0 Introduction

In many countries, the electric power industry is experiencing restructuring and consequently deregulation of power market due to ever rising demand for electricity. Earlier, all major activities, such as power generation, then transmission & distribution were managed by vertically integrated utility^[1]. Since then, restructuring of electric power industry has paved the ways to address the major challenges such as congestion in power network, ancillary services, inefficiencies in market power and reliability^[2]. Amongst all the challenges, the Congestion Management (CM) substantiates the management of congestion. Recently, various studies on CM have been reported: optimal power flow, load curtailment with redispatch and generation rescheduling. The state of occurrence of congestion in transmission network occurs when there is violation of physical and operational limits^[3]. However, different deregulation models are employed to alleviate the impact of congestion. Especially in deregulated markets, the price at each bus is determined by supply economics which leads to different Locational Marginal Price (LMP) for various buses in system^[4-5].

Importantly, congestion can be alleviated by implementing Flexible AC Transmission Systems (FACTs) devices. Their operation through transformer tap changers and system reconfiguration is analyzed and discussed in Ref.[6]. A real time-based hierarchical congestion management technique is discussed in Ref.[7] for rescheduling the operation of generators. The optimal schedule of demand response loads is estimated for solar and wind energy operated systems. A coordination scheme is presented for a distribution system operator for the mitigation of congestion management cost and revenue^[8]. The solution to the adopted strategy is obtained by implementing PSO(Particle Swarm Optimization) to a non-linear programming model. Importantly, congestion is resolved by the working coordination of dynamic tariff and scheduled reprofiling methods.

A dynamic tariff-based method for day ahead CM for distribution networks is highlighted in Ref.[9]. The proposed robust method is able to resolve congestion efficiently even under the uncertain dynamic conditions using the Roy Billonton test system. Rescheduling cost is minimized using monarch butterfly optimization technique in Ref.[10]. The basic aim is to reduce congestion in the proposed system with minimum network losses and line flows. IEEE-30 bus systems have especially been implemented to alleviate the issue of congestion. It ultimately increases the power system security and consequently reliability. In Ref.[11], it is revealed that with the accurate analysis of line contingency, FACTs devices can be mounted on any of IEEE-9 or IEEE-57 bus systems. This analysis effectively detects the effect of TL faults, while FACTs devices mitigate them without requiring the re-dispatching of generator and load shedding.

For universality and extensiveness, the optimization techniques are significant in CM for estimating optimal performance^[12]. These techniques are suitable for refining the power transfer capacity and voltage stability of FACTs devices with security constraints. In Ref.[13], the author has offered Krill Herd's optimization technique. It obtains placement and configurations of FACTs devices in an optimized manner in CM. Especially, the projected way is applied to IEEE-30 and IEEE-57 bus systems. It has been found that relieving of congestion was achieved with loss minimization in active power, as well as reductions in energy cost and operational cost of FACTs devices. In Refs.[14-15], the optimized placement of FACTs devices is reported. It especially reflects the line outage type of sensitivity factor. To address this, the PSO technique is coordinated with IEEE-14 and IEEE-57 bus systems. Acharyaet al.^[16] recognized the concept of rent for congestion and LMP difference to identify thyristor-controlled series compensation to reduce congestion. Refs.[17-18] proposed PSO technique for the optimal rescheduling of generators to improve active power generations. In this algorithm, the rescheduling cost is minimized. The algorithm is applied to the IEEE-30 and IEEE-118 bus systems. In this research work, the main objective functions in the optimization process are defined for the total real power and reactive power loss, along with their total CC. The critical objectives of present research are listed as follows:

• To apply and evaluate the performance of optimization techniques (PSO, and FPA(Flower Pollination Algorithm)) for minimizing the objective function;

• To estimate and minimize the CC for an IEEE-69 bus system using LMP optimization technique;

• To ensure that STATCOM is placed on the TL with minimum CC and minimum loss in real and reactive power flow;

• To establish superiority in FPA over PSO technique in estimating an optimal placement of a STATCOM.

In this paper, the implementation of two optimization techniques are presented for estimating an optimal location of STATCOM in CM: PSO and FPA. Using LMP technique, the CC is calculated and minimized for each line in an IEEE-69 bus system. TL with the highest CC is identified. Therefore, the STATCOM is placed on the TL using PSO algorithm for minimizing the objective functions. In order to justify the validity of the results, the system has been implemented and tested using FPA technique. The proposed methodology is helpful for future research engineers. It is revealed that the proposed technique is related to locate somewhat shunt type FACTs device and reduce the CC.

1 Size and Location of FACTs Device

STATCOM is a significant type of FACTs device in restructured power systems^[19]. Its connection at Point of Common Couplings (PCC) can vary the operating costs at various buses, which affects the rescheduling of generators-loads at minimum cost^[20]. However, the choice of its optimal location not only reduces congestion but also minimizes system losses. Importantly, when any system is under congestion, it is concluded that there is an increase in LMP at buses. In addition, the degree of difference in LMP at various PCCs indicates the amount of congestion. As the LMP differences increase, the more congestion in links is noticed. In specific, the link under congestion of neighboring lines forms the basis for identifying exact location of STATCOM to reduce the congestion. Thus, the appropriate size of STATCOM is estimated by implementing optimization algorithms on the bus system under study. The proposed dynamic compensator in Ref.[21] effectively maintained the voltage profile by the optimal size selection of a distributed generation source. Especially, the flow of reactive power is optimized using slightest error in voltage iterative approach. The chosen multi-constraints are: limits on voltage and their stability, as well as real and reactive power. In Ref.[22], the static and dynamic compensation are optimized for the supplied reactive power using the concept of marginal cost. Validations are obtained by comparing the performances of step changes under three types of load conditions. These validations are obtained during the presence and absence of reactive power compensation. From the mathematical model of STATCOM depicted in Fig. 1^[23], a brief analysis of the non-linear power equations of STATCOM is presented below.

The current drawn by the STATCOM and the converter power flow equations are given by:

$ E_{v R}=V_{v R}\left(\cos \delta_{v R}+j \sin \delta_{v R}\right) $

(1)

$ I_{v R}=Y_{v R}\left(V_{v R}-V_k\right) $

(2)

$ S_{v R}=E_{v R} I_{v R}^*=V_{v R} Y_{v R}^*\left(V_{v R}^*-V_k^*\right) $

(3)

With the help of circuit theory, the power flow equations are derived from Eqs.(1)-(3) for converter and bus k, respectively:

$ \begin{aligned} P_{v R}= & V_{v R}^2 G_{v R}+V_k V_{v R}\left[G_{v R} \cos \left(\delta_{v R}-\theta_k\right)+\right. \\ & \left.B_{v R} \sin \left(\delta_{v R}-\theta_k\right)\right] \end{aligned} $

(4)

$ \begin{aligned} Q_{v R}= & -V_{v R}^2 B_{v R}+V_k V_{v R}\left[G_{v R} \sin \left(\delta_{v R}-\theta_k\right)-\right. \\ & \left.B_{v R} \cos \left(\delta_{v R}-\theta_k\right)\right] \end{aligned} $

(5)

$ \begin{aligned} P_k= & V_k^2 G_{v R}+V_k V_{v R}\left[G_{v R} \cos \left(\delta_{v R}-\theta_k\right)+\right. \\ & \left.B_{v R} \sin \left(\delta_{v R}-\theta_k\right)\right] \end{aligned} $

(6)

$ \begin{aligned} Q_k= & -V_k^2 G_{v R}+V_k V_{v R}\left[G_{v R} \sin \left(\delta_{v R}-\theta_k\right)-\right. \\ & \left.B_{v R} \cos \left(\delta_{v R}-\theta_k\right)\right] \end{aligned} $

(7)

where V_vR represents voltage magnitude and δ_vR is phase angle (state variables).

The angle and source voltage magnitude of parallel converter are limited by

$ V_{v R}^{\min } \leqslant V_{v R} \leqslant V_{v R}^{\max } $

(8)

$ 0 \leqslant \delta_{v R} \leqslant 2 {\rm{\mathsf{π}}} $

(9)

where V_vR^min and V_vR^max are respective minimum and maximum limited in voltage V_vR.

2 Controlling Optimizing Techniques 2.1 Particle Swarm Optimization Technique

In this work, PSO control is implemented for CM which evaluates the optimal location of a STATCOM^[24]. In this controlling technique, each particle carries two vectors: one is position vector and the other is velocity vector. Also, the adjustment of each particle position is based on experience of P_best, and that of neighboring particle G_best. Additionally, both vectors have same size as problem dimension. To achieve an optimal solution, all variable particles should move randomly within the search space. It updates its constructed position on the basis of global best position and previous velocity vector, according to the equation below:

$ v_{i j}^{t+1}=v_{i j}^t+c_1 r_{1 j}^t\left(P_{\text {best }, \mathrm{i}}^t-x_{i j}^t\right)+c_2 r_{2 j}^t\left(G_{\text {best }}-x_{i j}^t\right) $

(10)

where v_ij^trepresents agent velocity at time t; x_ij^tis agent position at time t; P_{best, i}^t represents agent's best position starting from beginning through time t; G_best is agent's best global position starting from beginning through time t; c₁ and c₂ are determined cognitive and social components contribution level, named as positive acceleration constants, r_2j^t and r_1j^t are random numbers generated at time t.

It is observed in the PSO algorithm that some values of parameters with their choices have an important impact on output or efficiency of PSO. The basic parameters considered in this paper are: swarm size, iteration number, velocity component. Additionally, it has been proven that PSO algorithm is similarly inclined by velocity clamping, inertia weight, and other constraints. Pseudo code for this algorithm is presented in Appendix 1.

2.2 Flower Pollination Technique

In this controlling technique^[25], the main objective is the reproduction of flower through pollination. Importantly, this technique is based on the allocation of pollen grains formed by male gametes to stigma for their union^[26]. During this process, the birds and insects coordinate in the accurate transfer of pollen grains. Also, few flowers form strong bonds with mediators for their required pollination, and attract only specific species. Biotic and abiotic are the two ways for pollination. It has been found that nearly 90% of pollination happens in a biotic form, which helps in the allocation of pollen grains. Remaining 10% depends majorly on the presence of outside agents, namely wind and diffusion^[27]. The ultimate aim of implementing FPA in the present study is to ensure the survival of the fittest and optimal plants reproduction in terms of numbers and fittest. Therefore, based on the pollinating characteristics of flowers, four rules have been formulated for present work. Pseudo code of this technique is presented in Appendix 1. The steps of global pollination and flower constancy are represented by Eq.(11) as follows:

$ x_i^{t+1}=x_i^t+\gamma L(\lambda)\left(g_*-x_i^t\right) $

(11)

where,

$ L \approx \frac{\lambda \Gamma(\lambda) \sin \left(\frac{{\rm{\mathsf{π}}} \lambda}{2}\right)}{{\rm{\mathsf{π}}}}\left(\frac{1}{s^{1+\lambda}}\right)\left(s \gg s_0>0\right) $

(12)

For local pollination,

$ x_i^{t+1}=x_i^t+\in\left(x_j^t-x_k^t\right) $

(13)

2.3 LMP Difference Technique

Using LMP technique, the economic signals are used to estimate an optimal location of STATCOM. The value of marginal price at a bus contains significant information about the extent of congestion. Thus, the extent of line congestion is conveyed through the difference in LMP, with the highest difference in LMP indicating a congested and overloaded line. This LMP difference technique is used for optimized placement of STATCOM and minimization of objective functions. An important advantage of the proposed technique is that it operates the TL coordinated STATCOM by obtaining accurate results from Optimal Power Flow (OPF).

It is pertinent to mention here that the CC is a function of difference in LMP of two buses and their power flow. In this way, CC reveals the degree of congestion among various lines, and the line with the highest CC is marked as a need of STATCOM.

The modeling of multi-objective optimization problem of CM is done by running OPF algorithm on an IEEE-69 bus system. The optimal values for the total real and reactive power loss, along with LMP and power flow in each line, are computed. In order to minimize the total real and reactive power loss, the objective function in terms of total real power loss is given by,

$ P_l=\sum\limits_{k=1}^{N_l} G_k\left[V_i^2+V_j^2-2 V_i V_j \cos \left(\delta_i-\delta_j\right)\right] $

(14)

Similarly, the second objective function is total reactive power loss given by,

$ \begin{aligned} Q_l= & \sum\limits_{k=1}^{N_l} B_k\left[V_i^2+V_j^2-2 V_i V_j \cos \left(\delta_i-\delta_j\right)\right]- \\ & {\left[V_i^2 y_{i k}^0\right]+V_k^2 y_{k i}^0 } \end{aligned} $

(15)

where N_l represents total number of system lines; B_k is susceptance of line k; V_i and V_j are magnitudes of sending end and receiving end voltages of the line, respectively; δ_i and δ_j are angles of end voltages; y_ij⁰ is charging admittance. The third objective function which can have minimal value is the total CC ^[28] expressed as,

$ C_{\mathrm{C}}=\sum\limits_{i j}\left(L_{\mathrm{LMP}_j}-L_{\mathrm{LMP}_i}\right) * \operatorname{absolute}\left[p_f(i j)\right] $

(16)

where, L_LMPj is the locational marginal price of bus j; L_LMPi is the locational marginal price of bus I; p_f(ij) is the power flow in the branch ij connecting the two buses. For accuracy, the total real power loss has been computed from OPF, whereas total CC is computed after obtaining the optimal results from OPF. In this research work, the authors have utilized CC of line corresponding to where STATCOM has accurately placed with the minimization of objective function. Overall, the procedure for implementing the optimization techniques is summarized as below:

• Using OPF algorithm to estimate total real and reactive power loss for an IEEE-69 bus system.

• Calculating the CC and total CC, and identifying the TL with maximum CC amongst all.

• Appliying PSO and FPA techniques for identifying the optimal size and location of a STATCOM.

• Minimizing three pre-defined objective functions.

3 Load-Flow Analysis

To demonstrate the usefulness of the implemented optimization techniques, IEEE-69 bus system is chosen for identifying an optimal location of a STATCOM. To ensure a fair evaluation of the techniques, the minimization of the objective functions is obtained by implementing PSO and FPA optimization techniques. The performance of PSO and Genetic Algorithm (GA) based optimization techniques is implemented in Refs.[29-30] for wind-generator controlled microgrid. IEEE standards 1159: 1995 and 1250: 2011 are used for validating the operation of frequency in boundaries. In addition, the performance of GA, Artificial Neural (AN) and Adaptive Neuro Fuzzy Inference (ANFI) controlling techniques is validated for alteration of various gain parameters of STATCOM in Ref.[31]. It is found that these tuning methods are capable of maintaining the optimal performances of STATCOM even under transient conditions. However, a fuzzy based controller for governing the real and reactive power flow is successfully implemented for fixed speed wind energy conversion system^[32].The application of fixed capacitor and STATCOM is also reported in Ref.[33] which envisages the impact of genetic algorithm on the steady and dynamic conditions. Thus, the application of OPF reveals the optimal size and location of a STATCOM. Figs. 2(a) and 2(b) illustrate the real and reactive power flow just before the placement of STATCOM in IEEE-69 bus system. It is revealed that there is maximum power loss for line 56-57. Notably, reactive power compensation is also achieved for line 48 and line 56-57. Fig. 2 (c) depicts the variation of voltage magnitude in per unit and LMP at each bus of IEEE-69 bus system. It is oberved that the magnitude of voltage varies between 1.099 p.u. and 1.052 p.u., while the LMP (in $/MWh) varies between 2.072 and 2.235 in the same interval.

3.1 IEEE-69 Bus System with PSO Technique

To investigate the accuracy, three objective functions are minimized for an IEEE-69 bus test system. In it, there are total 69 buses and 67 lines sections, with line limits mentioned in Ref.[14]. Table 1 shows the total real power and reactive power losses computed by OPF algorithm, before the application of STATCOM. It also illustrates the TL with total CC. Clearly, the highest CC is computed as 111.07$/kVA. Notably, it is revealed that the branch with the highest CC is line 56-57. Among 67 lines in IEEE 69-bus system, the line 56-67 is the selected line which has the highest CC and is the most favorable optimal location of STATCOM. However, the proper sizing is determined by PSO algorithm when applied to bus number 57. Table 1 also illustrates the total real power, reactive power and total CC after the application of STATCOM. Notably, these results have been computed for optimal size and location of STATCOM at PCC. Additionally, it is demonstrated that minimal values of all objective functions are obtained.

Notably, from Table 1 it is observed that CC is considerably reduced after the placement of STATCOM using PSO technique. CC is found to be 288.05 $/kVA. Normally, it is observed that CC is the function of LMP, which is subjected to the constraints of existence of various transmission restrictions and TL losses. LMP is composed of three parts: reference bus marginal pricing, marginal pricing of TL losses and marginal pricing for a TL congestion. It is pertinent to mention that CC is the difference between LMP of the bus and summation of loss and energy components. Thus, the results presented in Table 1 clearly justify the mathematical framework analyzed in Eqs.(14) and (15).

3.2 IEEE-69 Bus System with FPA Technique

In order to validate the results of Section 3.1, FPA technique is implemented for IEEE-69 bus system. In addition, a sincere effort is made to compute the optimal location and size of STATCOM so that minimization of the defined functions can be achieved. Table 2 shows the total real power and reactive power loss before the application of STATCOM. In addition, total CC before the placement of STATCOM is also computed. Clearly, the highest CC is found to be 111.07$/kVA. It has been revealed that the branch with the highest CC is line 56-57. Table 2 also illustrates the total real power, reactive power and total CC after the application of STATCOM. Notably, these results are computed for optimal size and location of STATCOM at PCC. Additionally, it is demonstrated that minimal values of all objective functions are obtained.

The aforementioned analysis and Fig. 3 reveal that the reduction in total real power loss with PSO and FPA algorithm is 5.04% and 6.99%, respectively. Importantly, the CC is reduced with PSO and FPA algorithms by 1.21% and 2.58%, respectively.

4 Conclusions

Congestion is widely debated issue in the power system, and various methods are implemented to compensate it to a minimum level. The primary objective of the present research is to demonstrate the ability of faster rate of convergence of FPA. In this work, two optimization techniques implemented on a work are compared, and a new method for optimal placement of FACTs device is implemented in the selected power system. Congestion is widely debated issue in the power system, and numerous approaches have been implemented till date to compensate it to a minimum level. Thus, this study shows that installing FACTs device on the TL with the highest CC ensures minimization of total real power loss and total CC.

It is interpreted that the percentage decrease in total real power loss using PSO and FPA algorithms is 5.04% and 6.99%, respectively. However, the percentage reduction in CC using PSO and FPA algorithms is 1.21% and 2.58%, respectively. This comparative analysis proves that FPA technique has demonstrated its superiority over PSO technique in estimating the optimal placement of STATCOM and minimizing the objective function.

The study shows that installing FACTs device on the line that has the highest CC will ensure minimization of total real power loss, total reactive power loss and total CC. The CC collected by system operator is distributed to the Financial Transmission Right (FTR). FTR then uses it to compensate the risk of price differences between the injection and retrieval points. The rise in LMP is indicative of congested system, and the measure of congestion across a link is presented by the difference in LMP. Importantly, the degree of congestion is determined by the difference in LMP. With this difference, either the congested line or neighboring lines are marked as potential lines for the placement of shunt FACTs devices. This forms the basis of the proposed methods. However, CC is calculated to help identify the line with the highest CC, so that it helps in allocating STATCOM on either of the two buses containing the line. In this regard, the optimization technique helps in finding the optimal size of the device in order to reduce total real, reactive power loss, and total CC of the system. Major highlights of the present work are summarized below:

• Implementation of an OPF algorithm for estimating real and reactive power loss and CC for an IEEE-69 bus system is achieved.

• TL with the highest CC is identified.

• Minimization of objective functions for estimating the optimal location of STATCOM is demonstrated.

• The effectiveness of FPA technique is established by the PSO technique in reducing real power loss and total CC.

Overall, the results are sufficient for future research engineers to carry out extensive research using the proposed model for a deregulated market.

Appendix 1