GA and PSO Techniques Based Optimal Tuning of Power System Stabilizers in a PV Integrated Power Network

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 proportional-integral (PI) PSS. The preliminary investigations on the PV integrated standard IEEE power system reveal that the PSO and GA-based PSS are significantly better than the PI-PSS, and further, PSO-PSS is marginally more effective than the GA-PSS in damping the rotor angle oscillations.

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 high-gain and fast-response 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 GA-PSS and PSO-PSS to the traditional proportional-Integral (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.

Fig. 1. Schematic diagram of a synchronous generator with governor control and AVR.

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:

(1) $$\begin{array}{r}\dot{\delta }=\omega -{\omega }_s\end{array}$$

(2) $$\begin{array}{r}\dot{\omega }=\frac{1}{M}(P_m-P_e)\end{array}$$

(3) $$\begin{array}{c}\dot{E_q^{\prime }}=\frac{1}{T_{d0}^{\text{'}}}(E_{fd}-E_q^{\text{'}}-(X_d-X_d^{\text{'}})I_d)\end{array}$$

(4) $$\dot{E_d^{\prime }}=\frac{1}{T_{d0}^{\text{'}}}(-E_d^{\prime }-(X_q-X_q^{\prime })I_{\text{q}})$$

Fig. 2. Transfer function model of a synchronous generator.

The variation in the exciter output voltage is governed by:

(5) $$\begin{array}{r}\dot{E_{fd}}=\frac{1}{T_e}(-E_{fd}-K_e(V_t-V_{ref}))\end{array}$$

The currents equations of the synchronous generator in dq frame are:

(6) $$\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}{I}_q\\ I_d\end{array}\right]=\frac{1}{R_a^2+X_d^{\text{'}}{X}_q^{\text{'}}}\left[\begin{array}{cc}{R}_a& X_d^{\text{'}}\\ -X_q^{\text{'}}& R_a\end{array}\right]\left[\begin{array}{c}{E}_q^{\text{'}}-V\\ E_d^{\text{'}}-V\end{array}\right]\end{array}\end{array}$$

Transmission Line

Transmission lines are represented as the PI-Model shown in Fig. 3.

Fig. 3. Transmission line model.

Transformer

Transformers are represented as shown in Fig. 4.

Fig. 4. Transformer model.

PV System

Fig. 5 shows the representation of a PV cell. The power output of each such PV cell is given by:

Fig. 5. PV cell circuit diagram.

(7) $$\begin{array}{r}{P}_{PV}=I\cdot V\cdot \text{ FF }\cdot \eta \end{array}$$

With the overall efficiency of the PV system, the fill factor is given by:

(8) $$\begin{array}{r}FF=\frac{V_{mp}.I_{mp}}{V_{oc}\cdot I_{sc}}\end{array}$$

PSS

As shown in Fig. 6, the transfer function model of a PSS has four blocks, namely, gain, wash-out, phase compensation, and filter [16]. Gain block refines the input signal for PSS by filtering out unwanted disturbances, wash-out 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.

Fig. 6. Block diagram of PSS [17].

With speed deviation taken as the input signal, the transfer function of PSS can be written as:

(9) $$\begin{array}{r}{G}_{PSS}\left(s\right)=K_p\cdot \frac{s{T}_w}{1+s{T}_w}\cdot \frac{1+s{T}_1}{1+s{T}_2}\cdot \frac{1+s{T}_3}{1+s{T}_4}\end{array}$$

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 off-springs, which are then getting interacted with and thus generating the next generation off-springs. 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 off-springs.

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 GA-PSS and PSO-PSS are evaluated on the IEEE standard 14-bus 5-generator power system shown in Fig. 7 [34], with an integrated photo-voltaic energy conversion system at Bus-13. 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.

Fig. 7. IEEE 14-bus standard test system with integrated PV system and PSS.

Table I. Generators Data

Gen no.	Bus	S (MVA)	X′d (p.u.)	X′q (p.u.)	Td0′ (s)	Tq0′ (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

Table II. Automatic Voltage Regulator Data

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

Table III. Static Synchronous Compensator Data

Bus	Voltage magnitude (p.u.)	Qmax	Qmin
6	1.07	0.24	−0.6
8	1.09	0.24	−0.6

Table IV. PV System Data

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 14-bus system shown in Fig. 7 is performed corresponding to the following cases:

• Case 1: System-based 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 Generator-1 AVR provided with conventional PI-PSS.
• Case 4: Repetition of Case 3, replacing PI-PSS with the GA-tuned optimal PSS.
• Case 5: Repetition of Case 3, replacing PI-PSS with the PSO-tuned 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%.

Fig. 8. Case 1: Eigenvalues of the 14-bus system without PV integration.

Table V. Participation Factors of Oscillatory Modes

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

Fig. 9. Case 1: Variation of relative rotor angles following the 20% load increase at bus 12.

Fig. 10. Case 2: Eigenvalues of 14-bus system with PV integration.

Table VI. Eigenvalues of the 14-Bus System with Various PV Penetration Levels

		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.

Fig. 11. Case 2: Variation of relative rotor angle (${\mathrm{\delta }}_2$ − ${\mathrm{\delta }}_1$) following a disturbance with different levels of PV penetration.

• Case 3: When the AVR of generator 1 is provided with PI-PSS, 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 PI-PSS 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 GA-PSS and PSO-PSS.
• 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 Bus-1. 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:

Fig. 12. Case 3: Variation of relative rotor angle following a disturbance under 40% of PV penetration.

(10) $$\begin{array}{cc}Fitnessfunction& =\left(\frac{1}{Overshoot\times SettlingTime\times Steady-StateError}\right)\end{array}$$

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 ${\text{T}}_{1,}{\text{T}}_{2,}{\text{T}}_{3,}\text{and}{\text{T}}_{4,}$ 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 PI-PSS.

Table VII. A Typical Initial Chromosome of Genetic Algorithm with Corresponding PSS Parameters

	${\text{K}}_{\text{p}}$	${\text{T}}_1$ (s)	${\text{T}}_2$ (s)	${\text{T}}_3$ (s)	${\text{T}}_4$ (s)
String	1100	101	010	100	111
Fitness	0.3469

Table VIII. PSO Algorithm: Initial Particles Position Samples with Corresponding PSS Parameters

	${\text{K}}_{\text{p}}$	${\text{T}}_1$ (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

Table IX. A Typical Initial Chromosome of Genetic Algorithm with Corresponding PSS Parameters

	${\text{K}}_{\text{p}}$	${\text{T}}_1$ (s)	${\text{T}}_2$ (s)	${\text{T}}_3$ (s)	${\text{T}}_4$ (s)
PI-PSS	5	0.3800	0.0200	0.3800	0.0200
GA-PSS	13.8103	0.3221	0.0190	0.4801	0.0660
PSO-PSS	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 GA-PSS and PSO-PSS respectively with that of PI-PSS. It can be observed that both GA-PSS and PSO-PSS show better performance than the PI-PSS in terms of peak-overshoot and settling time and reduced oscillations. Further, the preliminary investigations reveal that the PSO-PSS is better than the GA-PSS in reducing the small signal oscillations.

Fig. 13. Relative rotor angle (${\mathrm{\delta }}_2$ − ${\mathrm{\delta }}_1$) following a disturbance with 60% PV penetration.

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 left-hand side of the s-plane, indicating stability in all three cases, irrespective of the type of PSS. However, the rotor angle oscillations are seen to be better damped with PSO-PSS when compared to PI-PSS and GA-PSS.

Table X. Critical Modes of 14-Bus System of Each Case of PSS Algorithms Under 60% Level of PV Penetration

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 GA-PSS and PSO-PSS have been compared with that of the PI-PSS. The dynamic performance of both GA and PSO tuned PSS were better than that of the PI-PSS. The simulation results of standard power systems have shown that the PSO-PSS shows better damping of small signal oscillations than that of the GA-PSS. 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

Table XI. List of Symbols and Abbreviations

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
${\text{I}}_{\text{d}}$	Direct-axis stator current (pu)
${\text{I}}_{\text{mp}}$	PV maximum power point current (pu)
${\text{I}}_{\text{q}}$	Quadrature-axis 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 (MJ-s/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)
$T_{d0}^{\prime }$	Direct-axis 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)
${\text{V}}_{\text{d}}$	Direct-axis stator voltage (pu)
${\text{V}}_{\text{mp}}$	PV maximum power point voltage (pu)
${\text{V}}_{\text{oc}}$	PV open circuit voltage (pu)
${\text{V}}_{\text{q}}$	Quadrature-axis stator voltage (pu)
${\text{V}}_{\text{ref}}$	Exciter reference voltage (pu)
${\text{V}}_t$	Exciter terminal voltage (pu)
${\text{X}}_{\text{d}}$	Direct-axis synchronous reactance (pu)
${\text{X}}_{\text{q}}$	Quadrature-axis synchronous reactance (pu)
${\text{X}}_d^{\prime }$	Direct-axis transient reactance (pu)
${\text{X}}_q^{\prime }$	Quadrature-axis 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