Flexible AC Transmission Systems (FACTS) play an important role in minimizing power losses and voltage deviations while increasing the real power transfer capacity of transmission lines. The extent to which these devices can provide benefits to the transmission network depend on their optimal location and sizing. However, finding appropriate locations and sizes of these devices in an electrical network is difficult since it is a nonlinear problem. This paper proposes a technique for the optimal placement and sizing of FACTS, namely the Thyristor-Controlled Series Compensators (TCSCs), Shunt VARs Compensators (SVCs), and Unified Power Flows Controllers (UPFCs). To find the optimal locations of these devices in a network, weak buses and lines are determined by constructing PV curves of load buses, and through the line stability index. Then, the whale optimization algorithm (WOA) is employed not only to find an ideal ratings for these devices but also the optimal coordination of SVC, TCSC, and UPFC with the reactive power sources already present in the network (tap settings of transformers and reactive power from generators). The objective here is the minimization of the operating cost of the system that consists of active power losses and FACTS devices cost. The proposed method is applied to the IEEE 14 and 30 bus systems. The presented technique is also compared with Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The findings showed that total system operating costs and transmission line losses were considerably reduced by WOA as compared to existing metaheuristic optimization techniques.
Keywords: FACTS; line stability index (Lmn); PV curves; whale optimization algorithm (WOA)
Electrical power generating stations are usually located far away from load centers, and utilities greatly rely on existing generation to satisfy load demand via power export-import arrangements. Therefore, practical power systems are highly interconnected. Due to excessive interconnections, network restructuring, and dynamic load patterns, some transmission lines operate well above their thermal and stability limits [[
Power system stability is usually described as the capability of the electrical system to remain in synchronism and continue its operation following a disturbance. The recent increase in the requirement of electrical energy and participation of private electricity producers has made the electric power industry very competitive. Therefore, utilities are more interested in transferring a large amount of power optimally through the existing network to gain more revenue instead of expanding the transmission system because of economic and environmental constraints.
Active power flow between two busses with voltages V
(
From Equation (
For short transmission lines, the maximum loading capabilities can be determined by thermal limits of conductor [[
Flexible AC Transmission Systems (FACTS) devices play a key role these days to enhance voltage stability, power transmission capacity and to decrease transmission line congestions [[
Each FACTS device controls its respective parameters to increase the overall efficiency of the power system. Control parameters of some commonly used FACTS devices are presented in Figure 1. A comparison of some basic types of FACTS devices on the basis of their impact on power system is presented in Table 1 [[
It can be seen from Table 1 that UPFC can be a strong candidate for all the mentioned applications. However, the cost of installation of FACTS should be considered before choosing the type of FACTS to be installed. The installation cost of the UPFC is higher than the SVC, hence if only voltage stability is required then SVC can be a better option than UPFC.
While placing FACTS controllers in power systems, some common questions arise, where is the best location to install FACTS? What sort of FACTS should be placed? How much capacity of FACTS is required? In this framework, numerous authors presented their own approaches. For example, genetic algorithm (GA) was utilized for the optimal placement of SVC, TCSC, and UPFC in [[
PSO was utilized to minimize the objective function of optimal placement and coordination of TCSC and SVC in [[
Indices which relate variations in loading parameter with regards to reactive power control and variation in loading parameter with regards to the reactance of respective line for optimal siting of SVC and TCSC were introduced in [[
None of the work has been carried out in the placement of multiple types of FACTS devices at weak lines and buses determined through sensitivity analysis and simultaneously determining the sizing and coordination of these devices in the power system network. Also, to the best of the authors knowledge whale optimization algorithm has not been tested yet for the optimal sizing and coordination of UPFC in the presence of other FACTS like TCSC and SVC.
In this paper, sensitivity analysis has been carried out initially through the Lmn index and by constructing PV curves of load buses to determine ideal locations for TCSC and SVC. Since the Lmn index is a very good indicator to determine critical lines, hence, the ideal location for TCSC can be determined in the network. Similarly, buses prone to voltage collapse can be easily determined by PV curves. Placing SVC at such buses can highly improve power system stability by providing reactive support and increasing loading capability. Locations of UPFC's are determined by finding out the lines that are carrying higher real power. After placement, sizing, and coordination of these devices are presented through the whale optimization algorithm (WOA), and the results are also compared with PSO and GA. The optimal coordination of FACTS devices is determined with existing reactive power sources in the network, which are tap settings of transformers and reactive power from generators. The objective that is being minimized for this purpose is the total system operational cost, which consists of the cost of active power loss and cost of installation of FACTS. As changing reactive power (VAr) delivered by generators and changing tap settings of transformers are independent of total system operational cost, they are only considered as fitness function variables and are not part of the objective function. The presented methodology is applied to IEEE-30 and 14 bus transmission systems. The rest of paper is arranged as follows. "Section 2" discusses steady-state modeling of SVC, TCSC, and UPFC in detail. "Section 3" explains the techniques used for optimal siting of SVC, TCSC and UPFC, "Section 4" discusses mathematical model of the problem. "Section 5" explains WOA in detail and how it is utilized for finding optimal rating and coordination of FACTS. "Section 6" explain the obtained results in detail. Paper is concluded under section "conclusion".
SVC is a shunt compensator, and TCSC is a series compensator. UPFCs combine both shunt and series compensation. The shunt compensators can inject or absorb reactive power, and series compensators can control the impedance of transmission lines.
Depending on the firing angles of thyristors, TCSCs can provide both capacitive or inductive compensations. They are positioned in series with the line and can influence impedance of transmission line. Thus, a TCSC can modify transmission line power carrying capabilities [[
In the presence of TCSCs, the true and reactive power flow equations from the respective buses can be given as:
(
(
(
(
Here,
(
(
SVCs are parallel connected devices, and they can modify reactive power flows at their point of coupling [[
The relationship between the injected or absorbed VAr at the bus by the SVC is given as follows:
(
where Bsvc and 'V' are the susceptance and bus voltage, respectively.
The unified power flows controller (UPFC) was first introduced by Gyugyi in 1992 [[
Real and reactive power flows between the buses are controlled through series and shunt transformers. CONV 1 and 2 are two, three-phase controllable bridges. The active and reactive power flow equations between the respective buses where UPFC are connected are given as:
(
(
(
(
The selection of the bus and the line for the placement of FACTS devices largely depends on the topology of the system and desired outcomes. In this work, SVCs are installed to modify the total reactive power flow between the buses. Series compensators TCSCs are used here to change the impedance of the line and, thus, controlling real power flow in the transmission lines. UPFCs are utilized for both voltage improvement and to increase transmission line power flow capacity. FACTS are placed at weak lines and weak buses, the main reason for that is they can modify the overall flow of reactive power in these lines and can indirectly redistribute the power flow to avoid transmission line overloading.
Weak lines and buses are identified using the Lmn Index and by constructing PV curves of all load buses. Although there are a number of ways of shortlisting weak lines and buses in the power system, most of them are either difficult to use or have different drawbacks. For instance, the fast voltage stability index (FVSI) [[
TCSC's are placed at weak lines which are determined using Lmn index, Lmn index is a very good indicator which shows how much the line is close to instability. The Lines having the Lmn index value close to 1 indicates that the line is critical, for the line to be stable its value is less the 1. The formula for calculating the Lmn index is shown in Equation (
(
where X is the reactance of the line Q
Optimal locations of SVC's are determined by analyzing PV curves. PV curves are constructed using the continuation power flow (CPF) technique. Conventional power flow methods fail at bifurcation point or maximum load-ability point because of singularity in the Jacobian matrix. Therefore, CPF was developed by Ajjarapu and Christy in 1992 to find power flows at all loading point by slightly modifying the power flow equations. CPF uses predictors and correctors scheme to find solutions for power flow at all loading points as shown in Figure 5.
Further details about CPF are given in [[
For optimal placement of the UPFCs those lines which are carrying higher active power are determined, and UPFCs are placed at the starting buses of these lines because their voltage magnitude and corresponding phase angle need to be controlled.
Optimal sizing and coordination of FACTs devices are determined with an objective to minimize operating cost (OC) of the power system, OC consists of active power loss cost, and FACTS device installation cost. For efficient coordination of FACTS devices, tap settings of the transformer and reactive power from generators are also considered as fitness function variables.
The total objective function that needs to be minimized is as follows:
(
where
(
(
(
(
(
Cost functions of TCSC, SVC, and UPFC based on Siemens database [[
- Bus voltages should be in their appropriate limits as (
20 ) - Thermal limits of transmission lines (
21 ) - Generator's reactive power supply limits (
22 ) - Limit on the arrangements of the transformer tap setting (
23 ) - SVC size constraints (
24 )SVC is the size of the SVC in pu. - TCSC size constraints (
25 )TCSC is the size of the TCSC in pu
The whale optimization algorithm was first presented by Lewis and Mirjalil in 2016 [[
- Encircling prey
- Exploitation phase
- Exploration phase
Details of each step and their respective modeling are discussed below.
Humpback-whales encircles the prey because they know the location of prey. The position of each humpback whales or search agents is updated according to the current optimal candidate solution. This prey encircling behavior can be expressed mathematically as:
(
(
where
(
(
As iterations proceed, 'a' is linearly decreased from [
Humpback-whales hunt their prey by bubble net mechanism. In this technique bubbles are released around the prey to make a trap then Humpback-whale move around the prey in the shrinking circle as well as come up and update its position in the spiral shape. This strategy can be mathematically modeled as follows.
To achieve this, the value of 'a' in Equation (
Here the total distance from prey to humpback-whales is determined, and a helix-shape movement equation for search agent is created, which can be written as follows.
(
where 'l' ranges from [−1, 1], 'b' is a constant and 'D' represent distance between jth whale and the prey (best-solution),
There is a 50% probability that the Humpback-whale will follow either shrinking circular or a helix-shaped movement as it moves around its prey. Thus, its total swim around the prey during its hunt down can be mathematically modeled as:
(
'p' represents a random value ranging from [0, 1]. Visual representation of how the position will be updated from Equation (
In order to provide a global search, the search-agent or Humpback-whale explores the best-solution and updates its position based on another randomly selected search-agent. This behavior can be mathematically expressed as follows:
(
(
This makes WOA a global search optimization algorithm. Where
Summarizing the above method, the WOA starts with a random particle having their respective fitness function values. As the iteration proceeds, each humpback-whale/search-agent upgrades its position in two different ways, either with regards to randomly selected particle or overall best particle present so far. A random particle is chosen when the magnitude of A > 1, while the best particle/solution is selected when the magnitude of A < 1. To switch between different phases (exploration or exploitation), values of parameter 'a' is varied, which ranges from [
To find optimal sizes of multiple FACTS devices (SVC, TCSC, and UPFC) as well as settings of power system components (transformers and generators) by the WOA, fixing dimensions of search-agent/whale is a major decision. In our case, there are a total of 18 decision variables for both IEEE 14 and 30 bus systems, as shown in Table 2. Therefore, in our implementation of WOA, each search-agent/whale has a row matrix A of 18 variables, such that A = [ A
Figure 8 presents a flowchart of proposed WOA for optimal sizing and coordination of FACTS devices.
Graph: Figure 8 Flow chart of the proposed WOA.
The proposed method of optimal siting, sizing and coordination of FACTS devices is applied on IEEE-14 and the 30 bus systems. The results and discussions of both the test systems are given below.
The IEEE-14 bus system [[
After defining FACTs positions using respective strategies, TCSCs, SVCs, and UPFCs are placed at their optimal locations, and different optimization algorithms like WOA, PSO, and GA are utilized to minimize the objective function. For WOA, the total of 100 whales or search agents are considered, and the iterations are run for 100 times. For IEEE-14 bus system total of 18 fitness function variables are considered here. The variables comprise of a total number of FACTS devices, total transformer taps settings, and reactive power generation units. Fitness function variables are nothing but just a string of variables carrying different ratings or sizes, which affect the fitness function value. The size of variables is changed in every iteration and optimized by an algorithm.
Figure 10 presents a comparison of reduction in true power losses using different techniques at various loading scenarios before and after the placement of the FACTS devices.
Table 3 shows the comparison of the operating-cost (OC) in the IEEE-14 system with and without FACTS using different techniques. Without FACTS devices, the OC only consists of cost due to active power loss. In this work, energy cost is considered as 0.092 $/kwh. PSO is run for 100 iterations with 100 particles. Similarly, WOA is run for 100 times, and the number of whales/search-agent taken are 100, GA is implemented in the MATLAB optimtool. After positioning of the FACTS devices, the total OC consists of the cost of the true power loss and the cost of installation of the FACTs. Variables here are the total number of tap settings of transformers, VAr generation units, and the total FACTS devices installed. Table 4 presents the optimal values of these variables using different techniques at different reactive power loadings.
From Table 3 net savings can be calculated as (A–B). A comparison of net saving for a whole year using PSO, GA, and WOA at different reactive loading is presented in Figure 11.
Figure 12, Figure 13 and Figure 14 show the variation of operation-cost by adding FACTS devices to the network using different techniques.
From Figure 11, we can see that at the base case reactive loading, WOA converged at 0.9995 × 10
The better performance of WOA is because of its highly effective exploration and exploitation. Equation (
IEEE-30 bus system [[
In IEEE-30 bus system lines 4, 7, 9 are carrying higher active power and are connected between buses 3-4, 4-6 and 6-7 thus UPFCs are connected at buses 3, 4, and 6 to control their voltage magnitude and respective angles. After connecting each FACTS device at their optimal locations WOA, PSO, and GA are utilized to determine optimal ratings of FACTS devices that minimize the objective function. The locations of the FACTS devices and one-line diagram of the IEEE-30 bus system are shown in Figure 15.
Total true power losses before and after placing FACTS at different reactive loadings are presented in Figure 16.
Table 5 shows a comparison of the OC in the IEEE-30 bus system using different techniques, OC without FACTS is just the cost of active power loss. After placing FACTS, the OC consists of real power loss costs and FACTS installation costs.
Comparison of the net savings (A–B) for a whole year with FACTS devices sized and coordinated using different techniques are shown in Figure 17.
Optimal variables setting at different reactive loading using PSO, GA and WOA is presented in Table 6.
Figure 18, Figure 19 and Figure 20 present variation of operating cost at different reactive loading using different techniques.
In this paper, a novel method was deployed for optimal placement and sizing of multiple types of FACTS devices as well as coordination with conventional reactive power sources. Initially, sensitivity analysis was carried out to find ideal locations for TCSC, UPFC, and SVC. Lines with a higher value of the Lmn index were considered weak lines and thus optimal locations for TCSC. Similarly, buses with a higher voltage deviation in response to increasing loading parameters are considered weak buses and suitable locations for SVC. Lines carrying higher active power are chosen as ideal for UPFC placement. After optimal placements of FACTS devices in the network, optimal settings of fitness function variables were determined by relatively newly introduced optimization techniques, namely whale optimization algorithm (WOA) and results were also compared with PSO and GA. The objective function included the cost of the active power loss and the total cost of FACTS devices. The proposed technique was applied to IEEE 14 and the 30 bus system. At 100% reactive loading in the 14 bus system, WOA saved $719,000 more as compared to PSO, and $125,000 more as compared to GA. Similarly, at the base case for 30 bus systems, WOA saved $957,000 and $67,000 more as compared to PSO and GA, respectively. Hence WOA delivered better results than PSO and GA in terms of minimization of total operating cost and active power losses. It was also noted that net savings obtained using WOA were higher than those obtained using PSO and GA for all loading conditions.
Future work may involve the improvement of the whale optimization algorithm by hybridizing it with other metaheuristic techniques. Furthermore, as integration of renewable energy sources is increasing very rapidly in the power system, the optimal coordination of FACTS in the presence of renewable sources is worth exploring.
Graph: Figure 1 Control parameters of commonly used FACTS devices.
Graph: Figure 2 Static model of TCSC.
Graph: Figure 3 Static model of the SVC.
Graph: Figure 4 Static model of Unified Power Flow Controller (UPFC).
Graph: Figure 5 Predictor and corrector scheme used in CPF [[
Graph: Figure 6 Shrinking-encircling technique [[
Graph: Figure 7 Spiral-updating position technique [[
Graph: Figure 9 FACTS devices placement on IEEE-14 bus system.
Graph: Figure 10 Comparison of real losses without and with FACTS in th IEEE-14 system using different.
Graph: Figure 11 Comparison of the net savings at various reactive loading.
Graph: Figure 12 Change in operating cost with the addition of FACTS at base-case using PSO, GA, and WOA.
Graph: Figure 13 Change in Operating cost with the addition of FACTS at 150% reactive loading using PSO, GA, and WOA.
Graph: Figure 14 Change in operating cost with the addition of FACTS at 200% reactive loading using PSO, GA, and WOA.
Graph: Figure 15 FACTS devices placement on the IEEE-30 bus system.
Graph: Figure 16 Comparison of real losses without and with FACTS in the IEEE-30 system using different techniques.
Graph: Figure 17 Comparison of the net savings at various reactive loadings using different techniques.
Graph: Figure 18 Change in operating cost with addition of FACTS at base-case using PSO GA and WOA.
Graph: Figure 19 Change in operating cost with the addition of FACTS at 150% reactive loading using PSO, GA, and WOA.
Graph: Figure 20 Change in the operating cost with the addition of FACTS at 200% reactive loading using PSO, GA, and WOA.
Table 1 Comparison of basic types of FACTS devices based on their impact on different applications.
FACTS Device Real Power Flow Transient Stability Voltage Control Dynamic Stability Thyristor-controlled series-compensators Medium Strong Small Medium Static synchronous compensators Small Medium Strong Medium Static-VAR-Compensators Small Small Strong Medium Unified-Power-Flow-Controllers Strong Medium Strong Medium
Table 2 Number of variables for the test systems in the presence of the FACTS devices.
Test-System No. of TCSC No. of SVCs No. of UPFCs No. of Tap Settings of Transformer Var Generation Units IEEE-14 bus system 4 4 3 3 4 IEEE-30 bus system 3 3 3 4 5
Table 3 Cost of power loss and operating cost of the system with FACTS in the IEEE-14 bus system using different techniques.
Percentage Total Cost of Power Loss of System (A) Algorithm Used to Minimize Objective Function FACTS Devices Cost Operating Cost with FACTS Devices (B) 200 1.1952 × 107 PSO 3.433 × 105 1.172 × 107 GA 2.988 × 105 1.1029 × 107 WOA 2.870 × 105 1.0891 × 107 150 1.1442 × 107 PSO 3.033 × 105 1.1199 × 107 GA 2.580 × 105 1.0318 × 107 WOA 2.525 × 105 1.021 × 107 100 1.1226 × 107 PSO 3.677 × 105 1.0714 × 107 GA 3.202 × 105 1.012 × 107 WOA 2.424 × 105 0.9995 × 107
Table 4 Optimal setting of variables by PSO, GA, and WOA at various loading scenarios in IEEE-14 bus system.
Fitness Function Variables Optimal Settings at 200% Reactive Loading Optimal Settings at 150% Reactive Loading Optimal Settings at 100% Reactive Loading SVC (7) 0.003 0.201 0.041 0.051 0.160 0.029 0.011 0.090 0.001 SVC (9) 0.022 0.170 0.034 0.012 0.120 0.290 0.061 0.052 0.001 SVC (11) 0.194 0.112 0.005 0.125 0.070 0.094 0.002 0.040 0.061 SVC (14) 0.127 0.061 0.002 0.102 0.102 0.081 0.194 0.150 0.001 TCSC (8) 0.001 0.014 0.050 0.086 0.124 0.005 0.034 0.102 0.060 TCSC (9) 0.002 0.008 0.050 0.018 0.001 0.059 0.011 0.016 −0.040 TCSC (15) 0.014 0.024 0.061 0.000 0.067 0.073 0.040 0.043 0.051 TCSC (18) 0.001 0.000 0.019 0.001 0.017 0.000 0.001 0.011 0.091 UPFC (3) 0.006 0.019 0.004 0.000 0.008 0.031 0.170 0.001 0.001 UPFC (2) 0.280 0.000 0.001 0.002 0.009 0.001 0.003 0.039 0.100 UPFC (5) 0.001 0.001 0.009 0.070 0.001 0.311 0.000 0.002 0.000 QG (2) 0.122 0.154 0.601 0.015 0.681 0.600 0.312 0.613 0.551 QG (3) 0.321 0.277 0.581 0.419 0.083 0.369 0.389 0.667 0.481 QG (6) 0.415 0.091 0.182 0.225 0.184 0.487 0.413 0.519 0.101 QG (8) 0.087 0.623 0.163 0.451 0.212 0.098 0.513 0.082 0.682 TAP (8) 0.982 1.018 0.900 0.913 0.976 1.0300 0.905 1.054 0.901 TAP (9) 0.995 0.919 0.991 0.985 0.992 0.902 0.991 0.981 0.978 TAP (10) 0.957 0.996 0.985 0.996 1.000 1.013 0.916 0.985 0.901
Table 5 Cost of power loss and operating cost of the system with the FACTS in the IEEE-30 bus system using different techniques.
Percentage Reactive Loading Total Cost of Power Loss of System (A) ($) Algorithm Used to Minimize Objective Function FACTS Devices Cost ($) Operating Cost with FACTS (B) ($) 200 1.4902 × 107 PSO 3.774 × 105 1.393 × 107 GA 3.517 × 105 1.325 × 107 WOA 3.527 × 105 1.318 × 107 150 1.443 × 107 PSO 3.479 × 105 1.3601 × 107 GA 3.082 × 105 1.215 × 107 WOA 3.028 × 105 1.205 × 107 100 1.4152 × 107 PSO 3.136 × 105 1.266 × 107 GA 2.908 × 105 1.177 × 107 WOA 2.791 × 105 1.1703 × 107
Table 6 Optimal settings of variables by PSO, GA, and WOA at various loading scenarios in IEEE-30 bus system.
Fitness Function Optimal Setting at 200% Reactive Loading Optimal Setting at 150% Reactive Loading Optimal Setting at 100% Reactive Loading SVC (26) 0.041 0.045 0.057 0.051 0.031 0.049 0.021 0.032 0.000 SVC (29) 0.194 0.211 0.049 0.143 0.117 0.240 0.078 0.046 0.010 SVC (30) 0.142 0.108 0.0418 0.072 0.093 0.140 0.055 0.048 0.018 TCSC (15) 0.026 0.014 0.050 0.086 0.124 0.005 0.034 0.102 0.060 TCSC (7) 0.015 0.024 0.061 0.000 0.067 0.073 0.040 0.043 0.051 TCSC (20) 0.048 0.008 0.050 0.018 0.001 0.059 0.011 0.016 −0.040 UPFC (3) 0.018 0.011 0.241 0.011 0.028 0.051 0.017 0.011 0.041 UPFC (6) 0.003 0.034 0.004 0.037 0.005 0.031 0.023 0.013 0.041 UPFC (4) 0.025 0.028 0.031 0.024 0.009 0.051 0.016 0.037 0.001 QG (2) 0.318 0.158 0.590 0.267 0.042 0.131 0.198 0.267 0.610 QG (5) 0.248 0.103 0.290 0.364 0.698 0.591 0.014 0.388 0.509 QG (8) 0.023 0.191 0.501 0.611 0.045 0.610 0.581 0.314 0.098 QG (11) 0.518 0.208 0.191 0.317 0.142 0.519 0.318 0.301 0.611 QG (13) 0.032 0.605 0.189 0.345 0.298 0.181 0.676 0.097 0.184 TAP (11) 0.914 0.904 1.034 0.982 0.914 0.991 0.948 0.903 1.032 TAP (12) 1.038 0.981 1.050 0.938 0.901 0.990 0.934 1.012 0.920 TAP (15) 0.902 0.958 1.043 0.984 0.918 1.020 0.931 0.907 1.018 TAP (36) 0.918 0.905 0.988 0.901 0.926 1.038 0.938 0.994 0.991
M.N. gathered required data, conducted simulations, wrote the initial draft of the paper and updated it many times. K.I. supervised this research work, edited multiple drafts of paper, and coordinated with all authors. A.P. contributed by supervising and editing the paper. The rest of the authors reviewed drafts of the paper and added value by revisions. All authors have read and agreed to the published version of the manuscript.
This research benefited from funding received from USAID for exchange visit to ASU.
The authors declare no conflict of interest.
The authors are grateful to NUST and USAID for supporting this research work. We would also like to acknowledge support and guidance provided by Pooja Gupta and Reetam Sen Biswas in the School of Electrical Computer and Energy Engineering, Arizona State University (ASU), USA.
By Muhammad Nadeem; Kashif Imran; Abraiz Khattak; Abasin Ulasyar; Anamitra Pal; Muhammad Zulqarnain Zeb; Atif Naveed Khan and Malhar Padhee
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