The Optimization

111

as in equation (6-1) as the system active power losses (𝑃𝐿𝑜𝑠𝑠). If there are n EVCSs, vector

𝑋 would be [𝑥₁, 𝑥₂, .. , 𝑥_𝑖, .. , 𝑥_𝑛], where 𝑥_𝑖 is the number of EVs at EVCS 𝑖.

𝑂𝐹(𝑋) = 𝑀𝑖𝑛(𝑃_{𝐿𝑜𝑠𝑠} (𝑋)) = 𝑀𝑖𝑛 (∑ 𝑃_𝑔 𝑁𝑔 𝑔=1 − ∑ 𝑃_𝑙(𝑋) 𝑁𝑙 𝑙=1 ) (6-1) such that ∑ 𝑥_𝑗 𝑁𝐶𝑆 𝑗=1 = ∑ 𝑥_𝑗𝑖𝑛𝑖𝑡 𝑁𝐶𝑆 𝑗=1 , 𝑗 = 1,2,3, … , 𝑁_𝐶𝑆 (6-2) 𝑥_𝑗𝑚𝑖𝑛≤ 𝑥𝑗 ≤ 𝑥𝑗𝑚𝑎𝑥 , 𝑗 = 1,2,3, … , 𝑁𝐶𝑆 (6-3) 𝑄𝑔𝑚𝑖𝑛 ≤ 𝑄𝑔 ≤ 𝑄𝑔𝑚𝑎𝑥 , 𝑔 = 1,2,3, … , 𝑁𝑔 (6-4) 𝑉_𝑖𝑚𝑖𝑛_{≤ 𝑉} 𝑖 ≤ 𝑉𝑖𝑚𝑎𝑥 , 𝑖 = 1,2,3, … , 𝑁𝑏 (6-5) 𝑆_𝑘 ≤ 𝑆_𝑘𝑚𝑎𝑥 , 𝑘 = 1,2,3, … , 𝑁_𝑇𝐿 (6-6) 𝑃_𝑖 + 𝑗𝑄_𝑖 = 𝑉_𝑖∑ 𝑌_𝑖𝑘𝑉_𝑘𝑒𝑗(𝛿𝑖−𝛿𝑘−𝜃𝑖𝑘) 𝑁𝑏 𝑘=1 , 𝑖 = 1,2,3, … , 𝑁𝑏 (6-7)

Where 𝑃_𝑔 and 𝑄_𝑔 are the active and reactive powers of g-th generator, respectively. 𝑃_𝑙 is the active power of the l-th load.

𝑥_𝑗𝑖𝑛𝑖𝑡 and 𝑥𝑗 in equation (6-2) are the numbers of EVs at the j-th EVCS in the initial and

each iteration of the optimization, respectively. The output of this optimization, 𝑋𝑜𝑝𝑡, is an optimal vector of EV distributions on the EVCSs as in (6-8):

𝑋𝑜𝑝𝑡 _{= [𝑥} 1 𝑜𝑝𝑡

112

The search space is represented by equation (6-3), where 𝑥_𝑗𝑚𝑖𝑛 and 𝑥_𝑗𝑚𝑎𝑥 are the lower bound and the upper bound of the range of parameters in the optimization variable vector 𝑋. They are the minimum and maximum number of EVs at the j-th EVCS, respectively. Equations (6-4) to (6-7) are the constraints of the optimization problem. 𝑉_𝑖 in equation (6-5) is the voltage magnitude at the i-th bus. 𝑆_𝑘 in equation (6-6) is the complex power of the k-th transmission line. 𝑃_𝑖 and 𝑄_𝑖 are the pure active and reactive power injected at the i-th bus. In equation (6-7), 𝑌𝑖𝑘 and 𝜃𝑖𝑘 are the magnitude and phase of the admittance between

buses i and k, respectively. 𝛿𝑖, 𝑎𝑛𝑑 𝛿𝑘 are the voltage angle at the i-th and k-th bus,

respectively. 𝑁_𝑔, 𝑁_𝑙, 𝑁_𝐶𝑆, 𝑁_𝑏, and 𝑁_𝑇𝐿 are the number of generators, loads, EVCSs, buses, and transmission lines, respectively.

6.3.1 Particle Swarm Optimization

In this work, the modified PSO algorithm is deployed to solve the optimization problem. PSO simulates the behaviors of bird flocking. In PSO [68], each single solution is a "bird" in the search space. It is called a "particle". All of the particles have fitness values which are evaluated by the objective function to be optimized, and have velocities which direct the flying of the particles. The particles fly through the problem space by following the current optimum particles. PSO is initialized with a group of random particles (solutions) and then searches for the optima by updating generations. In every iteration, each particle is updated by following two "best" values. The first one is the best solution (fitness) it has achieved so far. This value is called “pbest.” Another "best" value that is tracked by the particle swarm optimizer is the best value obtained so far by any particle in the whole swarm. This best value is a global best and is called “gbest”. As shown in the PS

113

Agent flowchart in Figure 6.3, after finding the two best values in the iteration, the particle updates its velocity and position. This is repeated while the condition of maximum iterations or minimum error criteria is not attained.

In this work, the number of iterations is 30 and the swarm size is 250 particles. If, during the PSO algorithm, a particle violates any of the constraints, it is ignored. The stopping criteria here is the maximum number of iterations. The PSO running time is recorded to be around 5.7 seconds.

The particle 𝑋 here is 6 dimensional, each dimension being the number of EVs at the corresponding EVCS. In the power flow, this number is translated into MW for varying the load representing the EVCSs. 𝑁_𝑗𝑚𝑖𝑛, the minimum number of EVs at the j-th EVCS, is set as 20% of the number of cars in the initial distribution of each round of optimization. 𝑁_𝑗𝑚𝑎𝑥, the maximum number of EVs at the j-th EVCS, is set as 200.

6.3.2 Normal Conditions

In the normal operating state, the system is said to be secure, where all constraints like maximum capacities of transmission lines, minimum and maximum voltage limits at buses, and the minimum and maximum real and reactive power generation are satisfied. In this condition, collective distribution of EVs is performed to minimize the active power losses in the system while meeting all normal operation constraints.

In this study, the minimum and maximum voltage limits at each bus for normal conditions are set at the standard limits of 0.95 and 1.05 p.u, respectively as set by the NEC standard.

114 6.3.3 N-1 Contingencies

The failing of one or several components in the power system may lead to transmission congestion, a situation in which the power system cannot operate safely. This occurs when there is not enough transmission capability to perform the needed power flow and keep the generation-loading balance. In these cases, overloading in some transmission lines or voltage violation beyond acceptable ranges might occur. If the constraints defined in equations (6-4) and (6-7) are violated in any post-contingency scenario, the system’s operating state is deemed insecure. This congestion may be alleviated by incorporating line capacity constraints in the real-time optimization.

The responsibility of the ISO is to maintain the system’s security and manage the congestion through a real-time congestion management (RTCM) algorithm. Optimal operation of FACTs devices, generation rescheduling, load demand response and load shedding are some tools for RTCM.

In this chapter, the collective distribution of EVs is used as the tool for the RTCM problem, while simultaneously minimizing the losses in the system. For this purpose, the minimum and maximum voltage limits are considered as 0.90 and 1.05 p.u. for contingency conditions, while the maximum loading for transmission lines would be 90% for N-1 contingencies and 100% for contingencies with higher order. In the RTCM problem, the dynamic and quasi dynamic thermal ratings of transmission lines can be considered, as described in [50]. However, in this work, only the static rating of the transmission lines are considered.

115

In the contingency conditions, the optimization is run with the new limits for voltage and transmission line loading. Although the objective function aims at minimizing the losses in the system, huge penalty factors are added to the objective function if any constraint is violated. Therefore, the optimal collective distribution of EVs is obtained to manage the congestion while minimizing the active power losses in the system. Numerical results will demonstrate the effectiveness of this method in mitigating the congestion in power systems.