PSO Tuned Power System Stabilizers for Damping Small Signal Oscillations in Saudi National Grid

This paper proposes the application of Particle Swarm Optimization (PSO) technique to tune Power System Stabilizers (PSS) for damping small signal oscillations in the high-voltage transmission network of the Western Region of Saudi Arabia. The system has six thermal generating stations and works at 380 kV, 230 kV, and 132 kV transmission voltage levels. The excitation systems are of IEEE type 1. This iterative process aims to fine tune the parameters of controllers to achieve the desired damping for the targeted oscillation modes. The preliminary investigations indicate that the system damping against three-phase short circuits is significantly better with PSO tuned PSS when compared to the conventional PSS. 

Introduction

Maintaining the stability of electric power systems is a fundamental requirement to ensure continuous and efficient electricity delivery. A significant challenge to system stability arises from power system oscillations, which, if not properly controlled, can lead to disturbances that disrupt the normal grid operations. Therefore, Power System Stabilizers (PSS) have become vital components integrated into generator excitation systems to mitigate such oscillations by enhancing system damping [1]. These stabilizers act as supplementary controllers, providing positive damping to both the interarea and local oscillation modes, thereby improving the dynamic stability of the system [2]. Traditional approaches for tuning power system stabilizers rely heavily on manual, heuristic, or trial-and-error techniques, which are often time-consuming and may fail to yield optimal results under diverse operating conditions [3]. This has motivated the exploration of advanced intelligent optimization techniques to automate and enhance the tuning process.

In this context, Particle Swarm Optimization (PSO) has emerged as a powerful and promising tool for PSS parameter tuning. PSO utilizes a population-based search inspired by social behaviors in nature, offering an adaptive, efficient, and robust means to identify optimal parameter sets that maximize system damping performance [4], [5]. Employing PSO enables the development of an intelligent tuning mechanism capable of accommodating varying operating conditions and system uncertainties, thereby enhancing the resilience and reliability of power systems [6]. This study proposes the application of PSO for tuning PSS parameters to improve the damping of the Saudi national grid under a wide range of operating conditions [7].

System Description

As of 2023, the Saudi electricity grid operated by the Saudi Electricity Company (SEC) is structured into four main regions: Central, Western, Eastern, and Southern, all of which are interconnected through a high-voltage transmission backbone. This study focuses on the Western Region, which includes major urban and industrial centers such as Jeddah, Makkah, and Madinah.

The region is characterized by high seasonal load variability and a mix of thermal and emerging renewable generation assets. The grid operates at multiple voltage levels, primarily 132, 230, and 380 kV, to ensure reliable transmission across substations, plants, and consumer nodes. As of 2022, Saudi Arabia’s total installed generation capacity has exceeded 80 GW, in alignment with the National Vision 2030 plan to diversify energy sources [8].

The system utilized represents the western region grid of Saudi Arabia, with major power plants and load centers. The grid consists of synchronous generators, transmission lines, transformers, and loads connected at different buses, representing the terminals of the substations. The remaining part of the grid outside the western region was represented as an equivalent set of generators at the boundary buses. As shown in Fig. 1, the system consists of six generation plants, producing a total of approximately 3258.6 MW, with a combined generator output capacity of 4153.3 MW. The demand side includes eight load centers, totaling 3200 MW, connected via 23 buses, 23 high-voltage transmission lines, and 11 transformers. Additionally, five fixed shunt devices provided a reactive support of nearly 350 MVAR.

Fig. 1. System diagram.

System Modeling

Synchronous Generator

A third-order model is utilized for the representation of synchronous generators to analyze the dynamic stability of the power system. It extends the classical second-order model by incorporating the dynamics of the generator’s internal voltage behind the transient reactance, which improves the accuracy without significantly increasing the model complexity. This model includes three state variables: rotor angle δ, rotor speed ω, and quadrature-axis transient internal voltage E^’_q.

The rotor angle and speed dynamics are governed by the swing equation given by two first-order differential equations:

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

(2) $$\begin{array}{r}\frac{d\omega }{dt}=\frac{{\omega }_s}{2H}(P_m-P_e)\end{array}$$

where H is the inertial constant, ω_s is the synchronous speed, and ω is the actual rotor speed.

The third equation models the electromagnetic dynamics of the internal voltage E^’_q, which reflects the generator response to excitation and internal field behavior. It captures the rate of change of the transient voltage on the q-axis as

(3) $$\begin{array}{r}\frac{d{E}_q^{\prime }}{dt}=\frac{1}{T_{do}^{\prime }}(E_f-E_q^{\prime })\end{array}$$

where E_f is the excitation voltage (field voltage) and T^’_do is the open-circuit d-axis transient time constant. This equation models how the internal voltage builds up or decays in response to changes in the excitation, which is critical for stability analysis.

To complete the model, the electrical output power P_e was calculated using the internal voltage E^’_q, terminal voltage V, and generator reactances. Assuming that the d-axis flux dynamics are neglected and that the stator resistance is negligible, the power output can be approximated as

(4) $$\begin{array}{r}{P}_e=\frac{E_q^{\prime }V}{X_d^{\prime }}sin(\mathrm{\delta }-\theta )\end{array}$$

where X^’_d is the d-axis transient reactance, δ is the internal voltage angle, and θ is the terminal voltage angle.

Excitation System

The excitation system is represented by a gain block and a first-order lag, as shown in Fig. 2. The input to the excitation system is the error between the reference voltage and measured terminal voltage, and the output is the field voltage. The output voltage of the exciter is

Fig. 2. Excitation system.

(5) $$\begin{array}{r}{E}_f(s)=\frac{K_A}{1+s{T}_A}(V_{ref}-V_t)\end{array}$$

where K_A is the AVR gain, T_A the AVR time constant, V_ref the reference voltage, and V_t the measured terminal voltage. Limiting blocks were added to prevent over- or under-excitation.

Power System Stabilizer

To enhance the system damping, a Power System Stabilizer (PSS) was integrated into the excitation system. As shown in Fig. 3, the PSS is modeled with a gain, washout filter, and lead-lag compensators. The washout filter ensures that the stabilizing signal responds only to dynamic changes, whereas the lead-lag stages adjust the phase of the signal to provide effective damping. The PSS takes the rotor speed deviation as its input and produces a stabilizing signal as its output, which is fed into the excitation system:

Fig. 3. Power system stabilizer block diagram.

(6) $$\begin{array}{r}{V}_{PSS}=G_{PSS}.\mathrm{\Delta }\omega \end{array}$$

where G_PSS is the overall transfer function of the PSS.

The washout filter acts as a high-pass filter, allowing only changes in speed to affect the output. Its transfer function is

(7) $$\begin{array}{r}{G}_{washout}(s)=\frac{s{T}_w}{1+s{T}_w}\end{array}$$

where T_w is the wash-out time constant. To provide proper phase compensation and improve damping, one or more lead-lag blocks are used. A single lead-lag compensator is modeled as

(8) $$\begin{array}{r}{G}_{lead-lag}(s)=\frac{1+s{T}_1}{1+s{T}_2}\end{array}$$

where T₁ and T₂ are the time constants that shape the phase response to match the dynamic behavior of the system. By combining these elements, the stabilizing signal becomes:

(9) $$\begin{array}{r}{V}_{PSS}\left(s\right)=K_{PSS}.\frac{s{T}_w}{1+s{T}_w}.\frac{1+s{T}_1}{1+s{T}_2}.\mathrm{\Delta }\omega \end{array}$$

where K_PSS is the PSS gain.

This signal is added to the reference input of the excitation system to modulate the generator excitation and provide damping to rotor oscillations. By doing so, the PSS enhances small-signal stability and helps maintain synchronism in multi-machine power systems.

Conventional Approach

The conventional approach for dynamic stability assessment involves eigenvalue analysis. Stable systems show eigenvalues with negative real parts, indicating good damping, whereas positive real parts show signal instability and growing oscillations.

Low-damped modes, which may arise from disturbances such as generation shifts or faults, require mitigation measures, such as Power System Stabilizers, to improve damping and maintain system resilience. The conventional method for tuning a Power System Stabilizer (PSS) involves identifying poorly damped electromechanical modes and adjusting PSS parameters to improve system damping. Each eigenvalue λ = σ ± jω corresponds to an oscillatory mode, where σ is the damping coefficient, and ω is the angular frequency. The damping ratio ζ of the mode is calculated as:

(10) $$\begin{array}{r}\zeta =\frac{-\sigma }{\sqrt{{\sigma }^2+{\omega }^2}}\end{array}$$

A damping ratio of less than 5% typically indicates inadequate damping and the need for stabilizer tuning. Once the critical modes are identified, the PSS is activated and tuned to specifically target them. The tuning objective is to adjust the PSS gain K_PSS,, and lead-lag time constants T₁ and T₂ so that the damping ratio of the targeted mode is increased beyond the minimum acceptable threshold.

The lead-lag blocks in the PSS are tuned to introduce a phase lead that counteracts the phase lag introduced by the generator and system dynamics at the critical frequency, typically 0.2–2 Hz. The rule of thumb is to tune T₁ and T₂ such that the maximum phase compensation occurs at modal frequency ƒ = ω/2π. This can be approximated as follows:

(11) $$\begin{array}{r}\sqrt{T_1{T}_2}=\frac{1}{\omega }\end{array}$$

After tuning, time-domain simulations and eigenvalue analysis were repeated to verify that the stabilizer improved damping without causing adverse effects, such as voltage instability and control loop interactions. Finally, the tuned PSS configuration was validated under various operating conditions involving different load levels, contingencies, and generation scenarios to ensure a robust performance. A sensitivity analysis was also conducted to examine how variations in the system parameters affect PSS performance.

Particle Swam Optimization Approach

Utilizing PSO algorithm, the goal is to determine the optimal values of the PSS parameters, such as the gain and lead-lag time constants. This is achieved by minimizing an objective function that reflects system performance. In this study, the objective function to be minimized is selected to be the inverse of the damping ratio ζ, and is given by:

(12) $$\begin{array}{r}\frac{1}{\zeta }=\frac{-\sqrt{{\sigma }^2+{\omega }^2}}{\sigma }\end{array}$$

To apply PSO, the problem was first formulated as a constrained optimization task. A group of particles (candidate solutions) is initialized with each particle representing a potential set of PSS parameters.

These particles were randomly distributed in the multidimensional search space defined by the allowable parameter ranges. Each particle’s position is evaluated using the objective function, which is calculated based on the system’s response in the frequency domain using eigenvalue analysis, or in the time domain through the simulation of disturbances. Particle velocity and position were updated iteratively using the following equations:

(13) $$\begin{array}{r}{v}_i^{(k+1)}=w{v}_i^{\left(k\right)}+c_1{r}_1(p_i-x_i^{\left(k\right)})+c_2{r}_2(g-x_i^{\left(k\right)})\end{array}$$

(14) $$\begin{array}{r}{x}_i^{(k+1)}=x_i^{\left(k\right)}+v_i^{(k+1)}\end{array}$$

where v_i and x_i are the velocity and position of particle i, p_i is the best position found by the particle, g is the global best position found by any particle, w is the inertia weight, and c₁, c₂ are acceleration constants with random numbers r₁, r₂ between 0 and 1, respectively.

The particles explore the search space collaboratively, sharing information about their best experiences to converge towards an optimal or near-optimal solution. The inertia weight w controls the balance between exploration and exploitation; it often decreases over iterations to fine-tune convergence.

This iterative process continues for a predefined number of generations, or until the objective function meets the desired tolerance level. Once PSO converges, the optimal PSS parameters are applied to the model and validated through simulation to confirm the improved damping and overall system performance under various operating conditions and disturbances.

Simulation Results

The effectiveness of the PSO approach in the tuning of the PSS for the improvement of dynamic stability was evaluated on the typical power grid discussed in Section 2. The system data and load flow results are provided in the Appendix. Table I lists data related to the PSO algorithm. The PSS parameters obtained through the conventional and PSO approaches are listed in Table II.

Table I. PSO Algorithm Data

Variable	Value
Number of particles	30
Search space bounds	[0], [10]
Iterations	100
Inertia weight (w)	0.5
Individual acceleration constant (c1)	1.5
Social acceleration constant (c2)	2.0

Table II. Tuned PSS Parameters

Variable	Unit	Conventional tuning approach	PSO tuning approach
Ks	[p.u.]	4.5	3.15
T1	[s]	1.2	0.76
T2	[s]	0.22	0.15
T3	[s]	1.0	0.66
T4	[s]	0.1	0.2

All generators in the system were provided with excitation systems. A three-phase fault was applied at bus 205 for 100 ms, followed by clearance of the fault by tripping the transmission line between buses 205 and 154. Fig. 4 shows the variation in the relative rotor angle of Generator 2 connected to bus 206 relative to the system average angle. The figure shows that compared to the case with no PSS, the rotor angle stabilizes, and the oscillations disappear within a few seconds.

Fig. 4. System performance with PSS and without PSS: Relative rotor angle of Generator 2 at bus 206.

Using the eigenvalue analysis function in PowerFactory, it was confirmed that the oscillation mode present in the RMS is of 2.24 Hz and in the absence of PSS, it was negatively damped. Table III lists the eigenvalues of the oscillation modes of the network.

Table III. Eigenvalues of Oscillation Modes

Mode	Real part	Imaginary part	Mode frequency	Damping ratio
	1/s	rad/s	Hz	%
Mode 1	1.25	14.04	2.24	−8.89
Mode 2	1.25	−14.04	2.24	−8.89
Mode 10	−0.27	5.69	0.91	4.67
Mode 11	−0.27	−5.69	0.91	4.67
Mode 8	−0.26	2.45	0.39	10.58
Mode 9	−0.26	−2.45	0.39	10.58
Mode 29	−1.35	9.76	1.55	13.73
Mode 30	−1.35	−9.76	1.55	13.73

Fig. 5 shows a comparison of the variation in the relative rotor angle of Generator 2 obtained with for both conventional and PSO-tuned PSS against those obtained with conventional PSS. From the figure, it can be observed that PSO-PSS exhibits significantly better damping performance than PSO-PSS. The peak overshoot was smaller and the settling time was shorter than that of the conventional PSS.

Fig. 5. Comparison of conventional-PSS and PSO PSS: Variation of relative rotor angle of Generator 2.

Eigen Value Analysis

Fig. 6 shows the distribution of the eigenvalues of the system with a tuned PSS implemented at the machine on bus 206. The figure shows all eigenvalues (green crosses) in the left half of the complex plane, indicating a well-damped and stable system with no unstable modes (no red crosses) when compared with the case of an untuned PSS.

Fig. 6. Tuned PSS - Eigenvalue modes plot.

Fig. 7 shows the controllability of Mode 16 of the system. From the eigenvalue distribution in the plot, Mode 16 is shown in the left-hand half of the complex plane, indicating that it is in a stable location that actively responds to control signals.

Fig. 7. Tuned PSS - Mode 16 analysis (Controllability).

The location of Mode 16 in the well-damped region with a sufficiently negative real part indicates effective damping by the tuned stabilizer, showing a successful control input influence on system states to reduce oscillations. Compared to an untuned PSS, where eigenvalues are closer to the imaginary axis with smaller damping ratios, the position of Mode 16 reflects reliable system stability and balanced oscillation frequency.

The action of the stabilizer suppresses mode 16 oscillations to maintain sustainable system operation, with no nearby unstable eigenvalues confirming its controllability. However, the observability of Mode 16 depends on the measurement details; modes with large imaginary parts are easier to detect, while mode 16’s relatively small imaginary component may limit its visibility in system responses. Fig. 8 shows the observability of Mode 16.

Fig. 8. Tuned PSS - Mode 16 analysis (Observability).

Furthermore, the detection of this eigenvalue relative to other modes can also be affected by the distribution of the eigenvalues. For aggregated eigenvalues with characteristics similar to those of Mode 16, distinguishing the effect of Mode 16 in system measurements may not be feasible. From a stability perspective, a well damped Mode 16 shows that the mode responds to disturbances but does not persist in oscillation. As such, Mode 16 does not dominate the measurement information relative to the poorly damp modes. Owing to the placement of Mode 16 in the plot and an eigenvalue presence with a significant imaginary part, it can be inferred that Mode 16 has a strong oscillatory property.

Fig. 9 shows the different state variables with the highest participation in the activity when exciting this particular oscillation mode. This provides an indication of the generators that can be considered as candidates for PSS tuning.

Fig. 9. Tuned PSS - Mode 16 analysis (Participation).

Low-frequency oscillatory modes may exhibit larger participation factors in generators with large inertia or generators connected to weak transmission corridors. In this figure, any eigenvalues with imaginary parts of 5 rad/s to 15 rad/s and real parts closer to −4 to −6, are likely to have more active oscillation participants.

Fig. 10 shows poorly damped modes indicated by eigenvalues clustering near the imaginary axis, particularly with real parts less than -0.25, reflecting weakly damped oscillations that allow disturbances to persist and increase the instability risk.

Fig. 10. Untuned PSS - Eigenvalue modes plot.

The eigenvalue distribution and damping of the system with an untuned stabilizer implemented at the machine on bus 206 in Fig. 11 show the contributions of different variables to the activity of this mode.

Fig. 11. Untuned PSS - Mode 1 analysis (Controllability).

Fig. 12 shows the observability of Mode 1, which indicates the degree of activity of a state variable when a particular mode is excited.

Fig. 12. Untuned PSS - Mode 1 analysis (Observability).

The eigenvalue distribution shows that some modes, including Mode 1, lie near the imaginary axis with weak damping, which is slightly improved compared to the untuned PSS case. Untuned stabilizers fail to provide sufficient damping, causing mode 1 oscillations to persist longer and impact rotor speed deviations in critical regions. The detection of Mode 1 effects is challenging when higher-frequency oscillations dominate, and noise mask signals or sensors are improperly placed.

An untuned PSS does not improve the oscillation observability, limiting operators’ ability to act promptly. Mode 1 is characterized by high participation factors from specific generators and buses, particularly those with weak excitation control or located in poorly damped transmission corridors. These components exhibit significant involvement in oscillations; generators showing rotor speed deviations contribute the most to mode 1. Enhancing Mode 1 observability and damping requires improved PSS tuning, strategic sensor placement, and advanced signal processing.

Fig. 13 shows the different state variables with the highest participation in the activity when exciting this particular oscillation mode.

Fig. 13. Untuned PSS - Mode 1 analysis (Participation).

The unstable configuration produces longer persistent oscillations, implying that generators under weaker damping conditions have a larger effect on mode 1. Such generators tend to connect to transmission lines with low impedance or operate near the maximum capacity, which means that they may also be more sensitive to disturbances created by oscillatory disturbances.

Fig. 14 shows the output signal of the tuned PSS over time, which exhibits smooth and effective damping characteristics. The tuned PSS minimizes oscillations by regulating the generator excitation, resulting in a stable system response after disturbances. Initially, the oscillations show a high amplitude, representing the natural transient system response before full stabilizer engagement.

Fig. 14. Tuned PSS - Output signal.

Over time, oscillations decay steadily, indicating an optimal damping action by the PSS, which absorbs oscillatory energy and prevents prolonged fluctuations. The tuning achieves rapid stabilization, significantly reducing the oscillation duration and magnitude and enhancing the overall dynamic stability. Compared with the untuned case, where oscillations persist longer with irregular decay, the tuned PSS facilitates superior system damping and stability.

As shown in Fig. 15, the system dynamics with the tuned PSS reveal well-damped modes shifted towards the left half of the complex plane, indicating enhanced voltage stability. The real-time excitation control modification by the tuned PSS mitigates excessive voltage swings during disturbances, promotes faster voltage recovery, and prevents instability. Without tuning, the simulations exhibited weak damping and extended oscillatory events, risking generator desynchronization. The tuned PSS shifts the eigenvalues further with higher real magnitudes, thereby enhancing the damping performance. Consequently, the rotor speed, generator output, and bus frequency oscillations decayed more rapidly, restoring a stable dynamic response. The improved eigenvalue distribution and increased damping ratios quantitatively confirmed the effectiveness of the stabilizer in dissipating disturbances and stabilizing the system.

Fig. 15. Tuned PSS - Network results.

Fig. 16 shows the signal output for the untuned PSS with no damping effects, as indicated by a waveform with longer-lasting oscillations due to less-than-optimum tuning. In comparison to the tuned stabilizer, which strongly dampens oscillations, the untuned PSS allows for some degree of oscillation, increasing the instability of the power system.

Fig. 16. Untuned PSS - Output signal.

Fig. 17 presents the network dynamics of the untuned PSS. It is clear that oscillations persist for longer times, posing greater issues to the network. The voltage oscillations last longer than ideal. The voltage varies significantly across the network in the absence of damping mechanisms, which is problematic because it has serious stability implications and higher risks of staying beyond acceptable operating limits.

Fig. 17. Untuned PSS - Network results.

Conclusion

Traditionally, Power System Stabilizer (PSS) tuning is performed using trial and error techniques, linearized small-signal models, or classical control approaches such as root locus and Bode plot analysis. These methods rely on offline studies and predefined system parameters to adjust PSS gain, phase compensation, and washout filter constants. Although effective for static system conditions, these techniques lack robustness under varying operating scenarios.

The methodology adopted in this study utilizes the Particle Swarm Optimization (PSO) algorithm to dynamically tune the PSS parameters of the generators in a typical Saudi National Grid by minimizing an objective function that reflects the system damping and oscillation performance. PSO enables global search capabilities, faster convergence, and adaptability to nonlinear and multimodal system behaviors, which are critical under contingency and fault conditions. Compared with conventional tuning, the PSO-based method provides more consistent and optimal damping ratios across multiple modes, leading to improved eigenvalue placement and better dynamic stability. Additionally, it reduces engineering effort, enhances repeatability, and is well suited for large-scale power systems with complex dynamics.

Conflict of Interest

The authors declare that they do not have any conflict of interest.