Power system stabilizers tuning for probabilistic small-signal stability enhancement using particle swarm optimization and unscented transformation

Abstract

In modern power systems, uncertainties related to loads and renewable energy sources increase the need for reliable tools for steady-state and dynamic studies. These uncertainties are commonly modeled using probability density functions. In the context of the probabilistic small-signal stability analysis of power systems, the challenge lies in developing computationally efficient tools that accurately calculate the probability of ensuring security (minimum damping ratio greater than or equal to a desired value) and stability (negative spectral abscissa) requirements. This paper proposes an optimization approach for designing power system stabilizers to maximize the probability of meeting these security and stability requirements. The innovation of this approach is the integration of the unscented transformation (UT) with the particle swarm optimization (PSO). The UT is advantageous, as it requires a smaller number of samples to compute the mean and standard deviation of the output variables, especially compared to the Monte Carlo simulation (MCS), whereas PSO provides high-quality solutions. The New-England test system is employed in a case study to validate the proposed approach. This case study highlights the method’s accuracy and computational efficiency advantages, showcasing its potential to address the challenges posed by increasing uncertainties in modern power systems.

1 Introduction

Low-frequency oscillations (LFO) in power systems are mainly caused by imbalances in mechanical and electrical torques at synchronous generators following minor disturbances, such as variations in load or changes in controller reference values [1]. These oscillations have been a significant topic of study in the power systems literature since the 1970 s [2,3,4,5]. The predominant strategy for damping these oscillations involves the implementation of power oscillation dampers (POD). Among various PODs, the power system stabilizer (PSS) is the most recognized, installed at synchronous generators. The PSS modulates the field voltage during transient events, thereby helping dampen the LFO [1, 6, 7].

Small-signal stability assessment (SSSA) is essential for studying LFO and PSS tuning. This assessment involves linearizing the power system model at a specific operating condition. Subsequently, the angular stability of the system is evaluated through modal analysis, which is essentially based on the calculation of eigenvalues, as detailed in [7].

In contemporary power systems, uncertainties in loads and renewable energy generation are critical in the planning and operational stages. Traditional deterministic small-signal stability assessment (D-SSSA) does not account for these uncertainties, which can lead to imprecise results, as highlighted in [8, 9]. Consequently, probabilistic approaches to small-signal stability assessment (probabilistic SSSA, P-SSSA) have become increasingly relevant. P-SSSA specifically addresses the challenge of computing the mean and standard deviation of the output variables (minimum damping ratio and spectral abscissa) from the mean and standard deviation of the input variables (loads and powers generated from renewable sources). Recently, [10] emphasized the importance of considering uncertainties in scenarios with a high penetration of renewable energy sources. This consideration is crucial regardless of the technology employed for power oscillation damping [10].

As described in [8] [11], approaches for performing probabilistic analyzes in power systems can be classified into three main categories: (i) analytical, (ii) numerical and sampling-based, and (iii) approximate methods.

Numerical and sampling methods, including Monte Carlo Simulation (MCS), Latin Hypercube Sampling, and Quasi-MCS, play a crucial role in probabilistic analyses in power systems. MCS, in particular, relies on generating many samples for random input variables (like loads and generations), applying the nonlinear model to each sample to obtain output variables, and then calculating statistical variables (mean and standard deviation) for each output. Despite its accuracy, MCS is known for its high computational demand. For example, in [12] a quasi-MCS technique is proposed for P-SSSA, considering uncertainties related to electric vehicles and wind generation. Similarly, [13] introduces an approach to the probabilistic design of conventional PSS in synchronous generators, focusing on maximizing the probability of meeting the system security requirements under uncertainties in wind generation. This method utilizes a combination of Genetic Algorithm and MCS. A Latin hypercube sampling-based approach has been proposed in [14] for the probabilistic design of conventional PSS at synchronous generators and POD at wind generators. This method aims to maximize the number of stable scenarios and employs differential evolution for problem-solving. A cooperative coevolutionary algorithm and the MCS for the probabilistic design of conventional PSS is proposed in [15]. This approach considers uncertainties in wind powers and loads. A framework based on Monte Carlo simulation for stochastic eigenvalue analysis of electric power systems is proposed in [16]. This framework, which addresses systems with high penetration of inertialess renewable generation, focuses on the influential factors affecting eigenvalue movement due to reduced inertia. In [17], Monte Carlo simulation is used to calculate the probability of occurrence of unstable scenarios resulting from network failures. In [18], the MCS is used to propose a framework for defining multi-stability operational boundaries of power systems. This framework takes into account varying penetration levels of power electronic interfaced generation and the uncertainty in system loading. Finally, [19] employs MCS to evaluate probabilistic stability, considering controllers designed by classical methods.

Analytical approaches such as the cumulant method have been proposed to minimize the computational cost of numerical-based methods. These methods rely on linearizing equations and provide results with a lower computational burden. However, significant fluctuations in input random variables can lead to substantial linearization errors. In [20], the differential evolution and an analytical method based on Taylor series are combined for the probabilistic design of conventional PSS considering load uncertainties. Conventional PSS for synchronous generators and power oscillation damping (POD) at static voltage compensators are designed in [9], taking into account uncertainties in wind power using the cumulant method and the fruit fly optimization algorithm. The paper [21] employs four bio-inspired optimization algorithms (bat algorithm, cuckoo search, firefly algorithm, and particle swarm optimization) along with the cumulant method for probabilistic tuning of conventional PSS and POD in wind generators and solar panels. The probabilistic design of conventional PSS is executed using the cumulant method and the directional bat algorithm in [22], considering the uncertainties of power load and wind generation. In [23], the cumulant method and the firefly optimization algorithm are utilized to design conventional PSS for synchronous generators and POD in energy storage systems. In [24], a surrogate method using deep learning, the bat algorithm-based optimization, and the cumulant method are integrated for a probabilistic design of POD controllers of wind and photovoltaic generators. In [25], the probabilistic small-signal stability in systems with uncertain wind power is evaluated by series expansion and maximum entropy theory, providing excellent accuracy and sufficient efficiency.

A balance between accuracy and efficiency in system analysis can be achieved by employing approximate methods like the unscented transformation (UT) [26] and two-point estimate method (2PEM) [27]. These methods determine a reduced number of samples in a deterministic way. The 2PEM is used in [28] to assess the power system small-signal stability analysis (P-SSSA). In [29], the impact of wind generation uncertainty in systems equipped with conventional PSS—not subject to tuning—is evaluated using the 2PEM. [30] discusses a probabilistic design of conventional PSS for synchronous generators and POD for wind generators and static synchronous compensators (STATCOM), utilizing the modified imperialist competitive algorithm and the 2PEM. In [31], the gradient optimization method and the 2PEM are utilized to optimize the virtual inertia and pitch angle parameters of wind turbines.

Finally, in [32], it is proposed an optimal procedure to design a robust PID-PSS using interval arithmetic for the single-machine infinite bus system. Interval methods, while helpful in ensuring safety under uncertainty, have notable disadvantages compared to probabilistic methods. They do not provide probability distributions within established ranges, leading to overly conservative solutions that may be less efficient and more costly. In addition, interval methods suffer from high computational complexity in large-scale problems, as they consider all possible values within limits.

Table 1 summarizes the publications mentioned above. This review highlights that the area of probabilistic design for power system stabilizers and power oscillation damping controllers, particularly employing bio-inspired optimization techniques together with the unscented transformation, remains unexplored. The UT is an approximate technique based on a reduced set of samples called sigma points, which are deterministically calculated.

Table 1 Publications on recent P-SSSA

1.1 Main contributions

This paper introduces a novel optimization approach for enhancing the probabilistic design of conventional power system stabilizers (PSS) in transmission systems. Leveraging particle swarm optimization (PSO) in tandem with unscented transformation (UT), the method aims to maximize the probabilities of meeting predefined thresholds for both the minimum damping ratio (small-signal security) and the spectral abscissa (small-signal stability). A key innovation lies in the utilization of UT to evaluate each individual of the PSO within a probabilistic small-signal stability assessment (P-SSSA), which constitutes the primary contribution of this research.

To capture the uncertainties inherent in loads and wind power generation, the model uses a normal distribution. The efficacy of the approach is demonstrated through a comprehensive case study using the New-England test system [33]. Furthermore, validation is conducted via nonlinear time-domain simulations.

Of particular significance, the proposed probabilistic tuning method exhibits superior performance compared to deterministic approaches, especially when subjected to scenarios unforeseen during the design phase. This underscores the robustness and practical applicability of the methodology in real-world transmission systems.

2 Background: probabilistic small-signal stability assessment

2.1 Deterministic small-signal stability

The small-signal stability analysis of transmission systems employs linearized models considering specific operational conditions. These conditions are provided by the power flow solution corresponding to a given load and generation profiles. The linearized models are represented through state-space formulations, both in open-loop and closed-loop configurations.

Figure 1 illustrates the static excitation system of a synchronous generator. This system’s primary function is to ensure that the terminal voltage, denoted as $\Delta V_{T}$, remains consistent with the reference value, $\Delta V_{REF}$, achieved by controlling the field voltage $\Delta E_{FD}$. In this static excitation system model, the parameters $K_A$ and $T_A$ correspond to the gain and the time constant, respectively [7]. Additionally, in Fig. 1, the transfer function of the power system stabilizer, PSS(s), is distinctly highlighted in red. This controller interacts with the excitation system during transient events, introducing a supplementary signal, $\Delta V_{PSS}$, to ensure the power system’s stability [2]. Notably, in this configuration, the stabilizer’s input signal is derived from the synchronous generator terminal velocity $\Delta \omega$, though alternative input types could also be adopted [7].

Fig. 1

Static excitation system

From Fig. 1, two distinct operational modes can be identified: (i) open-loop operation (without the PSS) and (ii) closed-loop operation (incorporating the PSS). The state-space representation corresponding to the open-loop operation is delineated in (1). For completeness, the differential equations associated with the generator and automatic voltage regulator models are presented in Appendix A.

$$\begin{aligned} \begin{aligned} \Delta {\dot{x}}=A_O \Delta x+B_O \Delta u \\ \Delta y=C_O \Delta x \end{aligned} \end{aligned}$$

(1)

where

• $A_O$, $B_O$, and $C_O$ are the state-space, input, and output matrices, respectively;

• $\Delta x$ represents the vector of states, encompassing angular speeds, internal angles, internal voltages, and field voltages. Similarly, $\Delta u$ denotes the vector containing input values, specifically $\Delta V_{REF}$, while $\Delta y$ signifies the vector associated with output variables, in this case, $\Delta \omega$.

The state-space model for closed-loop operation, denoted by the subscript C in (2), is derived from (1) using a feedback procedure that takes into account the transfer function of the PSS, as defined in (3) [7].

$$\begin{aligned}{} & {} \begin{aligned} \Delta {\dot{x}}=A_C \Delta x+B_C \Delta u \\ \Delta y=C_C \Delta x \end{aligned} \end{aligned}$$

(2)

$$\begin{aligned}{} & {} P S S_p(s)=K_p \frac{s T_w}{1+s T_w}\left( \frac{1+s \frac{\sqrt{\alpha _p}}{\omega _p}}{1 +s\frac{1}{\omega _p \sqrt{\alpha _p}}}\right) ^{n b} \end{aligned}$$

(3)

where (for the $p^{th}$ generator)

• $P S S_p(s)$ represents the transfer function of the PSS;

• $T_w$ denotes the time constant of the washout filter, ensuring that the PSS operates exclusively during transients;

• $K_p$ is the gain associated with the PSS;

• $\alpha _p$ and $\omega _p$ specify the parameters for the lead-lag stage of the PSS, which serves as the phase compensation stage;

• nb indicates the total number of lead-lag stages.

Each PSS comprises five parameters that require tuning: $K_p$, $\alpha _p$, $\omega _p$, $T_w$, and nb. Typically, $T_w$ and nb are predefined, leaving only the gain and the two lead-lag parameters to be fine-tuned. Consequently, for a system with ng generators equipped with PSS, a total of $3 \times ng$ parameters require design adjustments [34,35,36].

Once the state-space representation is obtained, it is possible to evaluate the small-signal stability through the modal analysis. It is done by calculating the m eigenvalues for a state-space matrix (order $m \times m$). Equation (4) brings the $i^{th}$ eigenvalue (with real $\sigma _i$ and imaginary $\omega _i$ components), whose damping ratio $\xi _i$ is calculated according to (5). The system is stable when all $\sigma _i$ are negative and $\xi _i$ are positive [7].

$$\begin{aligned} \lambda _i= & {} \sigma _i \pm j \omega _i \end{aligned}$$

(4)

$$\begin{aligned} \xi _i= & {} \frac{-\sigma _i}{\sqrt{\sigma _i^2+\omega _i^2}} \end{aligned}$$

(5)

In the context of deterministic small-signal analysis, establishing the minimum damping ratio $\xi _{\min }$ and the spectral abscissa $\sigma _{\max }$ is essential. It is made according to (6). In power system operation, we say that the power system is secure when its minimum damping ratio $\xi _{\min }$ is greater than or equal to a security level (10%, for instance) [37].

$$\begin{aligned} \begin{aligned} \sigma _{\max }&=\max \left( 1, \ldots , \sigma _i, \ldots , \sigma _m\right) \\ \xi _{\min }&=\min \left( 1, \ldots , \xi _i, \ldots , \xi _m\right) \end{aligned} \end{aligned}$$

(6)

In the context of power system analysis, we refer to deterministic small-signal analysis when all loads and generation from renewable sources are precisely known. However, a more comprehensive probabilistic analysis becomes essential when uncertainties are incorporated. This will be the focus of the subsequent subsection.

2.2 Probabilistic model of loads and renewable energy

In the present study, we perform a probabilistic analysis, accounting for uncertainties both in active and reactive power loads as well as in the generation from renewable sources, specifically wind generation. These uncertainties are modeled employing the Normal Distribution as depicted in (7) [15, 22, 23, 38].

$$\begin{aligned} f\left( P_{d k}\right) =&\,\frac{1}{\sigma _{P_{d k}} \sqrt{2 \pi }} e^{-\frac{\left( P_{d k}-\mu _{P_{d k}}\right) ^2}{2 \sigma _{P_{d k}}^2}} \nonumber \\ f\left( Q_{d k}\right) =&\,\frac{1}{\sigma _{Q_{d k}} \sqrt{2 \pi }} e^{-\frac{\left( Q_{d k}-\mu _{Q_{d k}}\right) ^2}{2 \sigma _{Q_{d k}}^2}} \nonumber \\ f\left( P_{g k}^{\text{ wind } }\right) =&\,\frac{1}{\sigma _{P_{g k}^{\text{ wind } }} \sqrt{2 \pi }} e^{-\frac{\left( P_{g k}^{\text{ wind } }-\mu _{P_{g k}^{\text{ wind } }}\right) ^2}{2 \sigma _{P_{g k}^{\text{ wind } }}^2}} \end{aligned}$$

(7)

where

• $\mu _{P_{d k}}$, $\mu _{Q_{d k}}$, and $\mu _{P_{g k}^{\text{ wind } }}$ represent the mean values of loads and wind generation at a node k;

• $\sigma _{P_{d k}}$, $\sigma _{Q_{d k}}$, and $\sigma _{P_{g k}^{\text{ wind } }}$ represent the standard deviations of loads and wind generation at a node k.

2.3 Procedure for probabilistic small-signal assessment using the Monte Carlo simulation

The probabilistic evaluation of the small-signal stability consists of calculating the mean ($\mu _{\sigma _{\max }}$ and $\mu _{\xi _{\min }}$) and standard deviation ($\sigma _{\sigma _{\max }}$ and $\sigma _{\xi _{\min }}$) values of the spectral abscissa and the minimum damping ratio, taking into account the uncertainties on loads and generation of renewable sources (see Sect. 2.2).

The Monte Carlo simulation is the predominant method for probabilistic analysis, utilizing a suite of random samples in its numerical approach. Figure 2 provides an illustrative example, showcasing a sample vector for a system comprising three buses and one wind generator.

Fig. 2

Illustrative example of a sample

Figure 3 outlines the procedure to evaluate probabilistic small-signal stability using the Monte Carlo simulation. In Step 1, the necessary number of samples, power system data (including nodes and branches), and stabilizer parameters are imported. Moving to Step 2, a sample, as exemplified in Fig. 2, is generated following Eq. (7). For Step 3, the state-space model in open-loop operation is established by solving the power flow and then linearizing the nonlinear system [7, 36]. In Step 4, a feedback procedure calculates the closed-loop state-space model, incorporating all the power system stabilizers. Step 5 then computes the minimum damping ratio and the spectral abscissa based on (6). After storing the values of $\sigma _{\max }$ and $\xi _{\min }$ in Step 6, Step 7 verifies if the desired number of samples has been achieved. If achieved, Step 8 calculates the mean and standard deviations, concluding the process in Step 9. If not, the procedure returns to Step 2 to generate another sample. For the sake of completeness, an illustrative example is presented in Appendix D.

Fig. 3

Flowchart—Monte Carlo Simulation

2.4 Procedure for probabilistic small-signal assessment using the unscented transformation

The unscented transformation method employs a reduced set of samples deterministically calculated so that the computational burden is significantly lower compared with the Monte Carlo simulation. Each sample is referred to as a sigma point $\varvec{\chi _{i}}$ with the same structure shown in Fig. 2. Figure 4 brings the steps required to assess the small-signal stability through the unscented transformation. Step 1 and steps 4 to 7 are similar to step 1 and steps 3 to 6 of Fig. 3, respectively.

Fig. 4

Flowchart—unscented transform based small-signal analysis

Firstly, the vector of means $z_m$ and the matrix with variances $P_z$ are defined according to Eq. (8) in Step 1. Considering nb load nodes and nren nodes with wind generators, the dimensions of $z_m$ and $P_z$ are ($(2nb + nren) \times 1$) and ($(2nb + nren) \times (2nb + nren)$), respectively.

$$\begin{aligned} \begin{aligned} z_m=\left[ \begin{array}{c} \mu _{P_{d k}} \\ \mu _{Q_{d k}} \\ \mu _{P_{g i}^{\text{ wind } }} \end{array}\right] \\ P_z=\left[ \begin{array}{ccc} \sigma _{P_{d k}}^2 &{} 0 &{} 0 \\ 0 &{} \sigma _{Q_{d k}}^2 &{} 0 \\ 0 &{} 0 &{} \sigma _{P_{g i}^{\text{ wind } }}^2 \end{array}\right] \end{aligned} \end{aligned}$$

(8)

where $k = 1, \cdots , nb$ and $i = 1, \cdots , nren$.

Being n uncertain variables (in this paper $n = 2nb + nren$), in Step 2 $2n+1$, sigma points are calculated. Each sigma point $\chi _{i}$ is a column vector representing a sample whose structure follows the one shown in Fig. 2. Different from the Monte Carlo simulation, sigmas points are deterministically calculated according to Eqs. (9)–(11).

$$\begin{aligned}{} & {} \chi _{1}= z_{m} \end{aligned}$$

(9)

$$\begin{aligned}{} & {} \chi _{i+1}=z_{m}+{u}_{i}, \quad i=1, \ldots , n \end{aligned}$$

(10)

$$\begin{aligned}{} & {} \chi _{i+1}=z_{m}-{u}_{i}, \quad i=1, \ldots , n \end{aligned}$$

(11)

where ${u}_{i}$ is a row vector of the matrix U, computed using the Cholesky decomposition, as presented in Eq. (12) [26].

$$\begin{aligned} {U^{T}} {U}=(n+\kappa ) {P}_{{z}} \end{aligned}$$

(12)

In (12), $\kappa$ is a parameter empirically tuned so that the results (means and variances of output variables) are close to those provided by the Monte Carlo simulation [39, 40]. In this paper, $\kappa = 0$ is considered.

For each sigma point, it is performed a deterministic small-signal analysis (Steps 4 to 7 in Fig. 4) to obtain the minimum damping ratio and the spectral abscissa based on (6): $\sigma _{\max i}$ and $\xi _{\min i}$ (here, i denotes the $i^{th}$ sigma point).

Once all sigma points are evaluated, the means ($\mu _{\sigma _{\max }}$ and $\mu _{\xi _{\min }}$) and standard deviations ($\sigma _{\sigma _{\max }}$ and $\sigma _{\xi _{\min }}$) of output variables are estimated in Step 9 by (13)-(14). For this purpose, weight factors $W_{i}$ are required.

$$\begin{aligned} \begin{aligned}&\mu _{\sigma _{\max }}=\sum _{i=1}^{2 n+1} W_i \cdot \sigma _{\max i} \\&\mu _{\xi _{\min }}=\sum _{i=1}^{2 n+1} W_i \cdot \xi _{\min i} \end{aligned} \end{aligned}$$

(13)

$$\begin{aligned} \begin{aligned} \sigma _{\sigma _{\max }}&=\sqrt{\sum _{i=1}^{2 n+1} W_i \cdot \left( \mu _{\sigma _{\max }}-\sigma _{\max i}\right) ^2} \\ \sigma _{\xi _{\min }}&=\sqrt{\sum _{i=1}^{2 n+1} W_i \cdot \left( \mu _{\xi _{\min }}-\xi _{\min i}\right) ^2} \end{aligned} \end{aligned}$$

(14)

The weight factors $W_{i}$ are obtained by (15)-(17). It is important to say that the sum of all weights must be equal to unity (see (18)).

$$\begin{aligned} W_{1}= & {} \frac{\kappa }{n+\kappa } \end{aligned}$$

(15)

$$\begin{aligned} W_{i+1}= & {} (2(n+\kappa ))^{-1}, i=1, \ldots , n \end{aligned}$$

(16)

$$\begin{aligned} W_{i+n+1}= & {} (2(n+\kappa ))^{-1}, i=1, \ldots , n \end{aligned}$$

(17)

$$\begin{aligned} \sum _{i=1}^{2 n+1} W_{i}= & {} 1 \end{aligned}$$

(18)

2.5 Confidence levels

According to [37], a system is considered to be secure from the angular small-signal point of view if its minimum damping ratio $\xi _{\text{ min } }$ is greater than or equal to a desired security margin $\xi _{d}$. In the context of probabilistic small signal, it is essential to define the security using probabilities as given in (19):

$$\begin{aligned} \begin{aligned}&P_r\left\{ \xi _{\text{ min } } \ge \xi _d\right\} \ge \gamma _{\xi _{\text{ min } }} \\&P_r\left\{ \sigma _{\text{ max } } \le 0\right\} \ge \gamma _{\sigma _{\text{ max } }} \end{aligned} \end{aligned}$$

(19)

where $\gamma _{\xi _{\text{ min } }}$ and $\gamma _{\sigma _{\text{ max } }}$ are the confidence levels. It is important to say that probabilities are calculated by using the values of means and standard deviations in (13)- (14).

3 Proposed approach for PSS tuning

3.1 Optimization approach

The main contribution of this paper is an optimization approach for the probabilistic design of power system stabilizers. The optimization uses the particle swarm optimization [41], and the unscented transformation performs the probabilistic analysis.

The mathematical formulation of the proposed approach is presented in (20)-(25). The objective function in (20) aims to maximize the probabilities of satisfying the minimum damping ratio requirement ($\xi _{\min } \ge \xi _d$) and the stability (given by the spectral abscissa $\sigma _{\max }$). Equation (21) ensures that the mean value of the minimum damping ratio $\mu _{\xi _{\text{ min } }}$ is greater than or equal to the security margin ($\xi _d$). Additionally, (22) ensures that the mean value of the spectral abscissa $\mu _{\sigma _{\text{ max } }}$ is negative (condition to stability). Finally, the parameters to be tuned ($K_p$, $\alpha _p$, and $\omega _p$) must be into the limits, as constrained in (23)-(25).

$$\begin{aligned} \max f(x)=&P_r\left\{ \xi _{\text{ min } } \ge \xi _d\right\} +P_r\left\{ \sigma _{\text{ max } }<0\right\} \end{aligned}$$

(20)

$$\begin{aligned} \text{ s.t. }&\mu _{\xi _{\text{ min } }} \ge \xi _d \end{aligned}$$

(21)

$$\begin{aligned}&\mu _{\sigma _{\text{ max } }}<0 \end{aligned}$$

(22)

$$\begin{aligned}&K_p^{\text{ min } } \le K_p \le K_p^{\text{ max } } \end{aligned}$$

(23)

$$\begin{aligned}&\alpha _p^{\text{ min } } \le \alpha _p \le \alpha _p^{\text{ max } } \end{aligned}$$

(24)

$$\begin{aligned}&\omega _p^{\text{ min } } \le \omega _p \le \omega _p^{\text{ max } } \end{aligned}$$

(25)

where $p=1, \ldots , npss$, being npss the number of stabilizers to be tuned. Each PSS has the transfer function defined in (3).

3.2 Fitness function calculation

Solving the optimization problem in (20)-(25) through an analytical approach is hard, and this paper uses particle swarm optimization for this task. It is a population-based method in which each individual represents a solution for the problem. In this section, the process to calculate the fitness function of each individual is presented.

In this paper, an individual $ind_{i}$ is a vector with the values of gains and parameters of the compensation stage (a possible solution) as defined in (26). The quality of the tuning provided by this individual is measured by its fitness function $fit_{i}$, calculated according to (27).

The fitness function $fit_{i}$ in (27) is composed of three parts: $F_1$, $F_2$, and $F_3$. The first component ($F_1$) is defined in (28) and deals with the objective function in (20). The second and third components ($F_2$ and $F_3$ defined in (29)-(30)) are necessary to deal with the constraints (21) -(22). They are applied penalizations in case of violations ($\xi _{\min }<\xi _d$ or $\sigma _{\max }>0$). Each component has its weight $\beta$, empirically defined.

$$\begin{aligned} {ind}_i=\left[ \begin{array}{lllllllll} K_1&\ldots&K_{n p s s}&\alpha _1&\ldots&\alpha _{n p s s}&\omega _1&\ldots&\omega _{n p s s} \end{array}\right] \end{aligned}$$

(26)

$$\begin{aligned} fit_{i} =\beta _1 \cdot F_1-\beta _2 \cdot F_2-\beta _3 \cdot F_3 \end{aligned}$$

(27)

where $\beta _1=1$ and $\beta _2=\beta _3=10^4$.

$$\begin{aligned} F_1=P_r\left\{ \xi _{\text{ min } } \ge \xi _d\right\} +P_r\left\{ \sigma _{\text{ max } }<0\right\} \end{aligned}$$

(28)

$$\begin{aligned} F_2= \left\{ \begin{array}{cc} \mid penal_{1} \mid &{} \text{ if } \mu _{\xi _{\text{ min } }}<\xi _d \\ 0 &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

(29)

where $penal_{1} = \xi _d - \mu _{\xi _{\text{ min } }}$.

$$\begin{aligned} F_3=\left\{ \begin{array}{cc} \mid penal_{2} \mid &{} \text{ if } \mu _{\sigma _{\max }} \ge 0 \\ 0 &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

(30)

where $penal_{2} = \mu _{\sigma _{\max }}$.

It is important to say that constraints (23)-(25) are directly handled by the PSO method for constrained optimization.

Finally, Fig. 5 brings an illustrative example regarding the Fitness Function $fit_i$ evaluation for a given individual $ind_i$. The first step is to perform a probabilistic small-signal stability analysis through the unscented transformation to calculate the means and standard deviations of the minimum damping ratio ($\mu _{\xi _{\min }} \text{ and } \sigma _{\xi _{\min }}$) and the spectral abscissa ($\mu _{\sigma _{\max }} \text{ and } \sigma _{\sigma _{\max }}$), as discussed in Sect. 2.4 and Fig. 4. These values are used to calculate the probabilities in (20) and (28) [42]. Finally, the fitness function $fit_{i}$ is calculated as defined in (27)-(30).

Fig. 5

Fitness function evaluation

3.3 Optimization methodology flowchart

As discussed in Sect. 3.2, the proposed approach in (20)-(25) is solved by the particle swarm optimization method. In addition, the calculation of the fitness function is illustrated in Fig. 5. Figure 6 illustrates the integration of the PSO method and the unscented transformation for the probabilistic analysis of small-signal stability.

The optimization method unfolds in a systematic manner. It commences in Step 1 with the definition of variable limits and the PSO and UT parameters. Step 2 initializes the iteration (or generation) counter, followed by the population initialization in Step 3. The population is randomly initiated, taking into account the constraints (23)-(25). Step 4 initializes the individual counter, and for each individual ${ind}_i$, a fitness function $fit_{i}$ is calculated, as explained in 5 to 7.

The first procedure to calculate the fitness function is, in Step 5, to execute the algorithm for the probabilistic analysis of the stability of the small signal discussed in Sect. 2.4 and Fig. 4. In this step, the uncertainties of wind and load generation are included in the analysis. Once the probabilistic analysis is complete, the probabilities of security ($P_r\left\{ \xi _{\text{ min } } \ge \xi _d\right\}$) and stability ($P_r\left\{ \sigma _{\text{ max } }<0\right\}$) are calculated in Step 6. The fitness function is calculated in Step 7 as discussed in Sect. 3.2.

In Step 8, it is checked if all individuals were evaluated. If not, the individual counter is increased in Step 9 and Steps 5 to 7 are carried out until all individuals have been evaluated.

Once all individuals are evaluated for a given generation/iteration t, the tuning process is finished in Step 13. On the other hand, the algorithm follows to Step 11 to update the iteration counter and Step 12 to update the population of individuals according to the PSO method discussed in [41].

Fig. 6

Flowchart—optimization appraoch

4 Case study

4.1 System description

The New-England test system, depicted in Fig. 7, will be used to discuss the results derived from the proposed approach. Comprising 39 nodes and ten generators, the detailed data for this system can be accessed in [33, 36, 43]. In the base case, this system has a total load of 6097.1 MW and 1408.9 MVAr, with active power generation of 6140.8 MW. In this specific case study, a wind generator with a dispatch capacity of 920 MW is placed at node 14, representing approximately 15% of the active power generation of the base case [44]. The unity power factor is adopted for the wind generator.

For the power system stabilizer (PSS) tuning process, both loads and wind generation are assumed to follow a normal distribution. The same assumptions have been made in [15, 22, 23, 38]. The mean values are set equal to the base case values, and the standard deviations are established at 5% of these mean values. In particular, the generator located at node 39 represents an equivalent system and is therefore excluded from receiving a PSS.

Fig. 7

New-England test system

4.2 Parameters of the optimization methods

In the specialized literature, it is known that parameterization of a metaheuristic is essential for securing high-quality solutions. Given that this parameterization is specific to a problem being addressed, the process is inherently empirical, often necessitating significant time investment from the user. In this study, the authors relied on the standard parameters delineated in the literature for PSO. To determine the optimal population size and the number of generations, we set a benchmark of 500 executions of the fitness function. Based on this, the following parameters were established:

• PSO: 25 particles, 20 iterations ($25 \times 20 = 500$), $c_1=c_2=2$ (acceleration constants), $\omega _{\max }=0.9$, $\omega _{\min }=0.4$ (inertia constants).

4.3 Open-loop eigenvalues in the base case

Deterministic small-signal analysis (see Sect. 2.1), considering the base case condition, reveals an unstable operation with a damping ratio of $-$6.0421% ($\xi _{min}= -6.0421\%$) and a spectral abscissa of +0.2045 ($\sigma _{max} = 0.2045$). Table 2 gives all electromechanical eigenvalues for the base condition that are plotted in Fig. 8.

Table 2 Open-loop electromechanical eigenvalues for the base case scenario

Fig. 8

Open-Loop eigenvalues for the base case scenario

Table 3 Operating conditions considered

The necessary action involves either a deterministic or probabilistic design of the power system stabilizers. The deterministic tuning is extensively discussed in the literature [34, 35]. Studies [9, 21, 45] indicate that the deterministic design may not be appropriate in the presence of uncertainties in power loads and renewable generations. In such instances, the probabilistic tuning presented in this paper is particularly compelling, given the low computational burden required by the unscented transformation.

In this paper, the design of power system stabilizers, for the base case, is carried out using two approaches: (i) the proposed probabilistic approach and the (ii) the deterministic one presented in Appendix B. A target damping ratio of 10% ($\xi _d = 10\%$) is considered.

It is important to emphasize that the deterministic approach aims at positioning the dominant eigenvalue (that with $\xi _{min}$) in the complex plane so that $\xi _{min}$ is close to $\xi _d$ in the base case. In this case, the deterministic method does not consider the probabilities of security and stability in the design stage.

Table 14, in Appendix C, presents the limits utilized during the design process. Both load and renewable generation are modeled with a mean equal to their nominal values and a standard deviation of 5% of these mean values.

4.4 Deterministic validation of the results

Tables 15 and 16, in Appendix C, displays the optimized parameters obtained using the probabilistic and the deterministic approaches, respectively. It is important to note that these designs must be validated considering other operating conditions to assess the robustness of the controller, as discussed in [35]. Table 3 presents the operating conditions used for tuning (the base case) and the other conditions used for the assessment stage.

Table 4 presents the dominant eigenvalue and the minimum damping ratio for each operating condition in Table 3. These values are obtained from a simple deterministic evaluation of the small-signal stability (when uncertainties are not considered in the analysis as discussed in Sect. 2.1). However, the PSS’s parameters are provided by the proposed probabilistic design. It is essential to notice that in both scenarios 01 (base case used in the tuning stage) and 02 and 03 (used for assessment), the system is stable and secure (with $\xi _{min} > \xi _d$). It shows the robustness of the proposed approach in ensuring security in scenarios not considered in the tuning stage.

However, the deterministic design, whose results are presented in Table 5, could not provide a secure condition (with $\xi _{min} < \xi _d$) in Scenario 02 (associated with the reduction of load and wind generation).

Table 4 Deterministic validation of the probabilistic design $^*$

Table 5 Deterministic validation of the deterministic design $^*$

To ensure completeness, the eigenvalues of the closed-loop operation are presented for each scenario in Table 3, as shown in Figs. 9 through 11. In these figures, the red crosses indicate the closed-loop eigenvalues considering the PSS tuned by the deterministic design, while the blue crosses are associated with the probabilistic tuning. A line in the complex plane indicates eigenvalues with a damping ratio exceeding 10%. Both probabilistic and deterministic controllers were analyzed for each scenario. Figure 10 correlates with the results in Table 5, revealing a complex eigenvalue with a damping ratio below the safe threshold of 10% (specifically, 7.8497%) when employing deterministic controllers.

Fig. 9

Closed-loop eigenvalues for the base case scenario: probabilistic (blue) and deterministic (red) (color figure online)

Fig. 10

Closed-loop eigenvalues for the second scenario: probabilistic (blue) and deterministic (red) (color figure online)

Fig. 11

Closed-loop eigenvalues for the third scenario: probabilistic (blue) and deterministic (red) (color figure online)

4.5 Probabilistic validation of the results

The obtained results using the deterministic and the probabilistic approaches were previously validated from the deterministic point of view. This validation shows that the probabilistic approach was able to provide a secure operation for scenarios that have not been considered in the design stage. To assess the controller’s robustness in an uncertain environment, the probabilistic small-signal analysis will be carried out using the unscented transformation (flowchart in Fig. 4 in Sect. 2.4). For this purpose, the following strategy is presented:

• 1 the three scenarios from Table 3 are considered;

• 2 the mean values of loads and wind generations are set equal to the updated powers in the item (1);

• 3 the standard deviations are established at 2.5%, 5% and 10% of these mean values;

• 4 it should be emphasized that only case 01 (base case) was considered in the design stage. In addition, for the probabilistic design, only one uncertainty level (standard deviation of 5%) was employed in the design stage.

Table 6 Impact of uncertainty on spectral abscissa (Scenario 01: base case)

Table 7 Impact of uncertainty on spectral abscissa (Scenario 02: load and wind generation reduction)

Table 8 Impact of uncertainty on spectral abscissa (Scenario 03: load increase)

Table 9 Impact of uncertainty on damping ratio (Scenario 01: base case)

Table 10 Impact of uncertainty on damping ratio (Scenario 02: load and wind generation reduction)

The following points summarize the observations in the probabilistic validation:

• the controller’s parameters designed by both design approaches (probabilistic and deterministic) allowed a stable operation no matter the operation condition and uncertainty level of input variables (loads and wind generation). It can be seen in Tables 6, 7, and 8 that the probability of stability ($P_r\left\{ \sigma _{\text{ max } }<0\right\}$) achieved the maximum value (100%);

• regarding the probability of security ($P_r\left\{ \xi _{\text{ min } } \ge \xi _d\right\}$), as one can see in Table 9, the proposed approach (probabilistic design), provided probabilities of 100% in the base case (scenario 01) for uncertainties levels of 2.5% and 5%. It is important to note that the likelihood of achieving the desired minimum damping ratio of 10% diminishes to 84.9013% when the uncertainty level is heightened. Figure 12 provides a comparative view of the (probability density functions) PDFs at varying levels of uncertainty when the parameters provided by the probabilistic approach are considered. However, for the deterministic design, the security probabilities are lower than those obtained in the probabilistic design, emphasizing the importance of the proposed approach in this paper;

• from Table 10, it is possible to see that under load and wind power generation reduction, the probability of security is nearly zero when the parameters provided by the deterministic approach are employed. Although the probability $P_r\left\{ \xi _{\text{ min } } \ge \xi _d\right\}$ is also reduced when the parameters from the proposed probabilistic approach are used, a safe level is obtained. Finally, as given in Table 11, under load increase, the PSSs adjusted by using a probabilistic approach also provide greater levels of security.

Table 11 Impact of uncertainty on damping ratio (Scenario 03: load increase)

Fig. 12

Comparison of PDF of minimum damping ratio for different levels of uncertainty (scenario 01 using the PSS’s parameters provided by the proposed approach)

4.6 Validation considering the Monte Carlo Simulation

As detailed in Sect. 2.4, the total number of uncertain variables is $n = 2nb + nren = 2 \times 39 + 1 = 79$. In this context, the unscented transformation utilizes a set of $2n+1 = 159$ sigma points, which are deterministically calculated samples. Consequently, 159 deterministic modal analyses are performed for each probabilistic small-signal analysis. This number is significantly lower than that typically used in Monte Carlo simulation, which ranges from $10^3$ to $10^4$. This reduction is notable, especially considering the number of individuals and generations in the particle swarm optimization (PSO) process, which involves 25 particles across 20 iterations ($25 \times 20 = 500$). Thus, the total number of deterministic modal analyses in the proposed approach is $159 \times 500$.

Considering the base case (scenario 01) and the PSS parameters given in Table 15, the mean and standard deviation of the damping ratio and spectral abscissa obtained from the unscented transformation are compared with those from the Monte Carlo simulation (MCS) in Tables 12 and 13. Despite its substantial computational demand, MCS is a benchmark method. Two distinct MCSs were conducted, one with $10^3$ samples and another with $10^4$ samples. The findings demonstrate that the proposed methodology, which relies on the unscented transformation using only 159 samples, yields results consistent with those from MCS. The effectiveness of the unscented transformation (UT) is further evidenced in Fig. 13, which displays a close match between the probability density functions of UT and MCS. The same conclusion can be drawn from Table 13 that brings the spectral abscissa.

Fig. 13

Comparison of PDF of minimum damping ratio

Table 12 Probabilistic analysis of damping ratio (scenario 01—base case)

Table 13 Probabilistic analysis of spectral abscissa (scenario 01—base case)

4.7 Nonlinear time-domain simulation considering the base case

This paper addresses the issue of small-signal stability. However, in agreement with the findings reported in the literature [7], it is also essential to perform a transient stability analysis. Typically, transient stability analysis is executed using nonlinear time-domain simulations. In this section, a simulation of a three-phase fault is performed, considering the base case and optimized controllers (see Tables 15 and 16). A short circuit is applied on bus 11 for 50 ms, and it is cleared by opening the transmission line 10–11 after the same duration, followed by a reclosure after another 50 ms.

Figure 14 illustrates the behavior of the terminal speed for generator 32, in the base case, since it is close to the faulted line. It is calculated with respect to generator 39 ($\Delta \omega _{32-39}$). It is possible to see that the probabilistic tuning provided a slightly better solution (damped oscillation).

Fig. 14

Terminal speed—generator 32

5 Conclusion

The proposed approach for the probabilistic design of conventional power system stabilizers, combining particle swarm optimization and unscented transformation, yielded promising results. First, it was observed that the unscented transformation requires fewer samples to evaluate probabilistic small-signal stability compared to the Monte Carlo simulation. This efficiency highlights the UT’s effectiveness. Second, when comparing the results of UT and MCS, including mean values, standard deviation, probabilities of satisfying security (minimum damping ratio) and stability (spectral abscissa) requirements, and probability density functions, a good agreement was found between both methods. This indicates the reliability of UT in probabilistic assessments. In addition, the probabilistic proposed approach provided better robustness than the deterministic approach, mainly for operating conditions not considered in the design stage. Since the probability of meeting the security requirement might be less than 100% for uncertainty levels not considered in the design, future works will be focused on considering a multi-scenario probabilistic approach. Lastly, nonlinear time-domain simulations confirmed the feasibility of the results in terms of angular transient stability, validating the practical applicability of the proposed approach.

Appendices

Appendix A

This appendix presents the generator model employed in this paper. It is a third-order model comprising Eqs. (31)–(33) (for a given generator k). Equation (34) describes the first-order model of the automatic voltage regulator in Fig. 1 [7].

$$\begin{aligned} \dot{\Delta \delta }_k= & {} \omega _s \cdot \Delta \omega _{k} \end{aligned}$$

(31)

$$\begin{aligned} \Delta {\dot{\omega }}_{k}= & {} \frac{\Delta P_{m k}}{2 H_k}-\frac{\Delta P_{g k}}{2 H_k}-\frac{D_k}{2 H_k} \cdot \Delta \omega _{k} \end{aligned}$$

(32)

$$\begin{aligned} \Delta {\dot{E}}_{q k}^{\prime }= & {} -\frac{\Delta E_{q k}^{\prime }}{T_{d 0 k}^{\prime }}-\frac{\left( X_{d k}-X_{d k}^{\prime }\right) }{T_{d 0 k}^{\prime }} \cdot \Delta I_{d k}+\frac{\Delta E_{F D k}}{T_{d 0 k}^{\prime }} \end{aligned}$$

(33)

$$\begin{aligned} \Delta {\dot{E}}_{F D k}\!= & {} \!-\!\frac{\Delta E_{F D k}}{T_{A k}}\!+\!\frac{K_{A k}}{T_{A k}} \!\cdot \!\left( \Delta V_{R E F k}-\Delta V_k\!+\! \Delta V_{P S S k}\right) \nonumber \\ \end{aligned}$$

(34)

where

• $D_k$ is the damping constant $(\textrm{pu})$;

• $\Delta E_{F D k}$ is the field voltage (exciter output) $(\textrm{pu})$;

• $\Delta E_{q k}^{\prime }$ is the internal voltage $(\textrm{pu})$;

• $H_k$ is the inertia constant $(\textrm{s})$;

• $\Delta I_{d k}$ is the d-axis stator current $(\textrm{pu})$;

• $\Delta P_{m k}$ is the mechanical power (pu);

• $\Delta P_{g k}$ is the electrical power (pu);

• $T_{d 0 k}^{\prime }$ is the d-axis open-circuit time constant $(\textrm{s})$;

• $X_{d k}^{\prime }$ is the d-axis transient reactance $(\textrm{pu})$;

• $X_{q k}$ is the q-axis synchronous reactance $(\textrm{pu})$;

• $\Delta V_{R E F k}$ is the reference voltage (pu);

• $\Delta V_k$ is the terminal voltage (pu);

• $\Delta V_{P S S k}$ is the PSS voltage (pu);

• $\omega _s$ is the synchronous speed $(\textrm{rad} / \textrm{s})$;

• $\Delta \omega _{k}$ is the rotor speed deviation $(\textrm{pu})$.

Appendix B

The deterministic approach for the design of power system stabilizers is presented in (35)-(40). It aims to allocate the dominant eigenvalue (associated with the minimum damping ratio) in a region where $\xi _{\text{ min }}$ is close to $\xi _d$, ensuring a level of security and stability, as given in (36)-(37).

$$\begin{aligned} \min f(x)=&\mid \xi _{\text{ min } } - \xi _d \mid \end{aligned}$$

(35)

$$\begin{aligned}&\xi _{\text{ min } } \ge \xi _d \end{aligned}$$

(36)

$$\begin{aligned}&\sigma _{\text{ max } } <0 \end{aligned}$$

(37)

$$\begin{aligned}&K_p^{\text{ min } } \le K_p \le K_p^{\text{ max } } \end{aligned}$$

(38)

$$\begin{aligned}&\alpha _p^{\text{ min } } \le \alpha _p \le \alpha _p^{\text{ max } } \end{aligned}$$

(39)

$$\begin{aligned}&\omega _p^{\text{ min } } \le \omega _p \le \omega _p^{\text{ max } } \end{aligned}$$

(40)

Table 14 Limits of PSS parameters

Table 15 PSS parameters—probabilistic design

Table 16 PSS parameters—deterministic design

The solution of (35)-(40) is carried out using PSO, where (38)-(40) are treated by the optimization method. Each individual is evaluated by employing the deterministic analysis of small-signal stability discussed in Sect. 2.1, where Eqs. (36)–(37) are integrated as penalties within the objective function of (35) in case of violations.

Appendix C

This appendix presents the limits of the optimization variables in Table 14. The parameters adjusted by the probabilistic and deterministic approaches are presented in Tables 15 and 16.

Fig. 15

Calculation of $\sigma _{max}$ and $\xi _{min}$ in MCS environment

Appendix D

To elucidate the probabilistic small-signal stability analysis process, detailed in Sect. 2.3 (Fig. 3), Fig. 15 is presented. This analysis considers a scenario in which two generators are equipped with a single power system stabilizer (PSS), as outlined in Step 1. The primary goal is to compute the mean and standard deviations of $\sigma _{max}$ and $\xi _{min}$. Accordingly, a sample is generated in Step 2, followed by the calculations for $\sigma _{max}$ and $\xi _{min}$ in Steps 3 to 6. Given the Monte Carlo simulation (MCS) framework, Steps 3 to 6 are repeated across multiple samples. The process culminates in the computation of the mean and standard deviations.