1. Introduction
Today, life without electricity is almost impossible to imagine. Generated electricity (active and reactive power) is usually transmitted over long distances to end users. However, due to the AC transmission line impedance and the voltage drop across the line, power losses always exist [
1,
2].
Modern power systems, besides classic generating units, consist of many renewables [
3]. There are also a lot of different devices based on power electronics, an example being Flexible Alternating Current Transmission System (FACTS) devices. At the moment, FACTS devices are used as the most advanced reactive energy compensation devices [
2]. They are also used to solve various problems in the power system such as power system stability, power transfer capacity, voltage profile, power system efficiency, and so on [
2]. This paper, in general, deals with one of the most used types of FACTS devices, known as SVC devices (Static VAr Compensator).
The SVC is the first FACTS device to be developed. This device connects in parallel to the system and is essentially a combination of parallel-connected capacitors (fixed or variable capacity) and coils so that it can operate in either inductive or capacitive mode [
2]. Therefore, it is dominantly used to regulate voltage and reactive power in a power system. Furthermore, these devices can regulate, i.e., produce and absorb reactive power over a wide range. As they are one of the cheaper FACTS devices, they are often observed in power studies. In the available literature, we can find many different studies on FACTS devices, especially on SVC devices, and their allocation in power systems with or without the presence of renewables [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25]. For example, the coordination of SVC devices and other FACTS devices for the management of power flow is presented in [
3]. Furthermore, the optimal placement and parameter setting of SVC devices for solving power system stability problems, the damping system-wide low-frequency oscillations, as well as power system loadability is presented in [
4,
5,
6,
7,
8,
9,
10,
11]. The loadability problem is often investigated along with the optimal system control for the minimization of total power system losses [
11]. The minimization of total power losses is one of the main goals of the implementation of FACTS devices in power networks; for this reason, almost all the papers dealing with FACTS devices observe this problem [
11,
12,
13,
14]. Finding the optimal setting of all control variables of FACTS devices, for different purposes, is investigated in [
15,
16,
17,
18,
19,
20,
21,
22]. In the literature, we can also find research looking into the improvement of voltage profiles in distribution networks with renewable power sources [
15,
17], the maximization of photovoltaic hosting capacities that minimize system operating costs [
16], the improvement in the reconstruction of power systems with wind generation [
17], voltage profile enhancement [
22], power quality improvement [
24,
25], and so on.
The enhancement of the voltage profile by using FACTS devices is strongly related to power loss minimization [
15,
22]. This fact is investigated in detail in [
22], observing both IEEE 9 and IEEE 30 bus systems. However, in [
22], the authors analyze voltage profile enhancement and power loss minimization only at constant load (a one-time step), which is not a realistic case. On the other side, in [
22], some data mismatching is evident. For example, system data do not correspond to the graphical view of analyzed networks, and so on. Also, the obtained results of optimal power system losses without SVC devices differ from results evident in many of the previously published papers that deal with optimal power flow (OPF). In [
22], it is observed that the minimal value of power losses in the IEEE 30 test bus network, without SVC devices, is 2.86 MW, while in [
26,
27,
28,
29,
30], the value of power system losses is about 3.1 MW or higher. The same situation occurs with the IEEE 9 test bus system. On the other side, in [
31], the authors analyze the impact of SVC devices on a IEEE 24 test bus system with three wind generators without any discussion about the power loss variation, the optimal node for an SVC or similar device.
Beside large power systems, the use of FACTS devices is also investigated for microgrid systems [
32,
33,
34,
35,
36,
37]. In general, microgrids are small energy grids in terms of scale, power, voltage level, and contain a different generator unit that can provide an adequate energy supply to demand [
36]. This design, in grid-connected and islanding modes, with SVC devices can significantly improve voltage stability, enhance power factor, reduce power system losses, mitigate the harmonic distortion, and so on [
37].
Different FACTS devices and power system stabilizers (PSS) can be also used for damping power system oscillations [
38,
39]. A PSS in control devices provides maximum power transfer and thus improves power system stability [
40]. Disturbances occurring in a power system cause electromechanical oscillations, and PSS have been widely used to damp these oscillations [
41]. By introducing additional signals into the excitation controllers of the generators, PSS can increase the damping torque of a system’s local mode [
42]. PSS are used as additional control devices to provide extra damping. The basic role of an SVC device is to control reactive power flow and fluctuations in system voltage. A supplementary role added to an SVC is to increase the power system damping. In order to enhance the damping of system oscillations, an SVC can be designed to modulate its bus voltage, especially for inter-area modes [
43].
The abovementioned studies have observed many different types of power systems. In some cases, the authors observed standard IEEE test bus systems [
4,
8,
12,
13], while in other papers, authors observed distribution systems [
15,
16]. Also, the problem of the optimal location of FACTS devices is predominantly solved by using some metaheuristic techniques. The techniques that can be found in the literature are Particle Swarm Optimization (PSO) [
4,
19,
20], Genetic Algorithm (GA) [
5,
10,
20], Hybrid Imperialist Competitive Algorithm genetic Algorithm (HICAGA) [
11], Whale Optimization Algorithm (WOA) [
12], Differential Evolution (DE) [
12], Grey Wolf Optimization (GWO) [
12], Quasi-Opposition based Differential Evolution (QODE) [
12], Quasi-Opposition based Grey Wolf Optimization (QOGWO) [
12], Differential Search (DS) [
13], Simulated Annealing (SA) [
20], Pattern Search (PS) [
20], Backtracking Search Algorithm (BSA) [
20], Gravitational Search Algorithm (GSA) [
20], and Reproduction Optimization (ARO) [
20]. An updated review of papers dealing with the optimal placement of different FACTS devices in power energy systems using metaheuristic optimization techniques is presented in [
21]. Besides metaheuristic methods, the optimal placement of SVC devices can be also solved by using the Newton–Raphson method [
22], as well as by using non-traditional optimization techniques [
23]. Many existing optimization methods dealing with the optimal location of SVC devices to obtain better system performance, such as smaller power system losses, point to the necessity of further research in this scientific field.
In this paper, we present the use of a CONOPT solver [
26,
31,
44,
45] embedded in the Generalized Algebraic Modeling Systems (GAMS) software package for this issue. The advantage of a CONOPT solver embedded in GAMS over other optimization techniques is its high-speed processing, as well as the fact that it always converges to the same optimal solution for any program starting [
26]. This paper also presents addition comparisons of the CONOPT solver and four metaheuristic methods (PSO, GSA—Gravitational search algorithm [
46,
47], ABC—Artificial bee colony algorithm [
29], and DE [
12,
48,
49,
50]). The optimal location of SVC devices is analyzed in IEEE 9 and IEEE 30 test systems. However, unlike [
22], in this paper, four cases are analyzed—the optimal SVC location in a power system with constant load; the optimal SVC location in a power system with variable load; the optimal SVC location in a power system with variable load in the presence of wind energy, and the impact of location of both a wind generator and SVC devices on power system losses. To the best of our knowledge, this kind of research has not been presented in any of the previously published papers.
The main contributions of this paper are as follows:
- -
The comparison of four metaheuristic algorithms applied to single-objective optimal power flow (with the objective function of minimization of power system losses) is presented and tested on both IEEE 9 and IEEE 30 test bus systems and comparison is made with similar studies in the literature;
- -
The comparison of minimum power system losses obtained using a CONOPT solver and the metaheuristics algorithms is presented;
- -
The impact of SVC location on power system losses is analyzed in both IEEE 9 and IEEE 30 test bus systems;
- -
The impact of the limited reactive power of SVC devices on power system losses is also analyzed in the same test systems;
- -
The impact of SVC location on power system losses in power networks with renewables is analyzed;
- -
The impact of different wind power generator connections in power networks on the SVC location is tested;
- -
For constant load data, the results obtained using the Newton–Rapson method for optimal SVC location and the minimal value of power losses are compared with known solutions from the literature
The remainder of the paper is organized as follows.
Section 2 gives basic information about SVC devices. The main equations related to the OPF problem are presented in
Section 3. Basic information about the GAMS-CONOPT solver and a comparison of the CONOPT solver with metaheuristic algorithms are presented in
Section 4. The impact of SVC location on power system losses that have been analyzed in four basic cases, their variations, and the simulation results are given in
Section 5. Concluding statements and directions for future work are presented in
Section 6.
3. Optimal Power Flow Formulation
Optimal power flow is one of the most crucial problems in power systems; it is used to determine the optimal settings for control variables while respecting various constrains [
51].
The standard formulation of an OPF problem is:
where
u is a vector of independent or control variables (such as active power generation at the PV buses except the slack bus, voltage magnitudes at PV buses, tap settings of the transformers),
x is a vector of dependent or state variables (such as active power at slack bus, voltage magnitude at PQ busses, reactive power output of all generator units),
J is the system’s optimization goal—objective function (such as total fuel cost, total emission, total power losses, voltage deviation, etc.),
g is set of equality equation, and
h is set of the inequality constraints.
The equality constraints of the OPF can be represented as follows:
where
,
and
are phase angle of voltage in node
i and
j, respectively,
NB is the number of buses,
Vi and
Vj are voltages in node
i and
j, respectively,
PGi is the active power generation in node
i,
QGi is the reactive power generation in node
i,
PDi is active power of demand in node
i,
QDi is reactive power of demand in node
i, and
Gij and
Bij are elements of admittance matrix representing conductance and susceptance between buses
i and
j, respectively.
The inequality constraints of the OPF reflect the limits of the devices present in the power system. The main inequality constraints are as follows:
In the above Equations (5)–(7), NG refers to the number of generators, NT is the number of transformers, NL is the number of lines, T is transformer-tap settings ought, VL is line voltage, VG is generator voltage, and Sl is line power.
Some OPF objective functions are:
- -
Minimization of total fuel cost:
where
ai,
bi, and
ci are the cost coefficients of generator
i.
- -
Minimization of total emission:
where
,
,
,
, and
are the emission coefficients of unit
i- -
Voltage deviation minimization:
- -
Total active power loss minimization:
In an OPF problem, one (single-objective OPF) or more (multi-objective OPF) objective functions can be optimized at the same time.