1. Introduction

Damping of small signal oscillations in power systems is a main problem that has been challenging engineers for years. These oscillations may be very poorly damped in some cases, resulting in mechanical fatigue at the machines and unacceptable power variations the across the important transmission lines. For this reason, the applications of the controllers to provide better damping for these oscillations are of utmost importance[1]. The goal of the electromechanical oscillations in power systems in an interconnected power system is to maintain the frequency and to damp small signal oscillations at short time so as to accommodate fluctuating load demands. By the development of interconnection of large electric power systems, there have been spontaneous system oscillations at very low frequencies in order of 0.2-3.0 Hz. In some cases, they continue to grow and presented a limitation on the amount of the power to be transmitted within the system if no adequate damping is available [1]. Power System Stabilizers (PSSs) are auxiliary control devices on synchronous generators, used in conjunction with their excitation systems to provide control signals toward enhancing the system damping of low frequency oscillations associated with the electromechanical modes and extend power transfer limits; thus maintaining reliable operation of the power system[2].

The conventional lead-lag compensators have been widely used as the PSSs. However, the problem of PSS parameter tuning is a complex exercise. These stabilizers have previously tuned both single and multiple operation points of the power system using various methods. The approaches used to the problem of PSS parameters tuning range from modern control theory[2-5], to the more recent one using many random heuristic techniques, such as like honey bee optimization technique[6], artificial bee colony search[7], chaotic optimization algorithm[8], genetic algorithm[9], rule based bacteria foraging[10] and strength Pareto evolutionary algorithm[11] for achieving high efficiency and search global optimal solution in the problem space.

Recently, the classical PSO based method[12, 13] has been represented to optimize of PSS parameters. The potential of this method was shown for PSS designing to improve low frequency oscillations damping at different operating conditions than the GA approach on a multi-machine power system. An Improved PSO with Time Varying Acceleration Coefficients (PSO-TVAC) has been undertaken to solve the PSS design problem by Shayeghi et al.[14] to improve the global search in the early stage of the optimization process and cheering the particles to converge toward the global optima at the end of it for effectively enhancement the convergence rate and the accuracy of the classical PSO. Using the numerical results it was shown that PSO-TVAC technique has better convergence property and can get superior low frequency oscillations than the classical one.

Despite the potential of the modern control techniques with different structure, Proportional Integral Derivative (PID) type controller is still widely used for industrial applications such as power systems control[15-17]. This is because it performs well for a wide class of process. Also, they give robust performance for a wide range of operating conditions and easy to implement. On the other hand, Shayeghi and Esmailnezhad[15] presented a comprehensive analysis of the effects of the different PID controller parameters on the overall dynamic performance of the PSS problem. It is shown that the appropriate selection of PID controller parameters results in satisfactory performance during system upsets. Thus, the optimal tuning of a PID gains is required to get the desired level of robust performance. Since optimal setting of PID controller gains is a multimodal optimization problem (i.e., there exists more than one local optimum) and more complex due to nonlinearity, complexity and time-variability of the real world power systems operation. Hence, local optimization techniques, which are well elaborated upon, are not suitable for such a problem. In order to overcome these drawbacks, a novel Iteration Particle Swarm Optimization (IPSO) technique is proposed for the optimal tune of the multiple PID type stabilizers in this paper to achieve desired level of low frequency oscillations damping and enhance dynamic stability of the multi machine power system. It should be noted that the performance of the classical PSO greatly depends on its parameters adjustments, and it often suffers the problem of being trapped in the local optima so as to be premature convergence. For this reason, in the proposed algorithm, a new term named, iteration best is incorporated in classical PSO to enrich the searching behavior, solution quality and to avoid being trapped into local optimum. The effectiveness of the proposed PID type stabilizer is tested on a multi-machine power system under different operating conditions in comparison with the PSO and PSO-TVAC[14] based tuned stabilizers through nonlinear time domain simulation and some performance indices. The simulation results reveal the robust performance of the proposed method for damping low frequency oscillations than the PSO, and PSO-TVAC one tuned PSS such that the relative stability is guaranteed and the time domain specifications concurrently secured. This presents a useful promising tool to choose enviable stabilizer from a set of optimally tuned PID type PSSs for the system operator, PSS manufacturer and customers.

2. Power System Modeling

A four-machine, two-area study system, shown in Fig. 1, is considered for the damping control design. Each area consists of two generator units. The rating of each generator is 900 MVA and 20 kV. Each of the units is connected through transformers to the 230 kV transmission line. There is a power transfer of 400 MW from Area 1 to Area 2. The detailed bus data, line data, and the dynamic characteristics for the machines, exciters, and loads are given in[18]. The loads are modeled as constant impedances. For the power system stability analysis a sufficient mathematical models considering a set of nonlinear differential-algebraic equations by assembling the models for each generator, load and other devices such as controls in the system is required. The two-axis model (fourth order)[2] given in is used for the time domain simulations study for each machine. The loads are modeled as constant impedances. A first order model of a static type automatic voltage regulator was used. Nonlinear dynamic equations of the each machine can be summarized as follows:

(1)

(2)

(3)

(4)

(5)

The nomenclature used is given in Appendix.

Figure 1. Single line diagram of a two area system

2.1. PSS Structure

The operating function of a PID type PSS is to produce a proper torque on the rotor of the machine involved in such a way that the phase lag between the exciter input and the machine electrical torque is compensated. The supplementary stabilizing signal considered is one proportional to speed. A widely speed based used PID is considered throughout the study[2]. The transfer function of the i^th PID type stabilizer is given by:

(6)

Where, T_D<<1 and usually is considered K_D/100. Δω_i is the speed deviation of the ith generator and U_i is the output signal fed as a supplementary input signal to the regulator of the excitation system. This type of PSS consists of a washout filter and a PID compensator. The washout filter, which really is a high pass filter, is regarded as to reset the steady-state offset in the output of the stabilizer. The value of the time constant T_w is usually not critical and it can range from 1 to 20 s. It should be noted that the PSS output must be limited to V_s^max and V_s^min for avoiding actuator system damaging. In this study, they are tunable. Thus, the optimized parameters of PID type PSS are:

• K_Pi, K_Ii and K_Di PID Gains

• T_Wi Time constant of washout filter

• V_s^max and V_s^min PSS output saturations

3. Iteration PSO Algorithm

3.1. Classical PSO Review

The PSO is a population-based method and is described by its developers as an optimization paradigm, which models the social behavior of the birds flocking or fish schooling for food. Therefore, PSO works with a population of potential solutions rather than with a single individual[12].

In the PSO technique a number of simple entities, the particles, are placed in the search space of some problem or function, and each evaluates the objective function at its current location. Each particle then determines its movement through the search space by combining some aspect of the history of its own current and best locations by those of one or more members of the swarm with some random perturbations. The next iteration takes place after all particles have been moved. Eventually the swarm as a whole, like a flock of birds collectively foraging for food, is likely to move close to an optimum of the fitness function[19].

In the PSO, the trajectory of each individual in the search space is adjusted by dynamically altering the velocity of each particle, according to its own flying experience and the flying experience of the other particles in the search space. The position and velocity vectors of the i^th particle in the D-dimensional search space can be represented as X_i = (x_i1; x_i2; …; x_id) and V_i = (v_i1; v_i2;…; v_id), respectively. According to a user defined fitness function, let us assume that the best position of each particle, which corresponds to the best fitness value (P_best) obtained by that particle at time, be P_i = (p_i1; p_i2; …; p_id), and the global version of the PSO keeps track of the overall best value (G_best), and its location, obtained thus far by any particle in the population. Then, the new velocities and the positions of the particles for the next fitness evaluation are calculated using the following two equations[12]:

(7)

Where, P_id and P_gd are P_best and G_best. The positive constants c₁ and c₂ are the cognitive and social components that are the acceleration constants responsible for varying the particle velocity towards p_best and g_best, respectively. Variables r1 and r2 are two random functions based on uniform probability distribution functions in the range[0, 1]. The inertia weight, ω, is responsible for dynamically adjusting the velocity of the particles, so it is responsible for balancing between local and global searches and hence requiring less iteration for the algorithm to converge[19]. The following inertia weight is used in Eq. (7):

(8)

Where, K_max is the maximum number of iterations and K is the current number of iteration. The Eq. (8) presents how the inertia weight is updated, considering ω_max and ω_min are the initial and final weights, respectively.

3.2. Iteration PSO Procedure

In this paper, a new index named, Iteration Best, is incorporated in Eq. (7) to enrich the searching behavior, solution quality and to avoid being trapped into local optimum, IPSO technique is proposed, Eq. (9) shows the new form of Eq. (7)[20]:

(9)

Where I_b is the best value of the fitness function that has been obtained by any particle in any iteration and c₃ shows the weighting of the stochastic acceleration terms that pull each particle toward I_b[20]. The motivation for using this method is enhancement the global search and to enrich the information sharing mechanism for effectively enhancement the convergence rate and the accuracy of the classical PSO.

4. IPSO Based Stabilizer Design

This section describes how the IPSO algorithm is employed to tune nonlinear parameters (V_s^max and V_s^min) and PID type PSS parameters for the two-area multi-machine power system as shown in Fig. 1. Just like any other optimization problem, a cost function needs to be formulated to determine optimal parameters of multiple PSSs. The goal of optimal tuning of stabilizer parameters task is to maximize low frequency oscillation damping; i.e.: minimize the settling time and overshoots in system response. In order to get the PSS design task a cost function that minimizes the time domain based system response characteristics is used in this study. It can be defined as follows:

(10)

(11)

Where, t_sim is the time range of simulation; Δω_ij is the relative rotor speed of the ith generator than the jth generator; OS_ij is the overshoot of Δω_ij. NP is the total number of operating points considered for optimization process. The salient feature of this objective function is that it needs the minimal dynamic plant information.

The design problem can be formulated as the following constrained optimization problem, where the constraints are the PSS parameters bounds:

(12)

Typical ranges of the optimized parameters are[0.1-20] for T_w and[0.01-50] for PID controller parameters (K_Pi, K_Ii and K_Di ) and[0.05-0.5] for V_smax and -V_smin. The IPSO approach was implemented in MATLAB software and used to solve this optimization problem and search for the optimal set of stabilizers parameters. To evaluate the effectiveness and robustness of the proposed optimization technique numerous operating conditions and the system configurations, simultaneously are considered. The multiple operation conditions are given in Table 1.

The optimal tuning of the PSS parameters is carried out by evaluating the fitness function as given in Eq. (12) for four operating conditions as given in Table 1 by applying a 6-cycle three-phase fault at the middle of one of the transmission line between bus-7 and bus-8. The fault is cleared by permanent tripping of the faulted line. In this study, the PSO module works offline. For the each PSS, the optimal setting of six parameters is determined by the IPSO, i.e. 24 parameters to be optimized.

Table 2. Control parameters for optimization PSO-TVAC IPSO PSO C1f 0.2 C1 0.01 C1 2.0 C1i 2.5 C2 0.01 C2 2.0 C2f 2.5 C3 c1 (1-ec1×k) ωmin 0.4 C2i 0.2 ωmin 0.4 ωmax 0.9 φ 4.1 ωmax 0.9 Population 40 ωmin 0.4 Population 40 Iteration 100 ωmax 0.9 Iteration 100 - - Population 40 - - Iteration 100 - -

In order to facilitate comparison with the PSO and PSO-TVAC methods, the design and tuning of the stabilizers parameters for this multi-machine power system, the PSO and PSO-TVAC methods (For more details see the Ref.[24]) were used. Figure 2 shows the block diagram of IPSO based tuned PID type stabilizers procedure. In order to acquire better performance, the control parameters of the proposed IPSO, PSO-TVAC[14] and PSO[12] techniques is given in Table 2. Optimized multiple stabilizers parameter set values corresponding to the best fitness achieved by each algorithm after 10 trials based on the objective function as given in Eq. (12) using the above methods are listed in Table 3. Figure 2 shows the minimum fitness functions evaluating process.

Table 3. The optimal parameter settings of the proposed stabilizers

Figure 2. Fitness convergence, Dotted (PSO), Dashed (PSO-TVAC) and Solid (IPSO)

5. Simulation Results

The effectiveness and robustness of the performance of the proposed PID type stabilizer under transient conditions is verified by applying a three-phase fault of 100 ms duration at the middle of one of the transmission lines between bus-7 and bus-8. The fault is cleared by permanent tripping of the faulted line. To evaluate the performance of the proposed stabilizer design approach the response of the proposed PSS are compared with the response of the PSO and PSO-TVAC damping controller individual design.

The inter-area and local mode of oscillations with the above stabilizers for deferent operating conditions as given in Table 1 is shown in Figs. 3-5, respectively. The performance of the IPSO based optimized multiple PID type stabilizer is quite prominent in comparison with the other PSSs and the overshoots and settling time are significantly improved with the proposed stabilizer.

To illustrate robustness of the proposed method, some performance indices based on the system performance characteristics are defined as:

(13)

(14)

(15)

(16)

Where, and T_s are mean overshoot, mean undershoot and mean settling time of four relative speed deviations of Δω₁₂, Δω₁₃, Δω₁₄ and Δω₃₄. It is merit mentioning that the lower the value of these indices is, the better the system response in terms of the time-domain characteristics. Numerical results of performance robustness for all system loading cases are shown in Table 4 with three PID type stabilizers by applying a three-phase fault of 100 ms duration at the middle of one of the transmission lines between bus-7 and bus-8. Evaluation of these Table reveals that the using the proposed IPSO the speed deviations of all machines are greatly reduced, has small overshoot, undershoot and settling time. Also, it achieves good robust performance compared to that of designed stabilizers using the PSO and PSO-TVAC methods.

Table 1. Four operating condition (pu) Operating Condition G1 G2 G3 G4 P Q P Q P Q P Q Case 1 (Base Case) 0.7778 0.1021 0.7777 0.1308 0.7879 0.0913 0.7778 0.0918 Case 2 (20% increase for system load in case 1) 1.084 0.3310 0.7778 0.4492 0.7879 0.1561 0.7778 0.2501 Case 3 (20% decrease for system load in case 1) 0.7778 0.0502 0.2333 0.0371 0.7989 0.0794 0.7778 0.0704 Case 4 (Loss of a line between bus 7 and bus 8) 0.7778 0.1021 0.7777 0.1308 0.7989 0.0903 0.7778 0.0981

Figure 3. Inter-area and local mode of oscillations for case 1: Dotted (PSO), Dashed (PSO-TVAC) and Solid (IPSO)

Figure 4. Inter-area and local mode of oscillations for case 2: Dotted (PSO), Dashed (PSO-TVAC) and Solid (IPSO)

Figure 5. Inter-area and local mode of oscillations for case 4: Dotted (PSO), Dashed (PSO-TVAC) and Solid (IPSO)

Table 4. Values of the performance indices under different conditions Algorithm Scenario Compare Index ITAE FD IAE ISE IPSO Base Case 16.533 12.342 7.1231 1.2532 20% increase 18.563 19.564 8.4573 2.3459 20% decrease 26.432 25.561 10.137 3.5466 2 line tripe 16.437 16.855 6.5232 1.5671 PSO-TVAC Base Case 20.562 35.643 11.349 2.2098 20% increase 22.560 21.778 9.4590 3.4633 20% decrease 32.451 27.400 10.354 3.8498 2 line tripe 17.674 20.638 8.5266 1.7733 PSO Base Case 54.563 101.33 36.658 8.6643 20% increase 60.756 123.34 47.789 15.223 20% decrease 63.675 103.42 55.897 19.264 2 line tripe 56.642 131.33 38.904 10.928

5. Conclusions

In this paper, a novel Iteration Particle Swarm Optimization (IPSO) algorithm has been successfully applied to the robust design of multiple PID type stabilizers to improve damping of the low frequency oscillation in the multi machine power system. The design problem of the robustly selecting stabilizer parameters is converted into an optimization problem according to time domain-based objective function over a wide range of operating conditions that is solved by the IPSO technique. It has stronger global search ability and more robust than PSO and PSO-TVAC heuristic methods. The effectiveness of the proposed strategy was tested on a two-area four machine power system under different operating conditions. The nonlinear time-domain simulation results demonstrate the effectiveness of the proposed PID type stabilizers and their ability to provide good damping of low frequency oscillations. The system performance characteristics in terms of ‘ITAE’, ‘IAE’, ‘ISE’ and ‘FD’ indices reveal that the proposed IPSO algorithm is superior that of the PSO and PSO-TVAC in terms of accuracy and computational effort.

Appendix: Numenclature