GA and PSO Techniques Based Optimal Tuning of Power System Stabilizers in a PV Integrated Power Network
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This paper involves the employment of Genetic Algorithm (GA) and Particle Swam Optimization (PSO) methods for the optimal tuning of gain and time constant parameters of power system stabilizers in a power network with integrated Photovoltaic (PV) energy conversion systems. To examine the critical modes of the PV integrated system and the impact of the PV integration on various modes of oscillations, eigenvalue analysis is carried out under various PV penetration levels. The investigations indicate that the rotor angle oscillations following a disturbance have less damping as the PV penetration level increases. To enhance the small signal stability, the excitation systems are provided with the signal from power system stabilizers (PSS). The gain and time constant parameters of the PSS are tuned utilizing the GA and PSO methods and compared with the conventional proportionalintegral (PI) PSS. The preliminary investigations on the PV integrated standard IEEE power system reveal that the PSO and GAbased PSS are significantly better than the PIPSS, and further, PSOPSS is marginally more effective than the GAPSS in damping the rotor angle oscillations.
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Introduction
Synchronous generators are equipped with automatic voltage regulators (AVR) to improve voltage regulation and to enhance the ability of power systems to ensure the synchronous operation of generators [1]. A highgain and fastresponse AVR is an effective way to improve the transient stability following sudden and large disturbances. However, it impairs the dynamic stability issues following small and slow disturbances. These small signal oscillations can be effectively damped by incorporating power system stabilizers (PSS) into the AVRs of appropriate machines.
Renewable energy sources (RES) such as solar photovoltaic (PV) energy conversion systems have been increasingly utilized to meet the growing power demand [2]. With high penetration of PV systems into a power grid, there is a possibility of stability issues [3], [4]. Hence, this paper proposes to evaluate the dynamic stability issues due to the increase in the penetration of solar PV systems into a power grid and the impact of PSS in damping small oscillations [5]–[8]. It is necessary to tune the PSS gain and time constants appropriately [9] for further enhancement of the damping provided by a PSS. This paper proposes the employment of genetic algorithm (GA) and particle swarm optimization (PSO) methods for the determination of the optimal gain and time constant parameters of the PSS [10]–[14]. It compares the performance of the GAPSS and PSOPSS to the traditional proportionalIntegral (PI) PSS. Fig. 1 shows the overall schematic diagram of a synchronous generator with the associated governor controllers and automatic voltage regulators (AVRs) [15]. It has been observed that the voltage regulators have a negative effect on the damping of small signal oscillations. To minimize this negative effect, an additional stabilizing signal generated by PSS from the generator speed deviation is injected into its excitation system.
System Modelling
Synchronous Generator
Transfer function model of a synchronous generator is shown in Fig. 2 [15]. The fourth order model is utilized to represent the synchronous generators in the system. Thus, the differential equations, corresponding to this model, are:
The variation in the exciter output voltage is governed by:
The currents equations of the synchronous generator in dq frame are:
Transmission Line
Transmission lines are represented as the PIModel shown in Fig. 3.
Transformer
Transformers are represented as shown in Fig. 4.
PV System
Fig. 5 shows the representation of a PV cell. The power output of each such PV cell is given by:
With the overall efficiency of the PV system, the fill factor is given by:
PSS
As shown in Fig. 6, the transfer function model of a PSS has four blocks, namely, gain, washout, phase compensation, and filter [16]. Gain block refines the input signal for PSS by filtering out unwanted disturbances, washout block reduces the steady time offset. The signal limits are used to optimize the output signal in order to prevent the undesirable saturation in the excitation system.
With speed deviation taken as the input signal, the transfer function of PSS can be written as:
PSS Tuning
Tuning of a PSS involves adjusting its gain and time constant parameters to ensure the desired damping of the system oscillations following a disturbance. PSS parameters tuned utilizing GA and PSO methods are discussed in this section.
Genetic Algorithm (GA) Based Tuning of PSS
The genetic algorithm operates based on the Darwinian principle of “survival of the fittest” [18]–[22]. An initial population is created, consisting of a specified number of individuals or solutions, where each individual is represented by a genetic string containing variable information. Each individual’s associated fitness is measured, so it typically represents an objective value. Individuals in a population generate fitter offsprings, which are then getting interacted with and thus generating the next generation offsprings. Chosen individuals are selected for reproduction or crossover at each generation, and appropriate mutation factors can develop the new population by randomly modifying the genes of an individual. The result is another group of individuals with better individual fitness. GA proceeds by replacing the individuals with lower fitness values in the previous generation with those having higher fitness values in the latest offsprings.
Particle Swarm Optimization (PSO) Based Tuning of PSS
Particle Swarm Optimization (PSO) is a fascinating optimization method that draws inspiration from the collective behaviors observed in nature, such as fish schooling, bird flocking, and swarming phenomena, as well as computational techniques [23]–[31]. This approach involves a dynamic process where a population of simple entities, known as particles, interact within a defined search space associated with a specific function or problem.
In the PSO method, there are several simple entities known as particles. Each particle then adjusts its movement based on its current and historical best positions, along with interactions with other particles and random changes. This process aims to guide the swarm towards optimal solutions over time, similar to how a flock of birds navigates towards a target point [32], [33].
Test System
The effectiveness of the GAPSS and PSOPSS are evaluated on the IEEE standard 14bus 5generator power system shown in Fig. 7 [34], with an integrated photovoltaic energy conversion system at Bus13. The system base is 100 MVA, with a voltage base of 69 kV in the zone covering buses 1, 2, 3, 4, 5, and 13.8 kV in the zone covering buses 6, 7, 9, 10, 11, 12, 13 and 14 while 18 kV in the zone with bus 8 [35]. The generator data and the data corresponding to the AVRs, SSC, and PV modules are shown in Tables I–IV, respectively.
Gen no.  Bus  S (MVA)  X′_{d} (p.u.)  X′_{q} (p.u.)  T_{d0}′ (s)  T_{q0}′ (s) 

1  1  615  0.6  0.646  7.4  0 
2  2  60  0.185  0.36  6.1  0.3 
3  3  60  0.185  0.36  6.1  0.3 
4  6  25  0.232  0.715  4.75  0.06 
5  8  25  0.232  0.715  4.75  0.06 
Gen no.  Maximum regulator voltage  Amplifier gain Ka  Time constant Ta (s) 

1  7.32  200  0.02 
2  4.38  20  0.02 
3  4.38  20  0.02 
4  6.81  20  0.02 
5  6.81  20  0.02 
Bus  Voltage magnitude (p.u.)  Q_{max}  Q_{min} 

6  1.07  0.24  −0.6 
8  1.09  0.24  −0.6 
Rated power (W)  Peak power (MW)  Peak power voltage (V)  Peak power current (A) 

150  150  34  4.4 
Open circuit voltage (V)  Short circuit current (A)  Minimum peak power (W)  
43.4  4.8  142.5 
Simulation Results
Dynamic stability analysis of the 14bus system shown in Fig. 7 is performed corresponding to the following cases:
 Case 1: Systembased case analysis with all the five generators with AVR.
 Case 2: Repetition of Case 1 with integrated PV at the load buses 12, 13, and 14, respectively.
 Case 3: Repetition of Case 2 with Generator1 AVR provided with conventional PIPSS.
 Case 4: Repetition of Case 3, replacing PIPSS with the GAtuned optimal PSS.
 Case 5: Repetition of Case 3, replacing PIPSS with the PSOtuned optimal PSS.
In each of these cases, the system is subjected to a disturbance of 20% increase (157 MW) in load at Bus 12 at the instant t = 1 second. These cases are evaluated on the basis of the respective sets of eigenvalues, participation factors of various oscillation modes, and the variation of relative rotor angles obtained through the time domain simulation. The simulation results are discussed in this section.
 Case 1: Fig. 8 shows the location of the eigenvalues in the complex plane. With 49 eigenvalues, 46 situated on the left half of the complex plane and one at zero, the system experiences instability due to the remaining two eigenvalues positioned on the right side of the complex plane. Table V shows the various oscillatory modes and the corresponding participation factors. The eigenvalues given in Table V show that the most critical modes are related to generator 1. Fig. 9 shows that the variation of the relative rotor angles of all the synchronous generators are without sufficient damping.
 Case 2: The impact of PV integration in the system is investigated by performing the dynamic stability simulation at various levels of PV penetration, focusing on the most critical modes of oscillation. Fig. 10 shows the location of eigenvalues corresponding to 20% PV penetration level. The PV is integrated into the system at the load buses 12, 13, and 14, respectively. Table VI shows the effect of PV integration on the oscillation modes of the system with 0%, 20%, 40%, and 60% PV penetration levels, respectively, of the total generation (785 MW). From this Table, it can be observed that the integration of PV increases the damping of oscillation mode except with modes 3, 4, and 5, where mode 3 is the most significantly affected one. The critical values of the system are obtained from the eigenvalue analysis, and the location of the eigenvalues is given in the complex plane. It is seen that the frequency modes vary from 0.01 to 2 Hz, the damping ratio is less than 20%.
Mode no.  Most associated states  Generator no. 

1, 2  ${\text{E}}_{q}^{\text{'}}$, ${\text{E}}_{\text{fd}}$  1, 1 
3  $\mathrm{\delta}$  1 
4, 5  ${\text{E}}_{q}^{\prime}$, ${\text{E}}_{\text{fd}}$  2, 3 
6  $\mathrm{\omega}$  1 
7, 8  ${\text{E}}_{q}^{\prime}$, ${\text{E}}_{\text{fd}}$  3, 2 
9  ${\text{E}}_{q}^{\prime}$  4 
10  ${\text{E}}_{d}^{\prime}$  5 
PV Penetration in % of total power generation (785 MW)  

0  20  40  60  
Mode  1, 2  0.9139 ± j8.4084  0.2955 ± j8.4183  −0.1307 ± j8.795  −0.1907 ± j7.5950 
3  0  0  0.0336  0.2750  
4, 5  −0.6044 ± j0.7577  −0.6306 ± j0.3553  −0.5976 ± j 0.7406  −0.5499 ± j 0.8222  
6  −0.1859  −0.1664  −0.2013  −0.2013  
7, 8  −0.6476 ± j0.3582  −0.6306 ± j0.3553  −0.6262 ± j0.2538  −0.6281 ± j0.2142  
9  −0.8987  −0.8983  −0.9594  −1.3043  
10  −0.9371  −0.9729  −0.9724  −1.3323 
Fig. 11 shows the variation of the relative rotor angle of generator 2 corresponding to 0%, 20%, 40%, and 60% PV penetration, respectively. It can be noted that the system becomes more oscillatory and tends to become more unstable with the increase in the level of PV penetration.
 Case 3: When the AVR of generator 1 is provided with PIPSS, the dynamic stability is getting improved, as seen in the variation of relative rotor angles of various generators, as shown in Fig. 12. It corresponds to 40% PV penetration. From the figure, it is evident that the PIPSS provides damping to the dynamic stability Oscillations. Further investigations are carried out to further reduce these oscillations by tuning the PSS using GA and PSO methods. Case 3 results are utilized as the base case for comparison of the effect of GAPSS and PSOPSS.
 Cases 4 and 5: GA and PSO algorithms are utilized to tune the gain and time constant parameters of the PSS connected to AVR of the generator at Bus1. These parameters are , , , and . The lower and upper limits of the time constants are 0.01 and 2, respectively, while the lower and upper limits of the gain are 1 and 50 respectively. The minimum and maximum weights of PSO are 0.4 and 0.9, respectively, the genetic algorithm employs a crossover probability of 0.95 and a mutation probability of 0.1, with a maximum of 250 generations allotted for each execution of the algorithm. fitness of each individual is evaluated as:
The optimization process is set to be stopped when one of the following criteria is met: reaching maximum number of generations; the second is when solution improvement is not significant for a number of generations. Table VII shows a typical initial chromosome of the genetic algorithm. Each chromosome has 16 digits with the first four bits assigned for the gain K and then three bits for the time constants ${\text{T}}_{1,}{\text{T}}_{2,}{\text{T}}_{3,}\text{}\text{and}\text{}{\text{T}}_{4,}$ respectively. Table VIII shows initial particles positions of PSO algorithm given that the population size is 10. Table IX shows the optimal parameters of PSS that are by GA and PSO methods along with those of the conventional PIPSS.
${\text{K}}_{\text{p}}$  ${\text{T}}_{1}$ (s)  ${\text{T}}_{2}$ (s)  ${\text{T}}_{3}$ (s)  ${\text{T}}_{4}$ (s)  

String  1100  101  010  100  111 
Fitness  0.3469 
${\text{K}}_{\text{p}}$  ${\text{T}}_{1}$ (s)  ${\text{T}}_{2}$ (s)  ${\text{T}}_{3}$ (s)  ${\text{T}}_{4}$ (s)  

Particle 1  40  1.8  0.3  1.2  0.4 
Fitness  0.0621  
Particle 2  5  0.8  1.5  0.4  0.8 
Fitness  0.0474  
Particle 3  18  1.4  0.9  1.1  0.6 
Fitness  0.1206 
${\text{K}}_{\text{p}}$  ${\text{T}}_{1}$ (s)  ${\text{T}}_{2}$ (s)  ${\text{T}}_{3}$ (s)  ${\text{T}}_{4}$ (s)  

PIPSS  5  0.3800  0.0200  0.3800  0.0200 
GAPSS  13.8103  0.3221  0.0190  0.4801  0.0660 
PSOPSS  25.1366  0.2753  0.0143  0.5350  0.0740 
Fig. 13 shows the variation of the relative rotor angle of generator 2 corresponding to 60% PV penetration. It compares the performance of the system with GAPSS and PSOPSS respectively with that of PIPSS. It can be observed that both GAPSS and PSOPSS show better performance than the PIPSS in terms of peakovershoot and settling time and reduced oscillations. Further, the preliminary investigations reveal that the PSOPSS is better than the GAPSS in reducing the small signal oscillations.
Table X shows the system eigen values corresponding to the provision of PSS to the AVR on generator at Bus 1, These values correspond to 60% PV penetration and a 20% (157 MW) increase in the load demand at Bus 13. All eigenvalues reside on the lefthand side of the splane, indicating stability in all three cases, irrespective of the type of PSS. However, the rotor angle oscillations are seen to be better damped with PSOPSS when compared to PIPSS and GAPSS.
Case  Mode 1, 2  Mode 3 

3  −0.9205 ± j7.667  −0.5524 
4  −2.3284 ± j5.7044  −2.9241 
5  −2.4439 ± j5.6818  −3.0966 
Conclusion
This paper has compared the effect of optimal tuning of power system stabilizers using genetic algorithm and particle swarm optimization method respectively to damp the small signal oscillations in PV integrated power systems. Both eigen value analysis and the time domain simulation have been performed to evaluate the effectiveness of both these methods. The dynamic response of the generator with the AVRs provided with GAPSS and PSOPSS have been compared with that of the PIPSS. The dynamic performance of both GA and PSO tuned PSS were better than that of the PIPSS. The simulation results of standard power systems have shown that the PSOPSS shows better damping of small signal oscillations than that of the GAPSS. This is contributed to its lesser sensitivity to parameters settings. Further, it converges faster than the genetic algorithm. Hence, the preliminary investigations on standard systems indicate that the PSO method is preferred over genetic algorithm for the optimal tuning of PSS gain and time constant parameters for more effective damping of small signal oscillations in a PV integrated power system.
Appendix
Symbol  Description 

${\text{E}}_{d}^{\prime}$  Generator internal voltage behind ${\text{X}}_{d}^{\prime}$ (pu) 
${\text{E}}_{\text{fd}}$  Exciter voltage (pu) 
${\text{E}}_{q}^{\prime}$  Generator internal voltage behind ${\text{X}}_{q}^{\prime}$ (pu) 
FF  PV fill factor 
${\text{I}}_{\text{d}}$  Directaxis stator current (pu) 
${\text{I}}_{\text{mp}}$  PV maximum power point current (pu) 
${\text{I}}_{\text{q}}$  Quadratureaxis stator current (pu) 
${\text{I}}_{\text{sc}}$  PV short circuit current (pu) 
${\text{K}}_{\text{e}}$  Exciter gain 
$\text{M}$  Inertia constant of the synchronous generator (MJs/elect rad) 
${\text{P}}_{\text{e}}$  Electrical power output of the generator (pu) 
${\text{P}}_{\text{m}}$  Mechanical power input of the generator (pu) 
${\text{R}}_{\text{a}}$  Armature resistance (pu) 
${T}_{d0}^{\prime}$  Directaxis open circuit transient time constant (s) 
${\text{T}}_{\text{e}}$  Exciter time constant (s) 
${\text{P}}_{\text{e}}$  Electrical power output of the generator (pu) 
${\text{V}}_{\text{d}}$  Directaxis stator voltage (pu) 
${\text{V}}_{\text{mp}}$  PV maximum power point voltage (pu) 
${\text{V}}_{\text{oc}}$  PV open circuit voltage (pu) 
${\text{V}}_{\text{q}}$  Quadratureaxis stator voltage (pu) 
${\text{V}}_{\text{ref}}$  Exciter reference voltage (pu) 
${\text{V}}_{t}$  Exciter terminal voltage (pu) 
${\text{X}}_{\text{d}}$  Directaxis synchronous reactance (pu) 
${\text{X}}_{\text{q}}$  Quadratureaxis synchronous reactance (pu) 
${\text{X}}_{d}^{\prime}$  Directaxis transient reactance (pu) 
${\text{X}}_{q}^{\prime}$  Quadratureaxis transient reactance (pu) 
$\mathrm{\delta}$  Rotor angle (°) 
$\mathrm{\omega}$  Generator angular velocity (rad/s) 
${\mathrm{\omega}}_{\text{s}}$  Synchronous angular velocity (rad/s) 
$\mathrm{\eta}$  Overall efficiency of PV system 
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