Static and Dynamic Security-Augmented Market ...

International Review of Electrical Engineering (I.R.E.E.), Vol. 7, N. 1 ISSN 1827- 6660 January-February 2012

Static and Dynamic Security-Augmented Market-Based Generation Scheduling Using Multi-Objective Artificial Bee Colony Algorithm M. Gitizadeh1, J. Aghaei2 Abstract – After some destructive blackouts, voltage stability and collapse have become worldwide concerned problems for electricity energy networks. Also, with increased loading and utilization of power transmission systems, voltage stability has become a rising concern in electric power utilities. Static aspects of power system security, such as voltage security margin, cannot guarantee the secure operation of the power system. Besides, dynamic modeling and evaluation of power system security is a complex and time consuming task. In this regard, to cope with such problems, a multi-objective framework is proposed which concurrently considers static and dynamic aspects of power system security in the scheduling problem of electricity energy markets. This paper presents a new approach based on the Multi-Objective Artificial Bee Colony (MOABC) algorithm to solve a security augmented market-based generation scheduling. In the proposed multi-objective framework, generation offer cost, Voltage Security Margin (VSM) (as static security index) and linearized Corrected Transient Energy Margin (CTEM) (as dynamic security index) of system are considered as objective functions of market-based generation scheduling. The New England test system is used as an example to illustrate the practical application of this security-augmented energy market clearing mechanism. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Market-Based Generation Scheduling, Multi-Objective Artificial Bee Colony (MOABC), Corrected Transient Energy Margin (CTEM), Voltage Security Margin (VSM)

PGmax,i and PGmin,i

Nomenclature i, j NB NU r n Zi PGi e i

NC SFj CTEM0 CTEMj

|Vi| |Vmax,i| and |Vmin,i| QGi

Indices of bus Number of system buses Number of system units Index for Pareto optimal solution Index of objective functions A binary variable indicating that the unit of ith bus accepted or not in the energy market Energy output of the unit at ith bus Bid price of the unit at ith bus for energy Number of credible contingencies The sensitivity of CTEM with respect to the generation of unit j (PGj) The system CTEM at the base case The system CTEM after a little change PGj applied in the generation of the unit j with respect to the base case Voltage magnitude of the ith bus Upper and lower limits of voltage magnitude of ith bus, respectively Reactive power output of the unit at ith bus

Manuscript received and revised January 2012, accepted February 2012

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QGmax,i and QGmin,i PDi and QDi ij

Yij

ij

Sij Sij

I.

Upper and lower limits of active power of the unit at ith bus, respectively Upper and lower limits of reactive power of the unit at ith bus, respectively Active and reactive loads of ith bus, respectively Difference between voltage angles of buses i and j Magnitude of element located in ith row and jth column of the admittance matrix of the power system Angle of element located in ith row and jth column of the admittance matrix of the power system Magnitude of apparent power flow of branch between ith and jth buses Apparent power flow capacity of branch between ith and jth buses

Introduction

In recent years, the electricity industry has experienced extreme changes due to a world-wide

Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

M. Gitizadeh, J. Aghaei

deregulation/privatization process that has considerably affected energy markets. In the new market-based environment, economic issues are the main responsibility of the system operators. However, due to the fact that an insecure network may lead to system collapse which results in significant financial and societal losses, power system security has taken a secondary role [1]. Hence, the power system security has became one of main responsibilities of market operators which should consider a simple, unambiguous and transparent way to evaluate system security, so that the “right” market signals can be conveyed to all market participants [2]. The power system security problems are classified as static and dynamic. The static security problem implies evaluating the system steady state performance for all possible contingencies. Besides, the dynamic analysis of security evaluates the time-dependent transition from the pre-contingent state to the post-contingent state [3]. In the new structure, the static security is usually measured through “system congestion” levels, which have a direct effect on market transactions and energy prices [1]. A wide range of optimization techniques have been applied in solving economic scheduling of generating units problems. Some of these techniques are based on mathematical optimization methods while others are based on artificial intelligence methods or heuristic algorithms. Guy [4] presented the early ideas of SCUC that the security constraints are usually relaxed by an additional set of associated multipliers such as the direct method reported in [5]. References [6]-[9] use Benders Decomposition (BD) for separating the UC in the master problem from the network security check in the sub-problem. Recently, Linear Programming (LP) [10] and Semi-Definite Programming (SDP) [11] for unified solution of SCUC have been presented. In [2], a multi-objective approach based on interior point method is used in an optimal power flow, so that the social benefit and the distance to a maximum loading condition based on a static analysis framework are maximized at the same time. Methods based on artificial intelligence techniques, such as artificial neural networks, are also implemented in many research works [12], [13]. Recently, many heuristic search techniques such as Particle Swarm Optimization (PSO) [14] and Genetic Algorithms (GAs) [15] are applied successfully to the market-based optimization problems. Considering static security cannot guarantee global security of a power system. Therefore, dynamic security constrained dispatch of an electric power network is a challenging task. Therefore, system dynamic security constraints have been concerned in economic dispatch [16], optimal power flow [17]–[20], and optimal operation [21]. For instance, transient stability constraints, such as the transient voltage constraint and steady-state thermal limit, are incorporated in these optimization problems. Also, transient stability constraints based on the theory of potential energy boundary surface (PEBS) have been used in [18]. In the deregulated framework where generation, transmission,


and distribution are separated entities, generation dispatch when dynamic security concerns have to be taken into account is a more complicated task [22]. To deal with this problem, the corrected hybrid method combining time-domain simulation [22] and corrected transient energy function (CTEF) have been proposed [23]. The CTEF is really a method for computing a stability index called the corrected transient energy margin (CTEM). An important feature of the CTEM is that it bears a linear relationship, within a usable range, to important control variables such as generator power exchanges [23]. Despite the broad studies about the power market issues, to the best of our knowledge, there is no work considering multi-objective market-based generation scheduling ensuring power system static and dynamic security, concurrently. In the present paper, a new multiobjective model for the market-based optimal scheduling of generating units is proposed considering dynamic security aspect of power system based on the linear formulations. The proposed multi-objective model includes generation offer cost, CTEM as dynamic security index and VSM as the voltage security index as competing objective functions. The paper illustrates how system security is accounted for and managed in optimal scheduling of generating units problem and discusses as well the effect that this mechanism has on the generating units’ dispatch levels and energy prices. Moreover, a new approach incorporating MOABC method is proposed to solve the multi-objective problem. The remainder of this paper is organized as follows: In Section II, the proposed multi-objective model for the market-based scheduling of generating units, considering dynamic and voltage security aspects, is formulated in the form of an optimization problem. Section III introduces the implemented MOABC approach for the multi-objective problem solution. In the next section, obtained results for the New England test system are presented and discussed to demonstrate the effectiveness of the proposed scheme. Some relevant conclusions are drawn in the Section V.

II.

Market-Based Generation Scheduling

To better illustrate the proposed multi-objective market-based generation scheduling problem, singleperiod scheduling without inter-temporal constraints (including ramp rate limits and minimum up/down time constraints) is considered, as this model is simpler to describe and analyze. However, in the proposed model, inter-temporal constraints can be easily incorporated based on the procedures presented in our previous work [24]. Considering these assumptions, the objective functions and constraints of the proposed model for market-based generation scheduling are presented in the following subsections, respectively.

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II.1.

Conventional Security-Constrained SingleObjective Market Clearing

II.2.

A market-clearing procedure is typically solved by the system operator to determine the set of accepted bids and offers and the resulting market clearing price [25]. Therefore, unit commitment, generation and consumption levels, and the price of energy are altogether outcomes of the optimization procedure. These results consequently depend on the bids and offers submitted by market agents, which are known by the system operator before running the market clearing procedure. Based on this approach security-constrained energy market-clearing problem using AC power flow formulation can be summarized as the following optimization problem: NB e i

min f1

(1)

PG i

i 1

subject to: NB

PGi

PDi

Vi V j Yij cos

ij

ij

j 1

(2)

i 1,..., NB NB

QG i

QDi

Vi V j Yij sin

ij

ij

j 1

(3)

i 1,...,NB Z i PG mini

PGi

Zi PG maxi

(4)

Z i QG mini

QGi

Z i QG maxi

(5)

for all branches

(6)

Sij

Sij

Security-Augmented Multi-Objective Market Clearing

As the first objective, it is desired to minimize offer cost of energy (f1). In the joint market clearing of energy and reserves auctions, the objective function is to minimize energy offer cost [26]: NB

Vi

Vmax,i for all buses

(7)

Equation (1) expresses the offer cost of the participating generating units in the energy auction. Also, equations (2) and (3) refer to AC power flow equations. Inequalities (4) and (5) permit PGi and QGi to be either zero (Zi =0) or at a value between the minimum and maximum range of the unit output (Zi =1), respectively. In other words, Zi binds generating units output to their operating limits. Moreover, in constraints (6) and (7) voltage of buses and branches’ flows, calculated by AC power flow formulations, are bounded by their allowable limits. Indeed, inequalities (6) and (7) form the so-called steadystate security constraints.


(8)

PG i

i 1

The second objective (f2) is corrected transient energy margin (CTEM) which needs to be maximized. Dynamic security characterizes a power system’s ability to withstand disturbances and ensure continuity of the service. For transient stability analysis, there are three methodologies which can be classified as: time domain simulation methods, direct or transient energy function (TEF) methods and dynamic reduction methods. Among the existing methods, in this paper Corrected Transient Energy Margin (CTEM) [23] as a well-known and widely-used transient stability index is employed. Indeed, CTEM belongs to the second classification of transient stability analysis methods. Hypothetically, CTEM is positive for a transient stable system and measures its stability margin. Moreover, it is proved in theory that CTEM ascertains a linear and proper criterion to estimate transient stability. The basic concepts and mathematical details of CTEM for the dynamic security enhancement in power-market systems can be found in [23]. CTEM approximately has a linear relationship with respect to several operating parameters (including fault clearing time, pool generation rescheduling and curtailment of a bilateral transaction) over a wide range of operating conditions [23]. This linear relationship of CTEM change can be expressed as follows: CTEM PG1 ,...,PG NU NU

Vmin,i

e i

f1

j 1

CTEM PGj

(9)

NU

PGj

SF j

PGj

j 1

where, SFj refers to the sensitivity of CTEM with respect to the generation of unit j (PGj). To obtain the sensitivity of the system CTEM with respect to each generation shift, the first order approximation of the Taylor series for the CTEM around the operating point of the power system is used. In other words, each sensitivity factor is approximated by its first order term based on the approximate linear relationship of CTEM with respect to generation shifts. To compute the sensitivity of the CTEM with respect to the generation of unit j, a small perturbation is applied to this generation. Then the change in CTEM is calculated. The CTEM sensitivity with respect to the generation of unit j (SFj) is obtained as:

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CTEM PGj

CTEM PGj

CTEM j

CTEM 0 PGj

(10) =SF j

where, CTEM0 is the network CTEM at the base case. Also, CTEMj shows the system CTEM after a little perturbation PGj is applied to the generation of the unit j with respect to the base case. It should be noted that, CTEM0 and CTEMj are calculated on the basis of the above step by step algorithm. In order to maintain power system dynamic security in the market clearing process, CTEM based on the linearized formulation of (9) is considered as the second objective function f2 of multi-objective model of the market-based scheduling of generating (f2 should be maximized): f2

CTEM

CTEM 0

CTEM

N

CTEM 0

SF j

voltage stability bifurcation analysis in the load domain. In CPF, loads and generations are increased by the loading parameter of as:

PGj

(11)

j 1

The objective function f2 in (11) is computed based on a single fault. However, in practice, a list of credible contingencies (here, credible faults) is usually considered for a power system obtained from a contingency ranking method [27]. So, we should consider a CTEM computed based on each credible contingency. In other words, we will have f2=CTEM1,…, fn+1=CTEMn, for a list of n credible contingencies. The other alternative is combining the CTEM objective functions of the credible contingencies (CTEM1,…, CTEMn) based on the weighted sum, with the weight value of each CTEM objective function determined according to the contingency ranking method [27], to construct a single objective function f2 for all credible contingencies. However, for the sake of conciseness and simplicity of the formulations, we will only consider a single credible contingency and compute f2 based on it in the subsequent sections, but our solution method can be extended for any number of objective functions. Voltage stability margin (f3) is the third objective which needs to be maximized. In order to determine whether or not the power system meets the voltage level and security requirements specified, it is necessary to maintain an adequate level of voltage stability in both normal and probable contingency operating states. The voltage magnitudes may be determined by solving the load flow equations for the system. To evaluate the security margin, SM, it is necessary to determine the system critical condition, i.e. Voltage Stability Limit (VSM). VSM in the load domain, which measures the distance from the current operating point to the voltage collapse point in terms of the load increment, is an efficient and widely used voltage stability index [1]. Continuation Power Flow (CPF), which is widely used in voltage stability studies, is applied here as a tool for the Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

PG

PD

PD

BC

PDstep

(12)

QD

QD

BC

QDstep

(13)

PG

BC

kG

PGstep

(14)

where, PD(BC), QD(BC), and PG(BC) are powers of loads and generators at the base case and PDstep, QDstep, and PGstep are generator and load power increments. The loading parameter of multiplies generator and load increments. KG balances network losses when load and generation are increased. As shown in Fig. 1, the maximum value of the loading parameter form the base case up to the Saddle Node Bifurcation (SNB) point represents the VSM and employed in this paper as the voltage stability index. The VSM in the load domain as defined above indicates the power system maximum loadability in terms of voltage stability [27]. In this paper, the VSM is normalized by the base case load and expressed in percent. The VSM as determined above can be written as a function of decision variables of energy market clearing optimization problem: VSM

f PDk ,QDk ,PGj

PSNB PBC PBC

(15)

In voltage stability studies, PDk and QDk of a demand are usually increased with a constant power factor. To do this, the power increments of PDstep and QDstep for each demand are considered with the same power factor of the demand.

Fig. 1. Bifurcation and P-V curves for voltage stability analysis

The third objective function of the MMP model of the market clearing problem is as follows (f3 should be maximized): International Review of Electrical Engineering, Vol. 7, N. 1

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f3

(16)

VSM

It is noted that in this paper VSM is calculated based on the CPF calculations. The multi-objective problem of market-based generation scheduling can be summarized as follows: NB e i Gi

min f1

, max f 2

CTEM ,

i 1

max f3

(17)

VSM

The optimization problem of (17) should be solved subject to constraints (2)-(7). In summary, the decision variable of the multi-objective optimization problem is: PGi.

III. Multi-Objective Artificial Bee Colony (MOABC) In Multi-objective Mathematical Programming (MMP) there is more than one objective function and there is no single optimal solution that simultaneously optimizes all the objective functions [28], [29]. In these cases, the decision makers are looking for the “most preferred” solution. In MMP, the concept of optimality is replaced with that of efficiency or Pareto optimality. Indeed, in multi-objective optimization, the goal is to find a set of solutions that represent a trade-off among the objectives; because there is no single solution available for the problem. Usually an external archive is used to maintain the solutions. The efficient (or Pareto optimal, non-dominated, non-inferior) solution is the solution that cannot be improved in one objective function without deteriorating its performance in at least one of the rest. In a Pareto-based approach, the nondominated solutions constitute a Pareto-front. It is popular that a subset of this Pareto-front or a single Pareto solution will be chosen for more considerations. It is because that without any further information, none of these solutions is better than the others. To solve this type of problems many algorithms have been introduced. Among them, the evolutionary and the new swarm intelligence (SI) algorithms which have high performance have become popular. One of these new SI algorithms is Artificial Bee Colony (ABC) algorithm which is based on the foraging behavior of honey bees. Multi-objective artificial bee colony (MOABC) was recently introduced in [30] and [31]. MOABC is mainly based on the Artificial Bee Colony (ABC) which is a single objective optimization algorithm [32]-[34]. MOABC has been tested on standard multi objective benchmarks and its performance has been evaluated upon two metrics. In single objective problems there is no need of metrics because, for example, in a minimization problem the result with the lowest value is the best one, but in multi objective problems since we have a set of solutions we cannot compare them in a straightforward manner with each other. That is where metrics show their


usefulness. These metrics are Generational Distance and Spacing. Generational Distance is a way to estimate the gap between the external archive (which is the output of the algorithm) and the optimal Pareto front. Spacing is a metric that measures the distribution of external archive. Performance analysis has shown that MOABC results are highly competitive based on those metrics with respect to other multi-objective algorithms [31]. In MOABC we have three types of bees which search in D-dimensional search space: employed bee, onlooker bee and scout bee. This algorithm assumes a number of food sources and works through optimizing them. Each of these food sources represents a solution of the problem. These food sources have been found by the employed bees and will be presented on the dance area. The onlooker bees are waiting on the dance area to choose a food source by its quality. This quality is relative to the objective functions we are optimizing. Note that in MOABC, the number of Employed bees and Onlooker bees are equal. When a food source could not get optimized in a few cycles, the Employed bee of that food source will become a Scout bee and it will do a random search on the problem space to find a new food source. Fig. 2 presents the MOABC algorithm in pseudo code. MOABC receives a set of parameters as inputs and employs the illustrated procedure to find the Pareto-front. Max_Trial is the parameter used to identify the food sources that should be abandoned. The Epsilon is a parameter determined heuristically for the problem at hand and it is used for maintaining diversity over the external archive. Generating units' scheduling is a realvalued unconstrained problem which has D parameters and each of these parameters can be represented by a dimension. Hence, we have a D-dimensional search space S that S R D . MOABC associates a position vector xi (i=1,2,…,D) with each food source. As it has been noted beforehand, each of these food sources represents a solution for the generating units' scheduling. Each dimension has a lower and upper bound that can be represented like Lbi