Attuned design of demand response program and M-FACTS for relieving congestion in a restructured market environment

Abstract

This paper addresses the attuned use of multi-converter flexible alternative current transmission systems (M-FACTS) devices and demand response (DR) to perform congestion management (CM) in the deregulated environment. The strong control capability of the M-FACTS offers a great potential in solving many of the problems facing electric utilities. Besides, DR is a novel procedure that can be an effective tool for reduction of congestion. A market clearing procedure is conducted based on maximizing social welfare (SW) and congestion as network constraint is paid by using concurrently the DR and M-FACTS. A multi-objective problem (MOP) based on the sum of the payments received by the generators for changing their output, the total payment received by DR participants to reduce their load and M-FACTS cost is systematized. For the solution of this problem a nonlinear time-varying evolution (NTVE) based multi-objective particle swarm optimization (MOPSO) style is formed. Fuzzy decision-making (FDM) and technique for order preference by similarity to ideal solution (TOPSIS) approaches are employed for finding the best compromise solution from the set of Pareto-solutions obtained through multi-objective particle swarm optimization-nonlinear time-varying evolution (MOPSO-NTVE). In a real power system, Azarbaijan regional power system of Iran, comparative analysis of the results obtained from the application of the DR & unified power flow controller (UPFC) and the DR & M-FACTS are presented.

Graphical abstract

Keywords

multi-converter flexible alternative current transmission systems (M-FACTS) / demand response / fuzzy decision making / multi-objective particle swarm optimization-nonlinear time-varying evolution (MOPSO-NTVE)

1 Introduction

Conventionally, system operators (SOs) under the traditional vertically integrated system try to operate and maintain system security by providing a minimum operation and maintenance costs to maximize the social welfare [1,2]. In the new era, the additional task of allocating a fair congestion cost to all market participations (MPs) poses a new challenge underneath the competitive environment [3].

The technology of flexible alternative current transmission systems (FACTS) all over the world is playing a key role in fostering the transmission network to be utilized to its full potential [4]. Many authors have developed the methodology to incorporate FACTS devices to manage the transmission congestion. A genetic algorithm (GA) has been used to determine the optimal placement of unified power flow controller (UPFC) in order to maximize the loadability of the system [5]. The one thyristor controlled series compensator (TCSC), four thyristor controlled phase shifters (TCPSs) and one UPFC have been located individually to alleviate the congestion. Simultaneous use of two thyristor controlled phase angle regulators (TCPARs) and two UPFCs have also been studied [6].

There are several possibilities of operating configuration by incorporating two or more converter blocks with flexibility. Among these configurations, there are two novel operating configurations, namely the interline power flow controller, the generalized unified power flow controller and generalized interline power flow controller, which are significantly extended to control power flows of multi-lines or a sub-network rather than control power flows of multi-lines or a sub-network rather than control power flow of single line by a UPFC or static synchronous series compensator (SSSC).

Demand response (DR) has been introduced as a potential solution to network problems [7]. DR generally refers to the adjustment of electricity usage by consumers in response to changes in market prices or when network jeopardized [8]. There are several studies around DR applications on different power system aspects. As it can be concluded from the definition of DR, most of the studies have been focused on the economic impacts of DR programs on electricity market [9]. On the other hand, in some studies DR programs have triggered to enhance network reliability [10]. In this regard, these two principle DR impacts have been compared and the present status and future trends of DR programs have been discussed [11]. Certain studies have aimed to develop DR models for evaluating the impact of DR programs on nodal or total power demand. DR modeling based on the idea of demand-price elasticity has been addressed [12,13]. An attempt has been made to validate the effects of DR programs based on analytical and technical methods [14]. Moreover, customer baseline load (CBL) has been applied in DR modeling [15]. The probabilistic DR model has been modified to estimate the behavior of thermostatic loads [16]. Furthermore, the impact of DR implementation has been modeled by a linear optimization technique [17], A multi-objective model has been employed based on DR price and a fuzzy method has been used to deal with this multiple objects model [18].

The main intent of the present paper is to combine the multi-converter flexible alternative current transmission systems (M-FACTS) with the DR program to eliminate congestion in a power market. The M-FACTS extends the concept of power and voltage control beyond what have been achieved with the known two-converter UPFC and interline power flow controller (IPFC) controller. The optimal power flow with the M-FACTS devices would be a very useful tool to operate, plan and manage power system effectively. So, the M-FACTS consisting of three converters, one connected in shunt and two in series with two transmission lines exiting a substation is considered. Besides, the DR defined as changes in electricity use by end-users from their normal consumption pattern in response to changes in market prices is considered. To locate the M-FACTS in order to relieve congestion in the deregulated power sector, a sensitivity-based approach has been used. In a real power system, the nonlinear optimization problem formulation with the M-FACTS and DR program to see the impact of these two tools on market cost, including market clearing and congestion cost, has been survived. The fitness functions used are the sum of the payments received by the generators for changing their output as compared to the original generation schedule, the total payment received by demand response participants to reduce their load and M-FACTS cost. With regard to the privilege of multi-objective particle swarm optimization-nonlinear time-varying evolution (MOPSO-NTVE) optimization method in encouraging particles to wander through the entire search space, instead of clustering around a local optimum, during early iterations of the optimization, this style is utilized to solve the problem and its performance is compared with that of the improved non-dominated sorting genetic algorithm-II (INSGA-II). A decision-making procedure based on FDM and TOPSIS method is adopted to rank the Pareto-optimal solutions from the best to worst and to determine the best solution in a deterministic environment with a single decision maker.

In summary, the main contribution of this paper is 4-fold: the integration of M-FACTS and DR for congestion management, the optimization of the congestion management cost and total cost of market, the solving of the multi-objective optimization problem of congestion management using M-FATS and DR by MOPSO-NTVE, and the use of the FDM approach to choose the best compromise solution from the set of Pareto solutions obtained through MOPSO-VTVE.

To validate the accuracy of the proposed method, case studies on the real power system of Azerbaijan regional electric of Iran are presented and simulation results of the M-FACTS and DR are compared with those of the UPFC and DR. The proposed method shows the benefits of M-FACTS and DR in a deregulated power market, and demonstrates the utilization of them by ISO to improve the total system social welfare and prevent congestion.

2 Multi-converter FACTS devices

In steady-state operation, the main objective of the M-FACTS is to control the voltage and power flow [19]. The equivalent circuit of the M-FACTS consisting of a controllable shunt injected voltage source and two controllable series injected voltage sources is shown in Figs. 1 and 2. The Z_{\mathrm{sh}i} and Z_{\mathrm{se}in} in Fig. 2 are shunt and series transformer impedances. The controllable injected voltage sources that are indicated in Fig. 2 are described as

(1)V_{\mathrm{sh}i}=V_{\mathrm{sh}i}\angle {\theta }_{\mathrm{sh}i},

(2)V_{\mathrm{se}in}=V_{\mathrm{se}in}\angle {\theta }_{\mathrm{se}in},

where n=j,\hspace{0.17em}k,\hspace{0.17em}\cdots.

2.1 Power flow equations of M-FACTS

According to the equivalent circuit of the M-FACTS shown in Fig. 2, the power flow equations can be derived as

(3)$$\begin{gathered} P_i=V_i^{\mathrm{2}}{g}_{ii}-V_i{V}_{\mathrm{sh}i}(g_{\mathrm{sh}i}\cos ({\theta }_i-{\theta }_{\mathrm{sh}i})+b_{\mathrm{sh}i}\sin ({\theta }_i-{\theta }_{\mathrm{sh}i})) \\ -{\displaystyle \sum _n{V}_i{V}_n(g_{in}\cos {\theta }_{in}+b_{in}\sin {\theta }_{in})} \\ -{\displaystyle \sum _n{V}_i{V}_{\mathrm{se}in}(g_{in}\cos ({\theta }_i-{\theta }_{\mathrm{se}in})+b_{in}\sin ({\theta }_i-{\theta }_{sein}))}, \end{gathered}$$

(4)$$\begin{gathered} Q_i=-V_i^{\mathrm{2}}{b}_{ii}-V_i{V}_{\mathrm{sh}i}(g_{\mathrm{sh}i}\sin ({\theta }_i-{\theta }_{\mathrm{sh}i})+b_{\mathrm{sh}i}\sin ({\theta }_i-{\theta }_{\mathrm{sh}i})) \\ -{\displaystyle \sum _n{V}_i{V}_n(g_{in}\sin {\theta }_{in}-b_{in}\cos {\theta }_{in})} \\ -{\displaystyle \sum _n{V}_i{V}_{\mathrm{se}in}(g_{in}\sin ({\theta }_i-{\theta }_{\mathrm{se}in})-b_{in}\cos ({\theta }_i-{\theta }_{\mathrm{se}in}))}, \end{gathered}$$

(5)$$\begin{gathered} P_{ni}=V_n^{\mathrm{2}}{g}_{nn}-V_i{V}_n(g_{in}\cos ({\theta }_n-{\theta }_i)+b_{in}\sin ({\theta }_n-{\theta }_i)) \\ \cdot V_n{V}_{\mathrm{se}in}(g_{in}\cos ({\theta }_n-{\theta }_{sein})+b_{in}\sin ({\theta }_n-\theta s{e}_{in})), \end{gathered}$$

(6)$$\begin{gathered} Q_{ni}=-V_n^{\mathrm{2}}{b}_{nn}-V_i{V}_n(g_{in}\sin ({\theta }_n-{\theta }_i)-b_{ij}\cos ({\theta }_n-{\theta }_i)) \\ \cdot V_n{V}_{\mathrm{se}in}(g_{in}\cos ({\theta }_n-{\theta }_{\mathrm{se}in})-b_{in}\cos ({\theta }_n-{\theta }_{\mathrm{se}in})), \end{gathered}$$

g_{\mathrm{sh}i}+j{b}_{\mathrm{sh}i}=\mathrm{1}/Z_{\mathrm{sh}i},g_{in}+j{b}_{in}=\mathrm{1}/Z_{\mathrm{se}in},g_{nn}+j{b}_{nn}=\mathrm{1}/Z_{\mathrm{se}in},g_{ii}=g_{\mathrm{sh}i}+{\displaystyle \sum _n{g}_{in}},b_{ii}=b_{\mathrm{sh}i}+{\displaystyle \sum _n{b}_{in}},

where n=j,\hspace{0.17em}k,\hspace{0.17em}\cdots.

2.2 Operating constraints of M-FACTS

According to the operating principle of the M-FACTS, the operating constraint representing the active power exchange between converters via the common DC link is

(7)\mathrm{PE}=\mathrm{Re}\left(V_{\mathrm{sh}i}{I}_{\mathrm{sh}i}^{\ast }-{\displaystyle \sum _n{V}_{\mathrm{se}in}{I}_{ni}^{\ast }}\right)=\mathrm{0},

where n=j,\hspace{0.17em}k,\hspace{0.17em}\cdots

The corresponding controllable injected voltage sources are restricted by their lower and upper limits. It is necessary to mention here that in this paper, the M-FACTS with a shunt converter and two series converters is considered in problem of integration of M-FACTS and DR for congestion management problem.

3 Cost of FACTS devices

For more practical optimal placement and sizing of FACTS devices, it is recommended to also consider their investment costs in the objective function. An example of the investment cost as a function of kvar for different FACTS devices is provided by Habur et al. [20]. Here, the cost of M-FACTS and UPFC device is presented in a quadratic form as

(8)C_{\mathrm{UPFC}}=\mathrm{0.0003}{S}_{\mathrm{UPFC}}^{\mathrm{2}}-\mathrm{0.2691}{S}_{\mathrm{UPFC}}+\mathrm{188.22},

(9)C_{M-FACTS}=\mathrm{0.00045}{S}_{M-FACTS}^2-\mathrm{0.40365}{S}_{M-FACTS}+\mathrm{282.33},

where C_UPFC, C_M-FACTS, S_UPFC and S_M-FACTS are the total investment cost (in US$/kvar) and the size (in Mvar) related to UPFC and M-FACTS. The cost of installation of the UPFC and M-FACTS devices has been mathematically formulated and is given by

(10){\mathrm{UPFC}}_{\cos t}=\mathrm{1000}\times C_{\mathrm{UPFC}}\times S_{\mathrm{UPFC}},

(11){M-FACTS}_{\mathrm{cost}}=\mathrm{1000}\times C_{M-FACTS}\times S_{M-FACTS},

where UPFC_cost and M-FACTS_cost are the cost of UPFC and M-FACTS in US$.

4 Location of M-FACTS

The severity of the system loading under normal and contingency cases can be described by a real power line flow performance index [21], as given by

(12)\mathrm{PI}={{\displaystyle \sum _{m=\mathrm{1}}^{N_l}{\displaystyle \frac{{\omega }_m}{\mathrm{2}{n}_p}}\left({\displaystyle \frac{P_{lm}}{P_{lm}^{\max }}}\right)}}^{\mathrm{2}{n}_p},

where P_{lm} is the real power flow and P_m^{\max } is the rated capacity of line m, n_p is the exponent, and {\omega }_m a real nonnegative weighting coefficient reflecting the importance of the lines. N_l is the total number of lines in the network. The value of the exponent has been taken as 2 and {\omega }_i=\mathrm{1}. The real power flow PI sensitivity factors with respect to the control parameters of M-FACTS can be defined as

c_{\mathrm{1}}^k={{\displaystyle \frac{\partial \mathrm{PI}}{\partial V_{\mathrm{se}ij}}}|}_{V_{\mathrm{se}ij}=\mathrm{0}},\hspace{0.17em}\mathrm{PI}\hspace{0.17em}\mathrm{sensitivity}\hspace{0.17em}\mathrm{with}\hspace{0.17em}\mathrm{respect}\hspace{0.17em}\mathrm{to}\hspace{0.17em}{V}_{\mathrm{se}ik},

c_{\mathrm{2}}^k={{\displaystyle \frac{\partial \mathrm{PI}}{V_{\mathrm{se}ij}\partial {\theta }_{\mathrm{se}ij}}}|}_{V_{\mathrm{se}ij}=\mathrm{0}},\mathrm{PI}\hspace{0.17em}\mathrm{sensitivity}\hspace{0.17em}\mathrm{with}\hspace{0.17em}\mathrm{respect}\hspace{0.17em}\mathrm{to}\hspace{0.17em}{\theta }_{\mathrm{se}ij},

c_{\mathrm{3}}^k={{\displaystyle \frac{\partial \mathrm{PI}}{\partial V_{\mathrm{se}ik}}}|}_{V_{\mathrm{se}ik}=\mathrm{0}},\mathrm{PI}\hspace{0.17em}\mathrm{sensitivity}\hspace{0.17em}\mathrm{with}\hspace{0.17em}\mathrm{respect}\hspace{0.17em}\mathrm{to}\hspace{0.17em}{V}_{\mathrm{se}ik},

c_{\mathrm{4}}^k={{\displaystyle \frac{\partial \mathrm{PI}}{V_{\mathrm{se}ik}\partial {\theta }_{\mathrm{se}ik}}}|}_{V_{\mathrm{se}ik}=\mathrm{0}},\mathrm{PI}\hspace{0.17em}\mathrm{sensitivity}\hspace{0.17em}\mathrm{with}\hspace{0.17em}\mathrm{respect}\hspace{0.17em}\mathrm{to}\hspace{0.17em}{\theta }_{\mathrm{se}ik},

c_{\mathrm{5}}^k={{\displaystyle \frac{\partial PI}{\partial I_{\mathrm{sh}}}}|}_{I_{\mathrm{sh}}=\mathrm{0}},\mathrm{PI}\hspace{0.17em}\mathrm{sensitivity}\hspace{0.17em}\mathrm{with}\hspace{0.17em}\mathrm{respect}\hspace{0.17em}\mathrm{to}\hspace{0.17em}{I}_{\mathrm{sh}}.

Using Eq. (12), the sensitivity of PI with respect to M-FACTS parameter X_k (V_{\mathrm{se}ij},{\theta }_{\mathrm{se}ij}, V_{\mathrm{se}ik}, {\theta }_{\mathrm{se}ik} and I_{\mathrm{sh}}) connected between bus-i and bus-j can be written as

(13){\displaystyle \frac{\partial \mathrm{PI}}{\partial X_k}}={{\displaystyle \sum _{m=\mathrm{1}}^{N_l}{\omega }_m{P}_{lm}^{\mathrm{3}}\left({\displaystyle \frac{\mathrm{1}}{P_{lm}^{\max }}}\right)}}^{\mathrm{4}}{\displaystyle \frac{\partial P_{lm}}{\partial X_k}}.

5 Congestion management formulation based on demand response and M-FACTS

At the beginning of deregulation, customers usually did not have active participation in power markets and therefore, they were unable to respond to the prices effectively. However, to have a complete competitive market, there should be enough motivations for customers to participate in power market operation. DR programs have created such opportunities for customers to be as players in the market [22]. DR programs can be defined as the changes in electricity use by end-users from their normal consumption patterns in response to the changes in the price of electricity over the time.

To manage the congestion caused by generation and demand re-dispatch, the amount of demand reduction by DR program is calculated [23]. The investment cost of M-FACTS devices are used to illustrate their impact on congestion management against their costs. The congestion management due to generation re-dispatch, demand re-dispatch and M-FACTS devices is formulated as

(14)\mathrm{Minimize}\hspace{0.17em}F=\left\{{\displaystyle \sum _{j=\mathrm{1}}^{N_G}\left|C_j(P_{g,j}^{\mathrm{0}}+\mathrm{\Delta }{P}_{g,j})-C_j(P_{g,j}^{\mathrm{0}})\right|},{\displaystyle \sum _{i\in \mathrm{red}}{\mathrm{PO}}_{D,i}^{\mathrm{down}}\mathrm{\Delta }{P}_{\mathrm{red},i}^{\mathrm{down}}}{u}_i{,M-FACTS}_{\mathrm{cost}}\right\},

subject to

(15)E(\left|\mathit{V}\right|,\hspace{0.17em}\mathit{\theta },\hspace{0.17em}\mathit{u})=\mathrm{0},

(16)H(\left|\mathit{V}\right|,\hspace{0.17em}\mathit{\theta },\hspace{0.17em}\mathit{u})\le \mathrm{0},

where \mathrm{\Delta }{P}_{g,j} is the change in the schedule of the jth generator, P_{g,j}^{\mathrm{0}} is the schedule of the jth generator in step 1, {\mathrm{PO}}_{D,i}^{\mathrm{down}} is the price offered by demand response i to decrease its demand, and u_i is the demand response commitment variable which has a binary value. M-FACTS_cost is the cost of M-FACTS devices. \left|\mathit{V}\right| is the vector of voltage magnitudes, \mathit{\theta } is the vector of phase angles, T is the dispatch time interval, and u is the vector of control variables. E and H are the sets of equality and inequality constraints. Vector u in Eqs. (15) and (16) is the control vector comprising of active-power generation changes, demand response commitments, input references to generator excitation controllers and network controllers including those of FACTS devices.

6 MOPSO-NTVE implementation to solve optimization procedure

The basic steps of a MOPSO-NTVE algorithm are initialization of the particles, computation of the velocity, updating of position and updating of leader’s archive. Let the position and the velocity of the ith particle in the n-dimensional search space be presented as {\mathit{x}}_i=\left[\begin{array}{llll}{x}_{i,\mathrm{1}}& x_{i,\mathrm{2}}& \cdots & x_{i,n}\end{array}\right] and {\mathit{v}}_i=\left[\begin{array}{llll}{v}_{i,\mathrm{1}}& v_{i,\mathrm{2}}& \cdots & v_{i,n}\end{array}\right], respectively. Meanwhile, according to a specific fitness function, let the local best of the ith particle be denoted as {\mathit{x}}_i^l=\left[\begin{array}{llll}{x}_{i,\mathrm{1}}^l& x_{i,\mathrm{2}}^l& \cdots & x_{i,n}^l\end{array}\right] and the global best found so far denoted as {\mathit{x}}^g=\left[\begin{array}{llll}{x}_{\mathrm{1}}^g& x_{\mathrm{2}}^g& \cdots & x_n^g\end{array}\right]. At each iteration, the new velocities of the particles are updated using Eq. (10) [24].

(17)$$\begin{gathered} {\mathit{v}}_i(k+\mathrm{1})=C(\phi )\{\omega (k){\mathit{v}}_i(k)+c_{\mathrm{1}}(k){\mathit{\varphi }}_{\mathrm{1}}({\mathit{x}}_i^l(k)-{\mathit{x}}_i(k))+c_{\mathrm{2}}(k){\mathit{\varphi }}_{\mathrm{2}}({\mathit{x}}^g(k)-{\mathit{x}}_i(k))\} \\ \hspace{0.17em}\mathrm{for}\hspace{0.17em}i=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}\cdots ,\hspace{0.17em}m\mathrm{.} \end{gathered}$$

In Eq. (17) the first term shows the current velocity of the particle, the second presents the cognitive part of PSO and the third corresponds to the social part of PSO. Each particle moves from the current position to the next one by the modified velocity expressed as

(18)\mathit{x}{}_i(k+\mathrm{1})=\mathit{x}{}_i(k)+{\mathit{v}}_i(k+\mathrm{1})\hspace{0.17em}\mathrm{for}\hspace{0.17em}i=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}\cdots ,\hspace{0.17em}m.

In MOPSO-NTVE, the inertia weight is given as described in Eq. (19). The cognitive parameter c₁ starts with a high value c_1max and the nonlinearity decreases to c_1min. Besides, the social parameter c₂ starts with a low value c_2min and the nonlinearity increases to c_2max using Eqs. (20) to (22) [25,26].

(19)\omega (k)={\omega }_{\min }+{\left({\displaystyle \frac{{\mathrm{iter}}_{\max }-k}{{\mathrm{iter}}_{\max }}}\right)}^{\alpha }({\omega }_{\max }-{\omega }_{\min }),

(20)c_{\mathrm{1}}(k)=c_{\mathrm{1}\min }+{\left({\displaystyle \frac{{\mathrm{iter}}_{\max }-k}{{\mathrm{iter}}_{\max }}}\right)}^{\beta }(c_{\mathrm{1}\max }-c_{\mathrm{1}\min }),

(21)c_{\mathrm{2}}(k)=c_{\mathrm{2}\max }+{\left({\displaystyle \frac{{\mathrm{iter}}_{\max }-k}{{\mathrm{iter}}_{\max }}}\right)}^{\gamma }(c_{\mathrm{2}\min }-c_{\mathrm{2}\max }),

(22)C(\phi )={\displaystyle \frac{\mathrm{2}}{\left|\mathrm{2}-\phi -\sqrt{{\phi }^{\mathrm{2}}-\mathrm{4}\phi }\right|}},\mathrm{where}\hspace{0.17em}\mathrm{4.1}\le \phi \le \mathrm{4.2},

where {\mathrm{iter}}_{\max } is the maximal number of iterations. \alpha, \beta and \gamma are constant coefficients.

To determine the optimal combination of \alpha, \beta and \gamma, all combinations must be tested. It is assumed that

(23)\alpha ,\beta ,\gamma \in \{\mathrm{0},\mathrm{0.5},\mathrm{1},\mathrm{1.5},\mathrm{2}\}.

There are 5³ possible combinations for the values of \alpha, \beta and \gamma. However, if \alpha, \beta and \gamma have many possible values, it may not be possible to perform the experiments of all combinations. Therefore, in order to sample a small but representative subset of this large number of experiments, an orthogonal design technique is adopted. The details of the orthogonal arrays and their application have been described in Ref. [27]. Equation (24) is an {\mathit{L}}_{\mathrm{25}}({\mathrm{5}}^{\mathrm{6}}) orthogonal array that can deal with at most six variables in five possible values with 25 experiments. Instead of 5³ possible combinations, only 25 experiments have to be performed to determine the optimal combination of \alpha, \beta, and \gamma.

(24){\mathit{L}}_{\mathrm{25}}({\mathrm{5}}^{\mathrm{6}})=\left[\begin{array}{llllll}\mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}\\ \mathrm{1}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{2}\\ \mathrm{1}& \mathrm{3}& \mathrm{3}& \mathrm{3}& \mathrm{3}& \mathrm{3}\\ \mathrm{1}& \mathrm{4}& \mathrm{4}& \mathrm{4}& \mathrm{4}& \mathrm{4}\\ \mathrm{1}& \mathrm{5}& \mathrm{5}& \mathrm{5}& \mathrm{5}& \mathrm{5}\\ \mathrm{2}& \mathrm{1}& \mathrm{2}& \mathrm{3}& \mathrm{4}& \mathrm{5}\\ \mathrm{2}& \mathrm{2}& \mathrm{3}& \mathrm{4}& \mathrm{5}& \mathrm{1}\\ \mathrm{2}& \mathrm{3}& \mathrm{4}& \mathrm{5}& \mathrm{1}& \mathrm{2}\\ \mathrm{2}& \mathrm{4}& \mathrm{5}& \mathrm{1}& \mathrm{2}& \mathrm{3}\\ \mathrm{2}& \mathrm{5}& \mathrm{1}& \mathrm{2}& \mathrm{3}& \mathrm{4}\\ \mathrm{3}& \mathrm{1}& \mathrm{3}& \mathrm{5}& \mathrm{2}& \mathrm{4}\\ \mathrm{3}& \mathrm{2}& \mathrm{4}& \mathrm{1}& \mathrm{3}& \mathrm{5}\\ \mathrm{3}& \mathrm{3}& \mathrm{5}& \mathrm{2}& \mathrm{4}& \mathrm{1}\\ \mathrm{3}& \mathrm{4}& \mathrm{1}& \mathrm{3}& \mathrm{5}& \mathrm{2}\\ \mathrm{3}& \mathrm{5}& \mathrm{2}& \mathrm{4}& \mathrm{1}& \mathrm{3}\\ \mathrm{4}& \mathrm{1}& \mathrm{4}& \mathrm{2}& \mathrm{5}& \mathrm{3}\\ \mathrm{4}& \mathrm{2}& \mathrm{5}& \mathrm{3}& \mathrm{1}& \mathrm{4}\\ \mathrm{4}& \mathrm{3}& \mathrm{1}& \mathrm{4}& \mathrm{2}& \mathrm{5}\\ \mathrm{4}& \mathrm{4}& \mathrm{2}& \mathrm{5}& \mathrm{3}& \mathrm{1}\\ \mathrm{4}& \mathrm{5}& \mathrm{3}& \mathrm{1}& \mathrm{4}& \mathrm{2}\\ \mathrm{5}& \mathrm{1}& \mathrm{5}& \mathrm{4}& \mathrm{3}& \mathrm{2}\\ \mathrm{5}& \mathrm{2}& \mathrm{1}& \mathrm{5}& \mathrm{4}& \mathrm{3}\\ \mathrm{5}& \mathrm{3}& \mathrm{2}& \mathrm{1}& \mathrm{5}& \mathrm{4}\\ \mathrm{5}& \mathrm{4}& \mathrm{3}& \mathrm{2}& \mathrm{1}& \mathrm{5}\\ \mathrm{5}& \mathrm{5}& \mathrm{4}& \mathrm{3}& \mathrm{2}& \mathrm{1}\end{array}\right]\mathrm{.}

7 Decision making tools

The decision making problem for alternative selection is usually called multiple attribute decision making, which has been proven to be an effective approach for ranking or selecting one alternative from a finite set of alternatives with respect to multiple, usually conflicting attributes [28]. A large number of methods have been developed for selecting the best compromise solution in multiple attribute or multiple criteria problems.

7.1 Fuzzy decision making

Each membership function is defined by the experiences and intuitive knowledge of the decision makers [29,30]. A simple linear membership function was considered for each of the objective functions. The membership function is defined as

(25)

for minimized objective functions and

(26)

for maximized objective functions, where {\mathrm{OF}}_i^{\min } and {\mathrm{OF}}_i^{\max } are the minimum and the maximum value of the ith objective function among all non-dominated solutions, respectively. The membership function \chi is varied between 0, 1, where \chi =\mathrm{0} indicates the incompatibility of the solution with the set, while \chi =\mathrm{1} means full compatibility. Figure 3 illustrates a typical shape of the membership function.

For each non-dominated solution k, the normalized membership function {\chi }^k is calculated as

(27){\chi }^k={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{ob}}}{\chi }_i^k}}{{\displaystyle \sum _{k=\mathrm{1}}^M{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{ob}}}{\chi }_i^k}}}},

where M is the number of non-dominated solutions and N_ob is the number of objective functions. The function {\chi }^k can be considered as a membership function of non-dominated solutions in a fuzzy set, where the solution having the maximum membership in the fuzzy set is considered as the best compromise solution.

7.2 TOPSIS method

The normalized decision matrix can be written as [31]

(28){\mathrm{NDM}}_{ij}={\displaystyle \frac{{\mathrm{DM}}_{ij}}{{\displaystyle \sum _{p=\mathrm{1}}^n{\mathrm{DM}}_{pj}}}},

where \mathrm{DM}=\{{\mathrm{DM}}_{ij}|i=\mathrm{1},\mathrm{2},\cdots ,n;j=\mathrm{1},\mathrm{2},\cdots ,m\} (n, m are the number of Pareto-optimal solutions and number of objectives respectively) is the n*m decision matrix and {\mathrm{DM}}_{ij} is the performance rating of alternative X_j (Pareto-optimal solution) with respect to attribute AT_i. The amount of decision information can be measured by the entropy value as

(29){\mathrm{EV}}_j={\displaystyle \frac{-\mathrm{1}}{\ln n}}{\displaystyle \sum _{i=\mathrm{1}}^n{\mathrm{NDM}}_{ij}\ln ({\mathrm{NDM}}_{ij})}.

The degree of divergence (D_j) of the average intrinsic information contained by each attribute {\mathrm{AT}}_j (j = 1, 2, ..., m) can be calculated as

(30)D_j=\mathrm{1}-{\mathrm{EV}}_j

and the objective weighted normalized value OW_ij is calculated as

(31){\mathrm{OW}}_{ij}=w_i\times {\mathrm{NDM}}_{ij}

To produce an overall performance index for each alternative, the positive ideal solution ({\mathrm{AT}}^{+}) and the negative ideal solution ({\mathrm{AT}}^{-}) are used, which are defined, respectively, by

(32)$$\begin{gathered} {\mathrm{AT}}^{+}=\left(\begin{array}{llll}\max ({\mathrm{OW}}_{i\mathrm{1}}),& \max ({\mathrm{OW}}_{i\mathrm{2}}),& \mathrm{...},& \max ({\mathrm{OW}}_{im})\end{array}\right)=({\mathrm{OW}}_{\mathrm{1}}^{+},{\mathrm{OW}}_{\mathrm{2}}^{+},\cdots ,{\mathrm{OW}}_m^{+}), \\ {\mathrm{AT}}^{-}=\left(\begin{array}{llll}\min ({\mathrm{OW}}_{i\mathrm{1}}),& \min ({\mathrm{OW}}_{i\mathrm{2}}),& \mathrm{...},& \min ({\mathrm{OW}}_{im})\end{array}\right)=({\mathrm{OW}}_{\mathrm{1}}^{-},{\mathrm{OW}}_{\mathrm{2}}^{-},\cdots ,{\mathrm{OW}}_m^{-}). \end{gathered}$$

The separation (distance) between alternatives can be measured by the n-dimensional Euclidean distance. The separation of each alternative from the ideal solution is given as

(33)D_j^{+}={\left\{{\displaystyle \sum _{i=\mathrm{1}}^m{({\mathrm{OW}}_{ji}-{\mathrm{OW}}_i^{+})}^{\mathrm{2}}}\right\}}^{\mathrm{1}/\mathrm{2}},j=\mathrm{1},\mathrm{2},\cdots ,n

Similarly, the separation from the negative ideal solution is given as [32]

(34)D_j^{-}={\left\{{\displaystyle \sum _{i=\mathrm{1}}^m{({\mathrm{OW}}_{ji}-{\mathrm{OW}}_i^{-})}^{\mathrm{2}}}\right\}}^{\mathrm{1}/\mathrm{2}},j=\mathrm{1},\mathrm{2},\cdots ,n

The relative closeness to the ideal solution of alternative X_j with respect to {\mathrm{AT}}^{+} is defined as

(35){\mathrm{RC}}_j={\displaystyle \frac{D_j^{-}}{D_j^{+}+D_j^{-}}},j=\mathrm{1},\mathrm{2},\cdots ,n

Since D_j^{-}\ge \mathrm{0} and D_j^{+}\ge \mathrm{0}, clearly, {\mathrm{RC}}_j\in [\mathrm{0},\mathrm{1}]. An alternative is chosen with the maximum RC_j, in descending order. It is clear that an alternative X_j is closer to {\mathrm{AT}}^{+} than to {\mathrm{AT}}^{-} as RC_j approaches 1.

8 Implementation

A case study based on the Azarbaijan regional power system of Iran as illustrated in Fig. 4 is presented in this section to prove the effectiveness of the proposed method. The Azarbaijan test system consists of 48 branches, 6 generators and 27 buses. The system data are available in Refs. [31,33]. The flowchart of the proposed method is depicted in Fig. 5.

The sensitivities of real power flow performance index with respect to M-FACTS control parameters are presented in Table 1. The congestion control with the injected voltage magnitude will not be as effective because the range of control is limited. Here, the phase angle control of M-FACTS is utilized for congestion management. The value of sensitivity c_{\mathrm{5}}^k is zero as I_sh has very little effect on power flow. It can be seen from Table 1 that among the values of c_{\mathrm{2}}^k and c_{\mathrm{4}}^k, line 19 has the highest negative sensitivity. Lines 15, 16, 21 and 22 are the lines related to buses 6 and 7. With regard to high sensitivity of line 15, M-FACTS shunt converter and M-FACTS series converter 1 and 2 respectively, is suitable to be installed in bus 6, lines 19 and 15. Similarly, according to the sensitivity of PI with respect to UPFC parameter in Table 2, line 35 is chosen as the best location for UPFC.

Seven load buses specified in Table 3 are selected for demand response participation based on their potential to reduce the transmission line congestion according to the generation shift factor.

The amount of incentive and penalty for the DR program are considered as fixed values which are $100 and $150 per MWh. The MOPSO-NTVE and INSGA-II algorithm (introduced in the Appendix) for the optimization problem has been implemented using Matlab software. The results of market clearing together with congestion management are obtained and discussed in this section. At the beginning of the procedure, the electricity market is cleared without considering the electricity network. The generator schedule following electricity market clearing is given in Table 4.

The Pareto-optimal set with MOPSO-NTVE and INSGA-II in two-dimensional and three-dimensional objective function in two states as DR & UPFC and DR & M-FACTS are represented in Figs. 6 and 7. It can be observed that the obtained solutions are well distributed on the trade-off surface, except some discontinuity. The trade-offs represented in Figs. 6 and 7 can help the decision maker to select suitable reference membership values. The decision making procedure based on FDM and TOPSIS theory is conducted to find the best compromise solution from the set of Pareto-solutions obtained using MOPSO-NTVE and INSGA-II.

The FACTS devices settings have been presented in Table 5. In addition, the total cost of market operation in two different options, that is DR & UPFC and DR & M-FACTS are listed in Table 6. The comparison of different options shows that using the combination of DR and M-FACTS can reduce the total market cost (including DR cost and re-dispatch cost). The re-dispatch, DR and FACTS costs are shown separately in Table 6 for comparison purpose. As indicated in Table 6 and Fig. 8, the total market cost is the lowest when the market operator deployed the combination of M-FACTS and DR programs where the M-FACTS cost is lower than the UPFC cost because of the lower requirement size. Also, the FDM based MOPSO-NTVE results in a better management with a lower cost.

9 Conclusions

Simultaneous utilization of the DR program and multi-converter FACTS devices to manipulate congestion in a restructured environment has been presented. The M-FACTS devices can construct a multi-terminal sub-grid, which can control the active and reactive power flows of a group of lines and selected bus voltage within a substation to their determined objectives. This has considerably extended the voltage and power flow control capability achieved by the independent STATCOM or SSSC or UPFC. Besides, the DR program as a new procedure for congestion management has been discussed. In the first stage, the market clearing process is conduced based on SW maximization while in the second stage, the congestion was relieved by decrement in the initial production of generators, reduction of demands and presence of the multi-converter FACTS devices. The problem associated with the DR program and multi-converter FACTS devices is a multi-objective nonlinear optimization problem. To solve the optimization problem, the MOPSO-NTVE method that has the capability of expediting convergence toward the global optimum than the ordinary PSO has been utilized and its efficiency is compared with that of INSGA-II. From the perspective of a decision maker, the FDM method is applied to determine the solutions with respect to all relevant attributes from the set of Pareto-solutions obtained using MOPSO-NTVE. The comparison of the two coordinated mechanisms, DR & UPFC and DR & M-FACTS, have been performed on the Azarbaijan regional power system of Iran. The results indicate that the congestion management by DR and M-FACTS can considerably reduce the congestion cost. The results confirm that the demand response with M-FACTS can play a major role in competitive market electricity. Moreover, comparative analysis in two states DR & UPFC and DR & M-FACTS has been conducted between two methods of problem solution as FDM based MOPSO-NTVE and FDM based INSGA-II. The simulation results from numerical tables and figures show that in the condition of MOPSO-NTVE based optimization, the best results have been obtained.

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