 6, again the offer prices (m1, m2 and m3) submitted by four generating units at bus no. 1 are intentionally increased by 30% and 60% respectively over Case 2 (multi-zone DA-RPMS) offer prices. Here, the four generating units at bus 1 are chosen for gaming because these providers are having some market power (price-setters) in Case 1 and Case 2 as clear from Table 6. A comparison of reactive power output schedules and offer prices in different gaming scenarios for evaluating market power in reactive power market is given in Table 6. The bold values represent the offer bids which are selected in different test cases as well as the base case values obtained in Case 2 and Case 2. In this table, the superscript 1 denotes that the particular offer is price-setting offer in Case 1, superscript 2 denotes a price-setting offer in Case 2, superscript 3 denotes pricesetting offer in Case 3 and so on up to superscript 6. All the simulation results in this case study are obtained using HFMOEA based optimization approach. The uniform market clearing prices obtained in all six test cases are compared in Table 7. The bold values again represent the base case results. It is noticed that the market clearing prices of Zone#2 and Zone#3 remain almost same even after a 30% or 60% increase in offer prices submitted by four generating units at bus no 1 in Case 5 and Case 6. In contrast, the market clearing prices are increased to

TPF ($)

r0

r1

r2

r3

0.96 0.96 0.96

0.48 0.75 0.48

0.86 1.118 1.376

0.38 0.38 0.38

488.60 654.87 766.12

0.96

0 0.89 0.48 0 0 0 0 0.89 0.48

0.86 0.81 0.73 1.118 0.81 0.73 1.376 0.81 0.75

0 0 0.38 0 0 0.38 0.35 0 0

465.74

0.96

0.96

476.15

549.10

very high values as obtained in Case 3 and Case 4. This shows that any attempt to game the prices by a market power holder in one zone will not affect the market prices of other zones, if market settlement is based on proposed multi-zone DA-RPMS model. 6. Conclusion In this work, a multi-objective localized/zonal DA-RPMS model is proposed in which three contradictory objective functions: TPF, TRTL and VSEI are optimized simultaneously subjected to various system constraints in order to clear a reactive power market on zonal basis. A new HFMOEA approach is applied for solving complex mixed integer nonlinear multi-objective optimization problems like DARPMS for taking critical decisions in competitive electricity market. The proposed multi-zone DA-RPMS model is compared with singlezone DA-RPMS model on IEEE 24 bus RTS. Further, the performances of NSGA-II and HFMOEA for the solution of both DA-RPMS models are tested and compared in terms of best compromise solutions and their Pareto-optimal fronts. The major findings of the work may be concluded in three perspectives. Firstly, the output simulation results prove the superiority of multi-zone DA-RPMS model over single-zone DA-RPMS model in terms saving of total payment for reactive power support services with smaller values of transmission loses. Secondly, as far as performance of proposed HFMOEA is concerned, it has been found that HFMOEA is able to generate superior Pareto-optimal front as compared to NSGA-II. Thirdly, in zone DARPMS model for reactive power market settlement, if a generator from one voltage control area/zone attempts to game the price, the market prices in other voltage control areas/zones will not be affected. This can prove that separating voltage control areas/zones will help the ISO to improve the fairness of market in economic views. Therefore, it is concluded that a HFMOEA based multi-zone DA-RPMS model provides a set of superior optimal solutions for taking better market clearing decisions by ISO. Acknowledgments The authors are grateful to the anonymous reviewers for their valuable and constructive comments along with helpful suggestions which greatly encouraged them to improve the paper’s quality. References [1] Federal Energy Regulatory Commission Staff. Report on principles for efficient and reliable reactive power supply and consumption. FERC Docket No. AD051-000; February 4, 2005. [2] Ahmed S, Strbac G. A method for simulation and analysis of reactive power market. IEEE Trans Power Syst 2002;15(3):1047e52. [3] Lin XJ, Yu CW, Chung CY. Pricing of reactive support ancillary services. IEEE Proc Gener Trans Distrib 2005;152(5):616e22.

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