Particle Swarm Optimization Solution for Power System Operation Problems Particle Swarm Optimization Solution for Power System Operation Problems

Application of particle swarm optimization (PSO) algorithm on power system operation is studied in this chapter. Relay protection coordination in distribution networks and economic dispatch of generators in the grid are defined as two of power system-related optimization problems where they are solved using PSO. Two case study systems are conducted. The first case study system investigates applicability of PSO on providing proper overcurrent relay settings in the grid, while in the second case study system, the economic dispatch of a 15-unit system is solved where PSO successfully provides the optimum power output of generators with minimum fuel costs to satisfy the load demands and operation constraints. The simulation results in comparison with other methods show the effectiveness of PSO against other algorithms with higher quality of solution and less fuel costs on the same test system.


Introduction
Electric power system is the most complex man-made system, and the modern society depends heavily on continuous and reliable operation of this system to supply electricity to commercial, residential and industrial consumers. Operation of the power grid involves a balanced platform in generation, transmission and distribution, which costs billions of dollars to run. The reliable and continuous availability of electricity with minimum costs is the major objective of utility grids and energy providers. Two of the important complex problems in power system are economic dispatch and power system protection.
The power plants and utility grids need to allocate the available generation units in an efficient and economical way to respond to the load demand in order to provide continuous power supply in stable conditions and with minimum power production costs. This is addressed as economic dispatch (ED) [1,2]. With the practical constraints on the generators, finding optimum power outputs with minimum fuel costs is challenging.
In addition, as the occurrence of failures and faults in the power grid is inevitable, the entire power system must be protected. The relay protection scheme is designed to detect faults and isolate the faulty parts of the grid from the healthy sections in order to mitigate the consequences of the faults and maintain continuity of service. If a fault occurs, the nearest corresponding relays must operate as fast as possible to clear the fault. If due to any reason these primary relays fail to react, their backup relays must operate and accomplish the task. Directional overcurrent relays (DOCRs) are a suitable and economical protection scheme for distribution systems [3]. The protection design of DOCRs is based on two parameters, time multiplier setting (TMS) and plug setting (PS). Proper settings of TMS and PS allow a primary relay to clear the faults in its protection zone as fast as possible and in case of failure, its backup relay operates immediately after a time interval to clear the fault. TMS and PS values of each relay must be coordinated with other backup relays, where again relays act with different current settings, which make the coordination a complex task. Each pairs of relays include four variables (TMS, PS) and the complexity of coordination will be intense in bigger systems with more relays and constraints. Due to the complex interconnection of the distribution systems and also nonlinear characteristics of operation time of relays, finding best relay settings could be very difficult.
In this chapter, PSO is applied as a solution to the introduced power system operation problems, namely ED and DOCR coordination. The rest of the chapter is organized as follows: in Section 2, these power system problems are defined and formulated as optimization problems. PSO algorithm is explained in Section 3. In Section 4, PSO is applied in two case study systems to conduct the performance and feasibility of this method. Finally, Section 5 concludes the chapter with the results in pervious sections.

Problem formulation
In this section, overcurrent relay coordination and economic dispatch problems are formulated separately as optimization problems.

Relay coordination problem
In a protection scheme, each primary relay should operate as fast as possible to clear the fault in a system. If the operation time of the relay takes longer than an acceptable time, the damage on the faulty equipment would be severe with serious consequences. In other words, minimizing the total operation time of relays decreases the risk and stress on the protected apparatus, which can be depicted as an optimization objective function: where t i is the operation time of relay R i , w i is the probability of the occurrence of fault on transmission line in the zone of protection, and it is normally set to 1; n is the total number of relays in the system.
Generally, the operation time of DOCRs is defined in (2): where I Fi is the fault current seen by the appropriate relay R i after being transformed through the secondary winding of corresponding current transformer (CT). Depending on the type of relays, the characteristic constants λ, L and η are selected [16]. In this chapter, continuous form of TMS and PS is considered with relay type of standard inverse definite minimum time (IDMT). Based on that, all the relays in the system are assumed identical with a common characteristic function approximated by: To ensure that the operation time of an individual relay is proper enough to mitigate the damage impact of faults on the apparatus, the time must be within an acceptable range: where, respectively, t imin and t imax are the minimum and maximum operating time of the relay R i . Each overcurrent relay has a manufactured TMS range to provide controllability of response to faults with different speeds. As shown in (3), t i is proportional to TMS values. Also, the PS has nonlinear effect on the operating time. Within a security margin and to avoid maloperation of an individual relay with normal load or slight overload current, the minimum pickup current setting is selected bigger than the maximum load current. The maximum plug setting is chosen not greater than the minimum fault current [19]. Therefore, there are constraints on TMS and PS as follows: where TMS imax , PS imax , TMS imin and PS imin are the maximum and minimum values of TMS and PS of relay R i . Although the constraints in Eqs. (4)-(6) seem to provide satisfactory limits on performance of each relay, they are not enough to guarantee correct performance of the protection scheme as the coordination between primary-backup relays has not been considered.
The constraints can only ensure the operation of an individual relay not a primary-backup pair. To coordinate adjacent relays as primary and backup relays, the primary relay should operate as fast as possible within its acceptable boundaries. If it fails to act, its backup relay needs to take over the tripping action with a minimum time. The minimum operating time of the backup relay must be small but yet bigger than the operating time of primary relay. Therefore, a coordination time interval (CTI) is added to the constraints to satisfy the proper coordination scheme.
where t i and t j are the operation time of primary and backup relays, respectively. CTI depends upon the relay type, circuit breaker speed, relay over-travel time and the safety factor time for CT saturation, setting errors, contact gaps and so on. According to the IEEE standard [21], CTI is set to 0.2 s for the digital relays. Figure 1 shows the backup-primary pair of relays in a radial network.

Economic dispatch
How to allocate the available generators in the grid to respond to the load demand with minimum fuel costs is an economical aspect of power dispatch in the electric power system that annually costs millions of dollars to operate. In a practical ED optimization, the generator constraints and network limits such as ramp rate limit, the prohibited zones of operation, generation capacity constraints and valve point effects are considered. Single quadratic equation is used to formulate the ED optimization problem: where α i , β i and γ i are the cost equation coefficients of unit i, P i is the output power in MWand F P i ð Þ is the cost function of that unit in $/h. The index m denotes the number of generators in a system.
The fuel cost function in (8) as the objective function of ED problem here is associated with practical constraints. Considering the ramp rate limits, the momentary output power of a generator cannot exceed its previously generated power more than a certain amount of UR i , the up-ramp rate limit and neither can it be less than that of the previously generated power by more than a certain amount of DR i , the down-ramp rate limit of the generator. Therefore: where P i is the current power output and P i0 refers to the previous power output of generator i. UR i and DR i represent the up-ramp limit and down-ramp limit of the generator i, respectively, in which the current generated power has the following constraint in MW/t: The input-output curves of the generation units have separate operation zones. The prohibited zones of operations are due to the operation of steam valve or the shaft bearing vibration of the generators. Therefore, the generated power is within the feasible zones of operation and outside the prohibited zones. For a generator i: where j is the number of prohibited zones of operation for unit i, P l i, j and P u i, j are the lower and upper boundaries, respectively, of prohibited zone j of generator i, P min i and P max i are minimum and maximum power capacity, respectively, of a generator that produces. From (11) and (12), it can be deduced that the operational power of a generator must be within the constraint in (13) and also be out of the prohibited zones in (12): Total delivered power from the units needs to meet the power demand and transmission loss in the grid.
where P D and P L represent the demand power and the power loss in the gird, respectively. The overall power loss of the committed units is based on the output power, which is formulated by B matrix coefficients known as Kron's formula: The power loss itself cannot be more than some permissible values: where the real power flow of line j is represented with P Lf , k and k is the number of transmission lines in a system. The power loss cannot be more than a maximum value of P max Lf , k .

Particle swarm optimization algorithm
PSO algorithm is a nature inspired method from social behavior of bird flocking and fish schooling, which is first introduced by Erberhart and Kennedy in 1995 [22]. As a populationbased stochastic optimization technique using swarm intellects in the search space, this technique is based on interaction of swarm of particles. Every particle includes two values of position and velocity that are updated during the iteration runs by considering each particle's best experience (best position) and the best achieved experience (global position) of all particles.
The update of position and velocity of the particles must be processed, and it has to follow Eqs. (17)- (19) for satisfying the constraint of an optimization problem. Each particle movement is based on the changes in its position and its velocity: where p kþ1 i and p k i are the position of particle i in the iteration k þ 1 and k, respectively, v kþ1 i is the velocity of the particle in k þ 1 iteration. A particle's velocity is defined as follows: where pbest i is the best so far position of the particle i as the best experience, while gbest is the best position among the whole swarm with all the particles in movement as global experience. c 1 and c 2 are the weighting factors, while rand 1 and rand 2 are two random numbers between zero and one. The parameter w is the inertia factor varying between w min ; w max ½ , as shown in (19), which is a linear decreasing inertia weight in this chapter.
where k and iter max are the current iteration and the maximum number of iteration during simulations, respectively.
The general steps of PSO on solving an optimization problem are as follows: 1. Set initial parameters w min , w max , c 1 and c 2 .
2. Generate initial populations having initial positions p and velocities v.

4.
Calculate fitness of particles F k i ¼ f p k i À Á , ∀i and find the index of the best particle b.

5.
Select pbest k i ¼ p k i , ∀i and set gbest k ¼ p k b .

7.
Update the velocity and position of particles.
∀i and obtain the index of the best particle b 1 .
if F kþ1 Update gbest of the population if F kþ1 b1 < F k b then gbest kþ1 ¼ pbest kþ1 b1 and b ¼ b1 else gbest kþ1 ¼ gbest k 11. If k < Iter max then k ¼ k þ 1 and go to step 6 else go to step 12.

Simulation results
Application of PSO on solving the defined problems in previous sections is validated here. In the first case study system, the overcurrent relay coordination problem in a distribution network is solved. The size of population and iteration numbers of PSO are 60 and 100, respectively. In the second case study system, the ED problem is addressed with population size of 100 and 500 iterations. c 1 , c 2 , w min and w max in both case study systems are the same and set to 2, 2, 0.4 and 0.9, respectively.

Case study 1: relay protection coordination
A 15-node radial network system including a total of 13 loads is considered, and PSO is employed to determine the optimal settings of all 28 digital over-current relays shown in Figure 2. Identical digital relays are used with same current transformer (CT) ratio 500:1. The constraint values of the relays are as follows: t imin and t imax for each relay is set to 0.1 and 4 s, respectively; TMS min and TMS max constraints are 0.1 and 1.1, respectively; PS min and PS max are 0.5 and 2.5, respectively [16].
The primary-backup relationships of the relays in the system are shown in Table 1. According to the objective function in (3), the maximum fault currents sensed by the relays are also required. Therefore, the case study system in Figure 2 has been modeled in DigSILENT PowerFactory software with simulating three-phase faults occurring in front of each relay. The collected data have been shown in Table 2. The load parameters of the radial network are shown in Table 3.
After data collection in DigSILENT PowerFactory software, the optimal settings (TMS and PS) of all 28 overcurrent relays are obtained by solving (3) subject to the constraints in (4)- (7) using the PSO in MATLAB software. The algorithm was executed 100 times to achieve accurate results. Table 4 shows the best TMS and PS settings of the relays to provide a reliable protection scheme Figure 2. A 15-node distribution system as case study one. in the distribution system. The total operation time of the relays in this system is 26.189 s. Figure 3 illustrates the convergence curve of PSO in solving the objective function.

Case study 2: economic dispatch
A 15-unit test system is used to investigate the feasibility of PSO in solving the nonsmooth economic dispatch considering transmission losses, ramp rate limits and the prohibited operation zones of the thermal generators. The cost curves data and operation limits of the 15-unit system are shown in Table 5. The B loss coefficients to calculate the power loss can be found in [4].  This system has many local minima with high dimensionality that draws realistic analysis for practical applications. PSO is applied on the 15-unit test system, and the results are compared with best results in literature on the same system: ACHS [8], SQPSO [11], ABC [6], IPSO [12], PSO-MSAF [13], DSPSO-TSA [14], ICA [5], GAAPI [9], MPSO [10], SOH-PSO [15], GCPSO [10], PC-PSO [15], BF [7], SPSO [15], PSO [4] and GA [4].
The simulation is tested for 100 times to ensure reliable analysis. Table 6 shows the optimum results of each method for the 15-unit system. Figure 4 illustrates the convergence of PSO method in 100 different trials, while the final fuel costs in 100 trials are shown in Figure 5. Table 7 shows the best economic dispatch of power using PSO for ED optimization in the 15unit system.
The optimum cost of 15-unit system with the proposed PSO solution is 32701.282 ($), which has better result than other methods. GA has the most deviating results compared with other hybrid and improved methods in this test system. Also, the mean value of final fuel cost of generators using PSO over 100 trials is less than minimum values obtained by other method, which indicates higher quality of solution and better performance of PSO compared with other algorithms on the same test system. The power loss in the grid obtained from PSO is less than other algorithms, which shows better dispatching scheme using PSO. The best convergence of PSO to the minimum fuel cost of generators in the grid while satisfying the constraints is shown in Figure 6.
In case study two, PSO achieves better results when compared with other hybrid or improved methods. It is worth mentioning that the maximum iteration number is 500 and the population     size is 100. In most of the quoted methods, the iteration and population sizes vary that can affect the final results. For example, the PSO in [4] has population size and iteration number of 100 and 500, respectively. The population size, iteration, crossover rate, mute rate and crossover parameter of GA [4] are 100, 200, 0.8, 0.01 and 0.5, respectively.
Different system configuration and programming language frameworks can also influence the results in which MATLAB 2015Ra was used for programming.

Conclusions
Distribution network relay coordination and the economic dispatch of generators in the electric power system were modeled as optimization problems. Particle swarm optimization (PSO) was successfully employed to solve the defined problems where two case study systems were conducted to validate the results. In the first case study system, PSO provided proper relay settings that allow all the relays in a system to perform with high reliability and accuracy. In the second case, the optimal power outputs of thermal generators in the grid were scheduled to satisfy the load demands and other practical constraints on the generators and the grid with minimum fuel costs. The compared results with other methods demonstrated higher quality of solution, and less fuel costs obtained by PSO. The general performance of PSO in this chapter indicates applicability of this method on practical power system-related problems that are difficult to be handled by conventional methods. Figure 6. Convergence of PSO with best fuel cost result for ED problem in case study two.