A Snake Optimization Algorithm-Based Power System Inertia Estimation Method Considering the Effects of Transient Frequency and Voltage Changes

Abstract

Inertia is the measure of a power system’s ability to resist power interference. The accurate estimation and prediction of inertia are crucial for the safe operation of the power system. To obtain the accurate power system inertia provided by generators, this paper proposes an estimation method considering the influence of frequency and voltage characteristics on the power deficit during transients. Specifically, the traditional swing equations-based inertia estimation model is improved by embedding linearized frequency and voltage factors. On this basis, the snake optimization algorithm is utilized to identify the power system inertia constant due to its strong global search ability and fast convergence speed. Finally, the proposed inertia estimation method is validated in four test systems, and the results show the effectiveness of the proposed method.

1. Introduction

With the increasing demand of loads and the wide integration of new energy in modern power systems, power systems encounter challenges in operation and control [1]. Particularly, many synchronous generator units have been replaced by non-synchronous generator units, leading to a great drop in power systems’ rotating inertia [2]. Under this circumstance, post-disturbance power system frequency tends to change drastically, which could cause cascading failures and even blackouts. Hence, the accurate estimation of power system inertia is crucial to anticipate the large deviation in power system frequency after a power imbalance accident [3,4].

Power system inertia estimation methods can usually be divided into two categories. One is to determine the system inertia offline with recorded measurements after a large power disturbance [5,6,7]. The other is to evaluate the system inertia online according to disturbance measurements in the power system operation [8,9]. This paper concerns the former approach. In [10], a polynomial approximation-based estimation method by fitting the post-disturbance wave forms of the transients is proposed. However, the estimation error of this method increases when the scale of the power system extends largely. In [11], the inertia of the Great Britain network is estimated by utilizing distributed installed phasor measurement units (PMUs). In [12], the inertia is also estimated by the ratio of the power deficit to the rate of change of frequency (RoCoF) and applied in the Taiwan power network, showing satisfying estimation accuracy. In [13], a practical inertia estimation scheme considering disturbance detection and RoCoF estimation is proposed based on the O-spline algorithm. In addition, swing equations are transformed to construct the relationship between the inertia constant and the oscillation modes of the frequency signal [14,15,16]. In [14], the effect of inertia distribution on inter-area oscillation is considered in a wind power-integrated power system area inertia estimation. In [17], the system inertia constant is estimated by the ARMAX algorithm, which can be modelled with the transfer functions of the power deficit and the system frequency. These kinds of methods usually rely on swing equations and obvious power deficits in power systems [18]. However, the effects of the damping coefficient on swing equations, post-disturbance voltage variations and primary frequency regulation are usually neglected in these methods. It has been noticed that the actual amount of power deficit is not always consistent with the assumed amount inferred from the disturbance event [19,20].

To overcome this drawback, improving the inertia estimation model by considering more frequency and voltage factors is the main approach. In [21], the damping effect caused by frequency deviation is considered in an online estimation model to improve inertia estimation accuracy. In [22], the effect of load model on the power deficit is considered when constructing the power system inertia estimation model. The application of this method in the Nordic power network shows promising results. In these methods, the load frequency and voltage effects are mainly considered, and the effects of generator regulations are ignored. Hence, the inertia estimation should be polished further when identifying the inertia of the generator side more accurately is targeted. In addition, it should be noticed that more model parameters are necessary to be identified when the inertia estimation model considers more power change factors, which makes the model’s complexity increase. To solve this problem, an intelligent optimization algorithm is available. In [23,24], the particle swarm optimization (PSO) and genetic algorithms are utilized to identify the inertia time constants of inertia estimation models, proving their effectiveness in parameter identification.

The contributions of this paper lie in two aspects. One is that an inertia estimation model considering the regulation effects caused by frequency and voltage deviations is proposed. In this model, the regulation effects of frequency and voltage are modelled as linearized factors and combined with mechanical power. On this basis, an intelligent optimization algorithm for parameter identification of the proposed inertia estimation model is necessary. Hence, the other contribution of this paper is utilizing the snake optimization (SO) algorithm to realize inertia estimation for the first time. The SO algorithm has been put forward recently and adopted in this paper for its global optimization ability and high efficiency [25].

The remainder of this paper is organized as follows. In Section 2, key issues about power system inertia estimation considering frequency and voltage characteristics are discussed. Furthermore, the SO algorithm is introduced and utilized to realize the proposed power system inertia estimation method in Section 3. The case studies and the conclusions are presented in Section 4 and Section 5, respectively.

2. Inertia Estimation Model and Its Transformation

In this section, the traditional inertia estimation model based on swing equations is presented at first. Afterwards, the transformed inertia estimation model considering the effects of transient frequency and voltage variations is proposed by modifying the traditional swing equations.

2.1. Inertia Estimation with Traditional Swing Equations

In power systems, swing equations describe the transition process of the generators. In the aspect of frequency stability, the generators in a power system can be equivalent to one generator, whose swing equations can be used to infer the post-disturbance frequency dynamics of power systems [26].

As the swing equations indicated, the relationship between the RoCoF and the power deficit can be expressed as follows:

$$2H_{\mathrm{est}}{\displaystyle \frac{d{f}_{coi}(t_0)}{dt}}=\Delta P(t_0),$$

(1)

where $H_{\mathrm{est}}$ is the equivalent inertia constant of the power system. ${\displaystyle \frac{d{f}_{coi}(t_0)}{dt}}$ is the instant RoCoF at the center of inertia when the disturbance occurred. In practical power systems, the instant RoCoF is usually approximated by the observed frequency measurements after the disturbance. $\Delta P(t_0)$ is the imbalanced power of the system, determined by mechanical power, P_m, and electromagnetic power, P_e, shown as follows:

$$\Delta P(t_0)=P_m(t_0)-P_e(t_0)=-\Delta P_{\mathrm{d}}(t_0),$$

(2)

where $\Delta P_{\mathrm{d}}$ is the magnitude of the power imbalance, which is usually assumed to be known when the disturbance occurred. In the actual inertia estimation process, $\Delta P_{\mathrm{d}}$ would change along with the power system’s post-disturbance dynamics. However, $\Delta P_{\mathrm{d}}$ is assumed to be fixed during the transient process in traditional inertia estimation models.

The frequency at the center of inertia $f_{coi}(t)$ is calculated according to the frequency of each generator and its inertia constant with the following formula [26]:

$$f_{coi}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{f}_i}{H}_i}{{\displaystyle \sum _{i=1}^n{H}_i}}},$$

(3)

where $n$ is the number of generators in the system. $f_i$ is the frequency at the bus of the $i$-th generator. $H_i$ is the inertia constant of the $i$-th generator, related to the rotor inertia time constant of the generator. Having obtained $f_{coi}(t)$, the RoCoF ${\displaystyle \frac{d{f}_{coi}(t)}{dt}}$ then can be calculated. The observation window is usually set as 0.05~0.2 s, when the primary frequency regulation process has not been activated.

In an actual power system, it could be difficult to know the inertia time constants, $H_i$, of all generators. Moreover, some renewable energy does not have a clear inertia time constant, which could be another challenge to evaluating the inertia provided by synchronous generators. Hence, $f_{coi}(t)$ is approximated with the mean values of the frequency measurements at all synchronous generators in this paper.

2.2. Transformed Inertia Estimation Model

In the traditional inertia estimation model, the power deficit is assumed to be fixed. However, the power deficit varies during transients due to the regulation effects of frequency and voltage characteristics. These regulation effects can be simplified, which can be represented by a linear relationship between power deficit changes and frequency and voltage deviations. Hence, the actual power deficit $\Delta P_a(t)$ can be expressed with the following formula:

$$\Delta P(t)=-\Delta P_R(t)-\Delta P_d(t),$$

(4)

where $\Delta P_R(t)$ is the actively and passively regulated power caused by frequency and voltage deviations after the disturbance. t is the time label when the frequency and voltage measurements are collected.

As described above, the regulation effects of frequency and voltage changes on the power deficit are simplified as the linear relationship. The reason lies in two folds. One is that the observation time window is short enough to approximate the load power change by the linear relationship of transient frequency and voltage. The other is that the regulation effect of the generators is usually realized with droop control, which can also be simplified as a linear relationship. Hence, the $\Delta P_R(t)$ has the following formula inferred from [16,17]:

$$\Delta P_R(t)=P_M((1+K\Delta V(t))(1+D\Delta f_{coi}(t))-1),$$

(5)

where $P_M$ is the total active power of the generators in the power system and t is the time label of the measurements. $\Delta V(t)$ is the weighted mean value of voltage deviation at the buses, which is then normalized with the initial voltage amplitude before the disturbance. $\Delta f_{coi}(t)$ is the deviation in frequency at the center of inertia, which is approximated by the mean value of frequency measurements at the synchronous generator-connected buses in this paper. K and D are voltage and frequency factors, respectively, which are undetermined parameters waiting for parameter identification.

As described above, the calculation of $\Delta V(t)$ and $\Delta f_{coi}(t)$ can be realized with the following formula:

$$\Delta V(t)={\displaystyle \frac{{\displaystyle \sum _{u=1}^m\left(V_u(0)\ast (V_u(t)-V_u(0))\right)}}{{\displaystyle \sum _{u=1}^m{V}_u(0)}}},$$

(6)

$$\Delta f_{coi}(t)=f_{coi}(t)-f_{coi}(0),$$

(7)

where m is the number of buses in the power system. $V_u(0)$ is the initial voltage amplitude at the u-th bus before the disturbance. $f_{coi}(0)$ is the initial value of the center of inertia frequency before the disturbance.

According to Equations (1), (4) and (5), the regulation effects caused by frequency and voltage deviations can be embedded to the traditional inertia estimation model, which finally has the following formula:

$$2H_{\mathrm{est}}{\displaystyle \frac{d{f}_{coi}(t)}{dt}}=-a{P}_M((1+K\Delta V(t))(1+D\Delta f_{coi}(t))-1)-\Delta P_d(t),$$

(8)

where a is a small uncertainty factor that reflects the variation in the power system generations caused by power system operation state changes.

With the transformed inertia estimation model, the frequency and voltage measurements of the power systems are assumed to be known. $\Delta P_d(t)$ is the initial power deficit, which should be known after the disturbance is determined. There are four parameters that need to be determined by parameter identification, i.e., H_est, a, K and D. It is obvious that the parameter identification can be realized by an intelligent optimization algorithm and the SO algorithm is selected in this paper for application.

3. Snake Optimization Algorithm-Based Inertia Estimation Method

On the basis of the inertia estimation model introduced in the previous section, the SO algorithm is introduced in this section to identify the unknown parameters.

3.1. Snake Optimization Algorithm

The SO algorithm is a heuristic optimization algorithm based on the biological behavior of snakes. It seeks the optimal solution by imitating snakes’ search strategy during predation, avoidance of danger and so on [25].

3.1.1. Basics of SO Algorithm

Mating behavior between male and female snakes is influenced by some factors. For example, snakes usually mate in late spring and early summer when the temperature is low. In addition, the mating process of snakes also depends on the supply of food. If the temperature is low and food is sufficient, male snakes will compete to attract the attention of female snakes. Otherwise, snakes will only search for food or eat the existing food. On this basis, the search process of the snake optimization algorithm will be divided into two stages: exploration and exploitation. Exploration reflects the environmental factors, i.e., snakes prefer to stay where food is sufficient. They will not search for food in environments where the temperature is low and food is insufficient.

Exploitation involves many transitional phases to improve the efficiency of the global optimization result. If food is available and the temperature is high, the snake will only focus on eating the available food. If food is available and the temperature is low, the mating process occurs. In the mating process, there are mainly two modes, i.e., combating mode and mating mode. In the combating mode, each male snake will fight to obtain the best female snake and each female snake will strive to select the best male snake. In the mating mode, the mating behavior for each pair of snakes is activated when food is available. If the mating process occurs during search space, the female snake may lay eggs and hatch new snakes.

3.1.2. Theoretical Foundation of SO Algorithm

Firstly, the initialization of the snake population is described mathematically as follows:

$${\mathrm{X}}_{\mathrm{i}}{=\mathrm{X}}_{\min }+\mathrm{r}\times ({\mathrm{X}}_{\max }{-\mathrm{X}}_{\min }),$$

(9)

where ${\mathrm{X}}_{\mathrm{i}}$ is the position of the $\mathrm{i}$-th snake, r is a random number within the range of [0, 1], and ${\mathrm{X}}_{\max }$ and ${\mathrm{X}}_{\min }$ are the upper and lower boundaries of the optimization problem.

Secondly, separate the population into two groups equally with the following formula, where female and male snakes account for 50% and 50%, respectively:

$${\mathrm{N}}_{\mathrm{m}}\approx \mathrm{N}/2,$$

(10)

$${\mathrm{N}}_{\mathrm{f}}{=\mathrm{N}-\mathrm{N}}_{\mathrm{m}},$$

(11)

where $\mathrm{N}$ represents the scale of the snake population. ${\mathrm{N}}_{\mathrm{m}}$ is the number of male snakes. ${\mathrm{N}}_{\mathrm{f}}$ is the number of female snakes.

Thirdly, evaluate each group and define the temperature and food quantity. Find the best individual in each group and obtain the best male ${\mathrm{f}}_{\mathrm{best},\mathrm{m}}$ and the best female ${\mathrm{f}}_{\mathrm{best},\mathrm{f}}$, as well as the food location ${\mathrm{f}}_{\mathrm{food}}$. The temperature can be defined by the following formula:

$$\mathrm{Temp}=\exp ({\displaystyle \frac{-k}{\mathrm{T}}}),$$

(12)

where $k$ represents the current number of iterations. $\mathrm{T}$ is the maximum number of iterations.

The quantity of food $\mathrm{Q}$ then is defined with the following formula:

$$\mathrm{Q}={\mathrm{c}}_1\cdot \exp \left({\displaystyle \frac{k-\mathrm{T}}{\mathrm{T}}}\right),$$

(13)

where ${\mathrm{c}}_1$ is assumed to be a constant, taken as 0.5.

Fourthly, during the exploration stage without food, if $\mathrm{Q}$ is less than a threshold, e.g., 0.25, snakes search for food by selecting any random location and update their positions with the following formula:

$${\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}+1\right){=\mathrm{X}}_{\mathrm{rand},\mathrm{m}}\left(\mathrm{t}\right)\pm {\mathrm{c}}_2\times {\mathrm{A}}_{\mathrm{m}}\times \left(\left({\mathrm{X}}_{\max }{-\mathrm{X}}_{\min }\right)\times {\mathrm{rand}+\mathrm{X}}_{\min }\right),$$

(14)

where ${\mathrm{X}}_{\mathrm{i},\mathrm{m}}$ represents the male snake position. ${\mathrm{X}}_{\mathrm{rand},\mathrm{m}}$ is the position of randomly selected male snake position. $\mathrm{rand}$ is a random number within the range of [0, 1]. ${\mathrm{c}}_2$ is a constant, taken as 0.05. ${\mathrm{A}}_{\mathrm{m}}$ is the ability of males to search for food, calculated with the following formula:

$${\mathrm{A}}_{\mathrm{m}}=\exp \left({\displaystyle \frac{-{\mathrm{f}}_{\mathrm{rand},\mathrm{m}}}{{\mathrm{f}}_{\mathrm{i},\mathrm{m}}}}\right),$$

(15)

where ${\mathrm{f}}_{\mathrm{rand},\mathrm{m}}$ is the fitness value of the randomly selected male snake position ${\mathrm{X}}_{\mathrm{rand},\mathrm{m}}$. ${\mathrm{f}}_{\mathrm{i},\mathrm{m}}$ is the fitness value for male snake position ${\mathrm{X}}_{\mathrm{i},\mathrm{m}}$.

$${\mathrm{X}}_{\mathrm{i},\mathrm{f}}(\mathrm{t}+1){=\mathrm{X}}_{\mathrm{rand},\mathrm{f}}(\mathrm{t})\pm {\mathrm{c}}_2\times {\mathrm{A}}_{\mathrm{f}}\times (({\mathrm{X}}_{\max }{-\mathrm{X}}_{\min })\times {\mathrm{rand}+\mathrm{X}}_{\min }),$$

(16)

where ${\mathrm{X}}_{\mathrm{i},\mathrm{f}}$ represents the female snake position. ${\mathrm{X}}_{\mathrm{rand},\mathrm{f}}$ is the randomly selected female snake position. $\mathrm{rand}$ is a random number within the range of [0, 1]. ${\mathrm{A}}_{\mathrm{f}}$ is the ability of females to search for food, calculated with the following formula:

$${\mathrm{A}}_{\mathrm{f}}=\exp \left({\displaystyle \frac{{-\mathrm{f}}_{\mathrm{rand},\mathrm{f}}}{{\mathrm{f}}_{\mathrm{i},\mathrm{f}}}}\right),$$

(17)

where ${\mathrm{f}}_{\mathrm{rand},\mathrm{f}}$ represents the fitness value of the randomly selected male snake position ${\mathrm{X}}_{\mathrm{rand},\mathrm{f}}$. ${\mathrm{f}}_{\mathrm{i},\mathrm{f}}$ is the fitness value for the male snake position ${\mathrm{X}}_{\mathrm{i},\mathrm{f}}$.

Fifthly, during the exploration stage with food and if $\mathrm{Q}$ is larger than a threshold, if the temperature is larger than a threshold, e.g., 0.6, the temperature is seen as a hot state. Snakes only search for food, and the position is updated with the following formula:

$${\mathrm{X}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}+1\right){=\mathrm{X}}_{\mathrm{food}}\pm {\mathrm{c}}_3\times \mathrm{Temp}\times \mathrm{rand}\times \left({\mathrm{X}}_{\mathrm{food}}{-\mathrm{X}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)\right),$$

(18)

where ${\mathrm{X}}_{\mathrm{i},\mathrm{j}}$ represents the position of the individual male or female snake. ${\mathrm{X}}_{\mathrm{food}}$ is the optimal position for individual snakes. $\mathrm{rand}$ is a random number within the range of [0, 1]. ${\mathrm{c}}_3$ is assumed to be a constant, taken as 2.

If Q is larger than a threshold and temperature is smaller than a threshold, e.g., 0.6, the temperature is seen as a cold state. The snake will be in combating mode or mating mode.

For combating mode, the position is updated with the following formula:

$${\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}+1\right){=\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}\right){+\mathrm{c}}_3\times \mathrm{FM}\times \mathrm{rand}\times \left(\mathrm{Q}\times {\mathrm{X}}_{\mathrm{best},\mathrm{f}}{-\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}\right)\right),$$

(19)

where ${\mathrm{X}}_{\mathrm{i},\mathrm{m}}$ represents the position of the $\mathrm{i}$-th male snake. ${\mathrm{X}}_{\mathrm{best},\mathrm{f}}$ is the best position in the female snake group. $\mathrm{rand}$ is a random number within the range of [0, 1]. $\mathrm{FM}$ is the male combat ability.

$${\mathrm{X}}_{\mathrm{i},\mathrm{f}}\left(\mathrm{t}+1\right){=\mathrm{X}}_{\mathrm{i},\mathrm{f}}\left(\mathrm{t}\right){+\mathrm{c}}_3\times \mathrm{FF}\times \mathrm{rand}\times \left(\mathrm{Q}\times {\mathrm{X}}_{\mathrm{best},\mathrm{m}}{-\mathrm{X}}_{\mathrm{i},\mathrm{f}}\left(\mathrm{t}\right)\right),$$

(20)

where ${\mathrm{X}}_{\mathrm{i},\mathrm{f}}$ represents the position of the $\mathrm{i}$-th female snake. ${\mathrm{X}}_{\mathrm{best},\mathrm{m}}$ is the best position in the male snake group. $\mathrm{rand}$ is a random number within the range of [0, 1]. $\mathrm{FF}$ is the female combat ability. $\mathrm{FM}$ and $\mathrm{FF}$ can be calculated using the following formula:

$$\mathrm{FM}=\exp \left({\displaystyle \frac{-{\mathrm{f}}_{\mathrm{best},\mathrm{f}}}{{\mathrm{f}}_{\mathrm{i}}}}\right),$$

(21)

$$\mathrm{FF}=\exp \left({\displaystyle \frac{-{\mathrm{f}}_{\mathrm{best},\mathrm{m}}}{{\mathrm{f}}_{\mathrm{i}}}}\right),$$

(22)

where ${\mathrm{f}}_{\mathrm{best},\mathrm{f}}$ represents the fitness value of the best position ${\mathrm{X}}_{\mathrm{best},\mathrm{f}}$ in the female snake group. ${\mathrm{f}}_{\mathrm{best},\mathrm{m}}$ is the fitness value of the best position ${\mathrm{X}}_{\mathrm{best},\mathrm{m}}$ in the male snake group. ${\mathrm{f}}_{\mathrm{i}}$ is the fitness value of individual snakes.

For mating mode, the position is updated with the following formula:

$${\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}+1\right){=\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}\right){+\mathrm{c}}_3\times {\mathrm{M}}_{\mathrm{m}}\times \mathrm{rand}\times \left(\mathrm{Q}\times {\mathrm{X}}_{\mathrm{i},\mathrm{f}}\left(\mathrm{t}\right){-\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}\right)\right),$$

(23)

$${\mathrm{X}}_{\mathrm{i},\mathrm{f}}\left(\mathrm{t}+1\right){=\mathrm{X}}_{\mathrm{i},\mathrm{f}}\left(\mathrm{t}\right){+\mathrm{c}}_3\times {\mathrm{M}}_{\mathrm{f}}\times \mathrm{rand}\times \left(\mathrm{Q}\times {\mathrm{X}}_{\mathrm{i},\mathrm{m}}\left(\mathrm{t}\right){-\mathrm{X}}_{\mathrm{i},\mathrm{f}}\left(\mathrm{t}\right)\right),$$

(24)

where ${\mathrm{X}}_{\mathrm{i},\mathrm{m}}$ represents the position of the $\mathrm{i}$-th male snake. ${\mathrm{X}}_{\mathrm{i},\mathrm{f}}$ is the position of the $\mathrm{i}$-th female snake. $\mathrm{rand}$ is a random number within the range of [0, 1]. ${\mathrm{M}}_{\mathrm{m}}$ and ${\mathrm{M}}_{\mathrm{f}}$ are the mating abilities of male and female snakes, which can be calculated with the following formula:

$${\mathrm{M}}_{\mathrm{m}}=\exp \left({\displaystyle \frac{-{\mathrm{f}}_{\mathrm{i},\mathrm{f}}}{{\mathrm{f}}_{\mathrm{i},\mathrm{m}}}}\right),$$

(25)

$${\mathrm{M}}_{\mathrm{f}}=\exp \left({\displaystyle \frac{{-\mathrm{f}}_{\mathrm{i},\mathrm{m}}}{{\mathrm{f}}_{\mathrm{i},\mathrm{f}}}}\right),$$

(26)

where ${\mathrm{f}}_{\mathrm{i},\mathrm{m}}$ represents the fitness value of the $\mathrm{i}$-th male snake position. ${\mathrm{f}}_{\mathrm{i},\mathrm{f}}$ is the fitness value for the $\mathrm{i}$-th female snake position. If the eggs hatch, select the worst male and female and replace them.

$${\mathrm{X}}_{\mathrm{worst},\mathrm{m}}{=\mathrm{X}}_{\min }+\mathrm{rand}\times ({\mathrm{X}}_{\max }{-\mathrm{X}}_{\min }),$$

(27)

$${\mathrm{X}}_{\mathrm{worst},\mathrm{f}}{=\mathrm{X}}_{\min }+\mathrm{rand}\times ({\mathrm{X}}_{\max }{-\mathrm{X}}_{\min }),$$

(28)

where ${\mathrm{X}}_{\mathrm{worst},\mathrm{m}}$ is the worst position in the male snake group. ${\mathrm{X}}_{\mathrm{worst},\mathrm{f}}$ is the worst position in the female snake group.

3.2. Application of SO Algorithm in Inertia Estimation

As described in Section 2, the inertia estimation can be applied in both the traditional model and the proposed transformed inertia estimation model. When applying the SO algorithm in inertia estimation, the kernel is to design the fitness function.

According to the traditional model, the fitness function can be obtained with the following formula, where H_est is the only parameter to be identified:

$$F_1=(2{\displaystyle \frac{H_{\mathrm{est}}}{f_N}}{\displaystyle \frac{d{f}_{coi}(t)}{dt}}+\Delta P_d(t){)}^2,$$

(29)

As for the proposed transformed inertia estimation model, the fitness function can be obtained with the following formula. It can be inferred from the formula that the model is more complex, containing voltage and frequency factors K and D in addition:

$$F_2=(2{\displaystyle \frac{H_{\mathrm{est}}}{f_N}}{\displaystyle \frac{d{f}_{coi}(t)}{dt}}+\Delta P_d(t)+a{P}_M((1+K\Delta V(t))(1+D\Delta f_{coi}(t))-1){)}^2$$

(30)

where H_est, $a$, $K$ and $D$ are parameters to be identified.

When applying the snake optimization algorithm in the proposed inertia estimation model, the searching ranges of $a$, $K$ and $D$ can be [0.95, 1.05], [0, 2] and [0, 6], respectively [27]. The searching range of $H_{\mathrm{est}}$ is determined to be [0, $2\cdot H_{sys}$], where $H_{\mathrm{est}}$ is the theoretical value of the power system inertia constant. The implementation scheme of SO algorithm is presented in Figure 1.

4. Case Study

In this paper, the proposed method is verified with several test systems by Power System Analysis Software Package (PSASP) v7.41.04. Comprehensive load models are configured in the test power system. The disturbance is set as an increase of load at a randomly selected bus. It is also assumed that the frequency at the generator buses and the voltage at the buses are measurable. For comparison, the PSO algorithm, the genetic algorithm (GA) and the differential evolution (DE) algorithm are also selected for inertia estimation.

4.1. Test System 1: Two Generators Test System

In Figure 2, the structure of this test system is presented. In this test system, the amount of load at bus 3 is set as 230 MW and the disturbance amount of load increase is set as 20 MW. Two generators in this test system have the same rated power and their inertia time constants are set as 4 s and 6 s, respectively. Hence, the equivalent system inertia of the generator side is 5 s, which can be seen as the theoretical value.

The upper and lower limits of the unknown parameters in the inertia estimation models are summarized in

Table 1. Ranges of unknown model parameters in traditional and proposed models.

Inertia Estimation Model	Unknown Parameters	Range
Traditional Model	$H_{\mathrm{est}}$	[0, 10]
Proposed Transformed Model	$H_{\mathrm{est}}$	[0, 10]
	$a$	[0.95, 1.05]
	$K$	[1, 2]
	$D$	[1, 2]

.

By applying the SO algorithm in the inertia estimation models, the inertia estimation results with different observation time windows are summarized in

Table 2. Performance of SO algorithm under different models and observation time window in test system 1.

Observation Time Window	Traditional Model	Proposed Transformed Model
	Result/Error	Result/Error
0.1 s	7.2464 s/2.2464 s	5.1636 s/0.1636 s
0.2 s	6.9042 s/1.9042 s	5.0737 s/0.0737 s
0.3 s	6.8231 s/1.8231 s	4.9727 s/0.0273 s
0.4 s	6.7889 s/1.7889 s	4.5924 s/0.4076 s
0.5 s	6.7502 s/1.7502 s	4.9574 s/0.0426 s
0.6 s	6.7024 s/1.7024 s	4.7194 s/0.2806 s
0.7 s	6.6636 s/1.6636 s	5.0877 s/0.0877 s
0.8 s	6.6485 s/1.6485 s	5.2210 s/0.2210 s
0.9 s	6.6540 s/1.6540 s	4.8786 s/0.1214 s
1.0 s	6.6628 s/1.6628 s	5.1089 s/0.1089 s

.

It can be inferred from Table 2 that the performance of the SO algorithm on the proposed transformed inertia estimation model is obviously better than that on the traditional model, whose estimation results are more approximate to the equivalent system inertia of the generator side, i.e., 5 s. The accuracy difference comes from transient voltage and frequency variations after the disturbance, which makes the actual power deficit different from the pre-known disturbance power.

Furthermore, the PSO, GA and DE algorithms are also applied based on the proposed transformed model. The average estimation errors of the PSO, GA and DE algorithms under all observation time windows are 0.2743 s, 0.2987 s and 0.2870 s, respectively. As presented in Table 2, the average estimation error of the SO algorithm under all observation time windows is only 0.1534 s, which shows a great advantage in inertia estimation accuracy. Hence, the SO algorithm shows an overall better performance in terms of accuracy.

Specifically, the fitness value variation processes of the SO- and PSO-based methods are demonstrated in Figure 3. It indicates that the fitness value of the SO algorithm decreases sharply in only several steps, while the PSO algorithm takes about 20 steps to realize a similar decrease, which shows the great convergence speed of the SO algorithm.

4.2. Test System 2: Three Generators and Nine Buses Test System

This test system is an IEEE standard test system, whose total loads are 315 MW at buses 7, 8 and 9. Three generators in this test system also have the same rated power, and their inertia time constants are set as 6.4 s, 3.01 s and 23.64 s, respectively. Hence, the theoretical system inertia of the generation side is 11.1067 s, which is referred to as the actual system inertia. The amount of load increase is set as 50 MW.

In this test system, the upper and lower limits of the inertia constant are set as 20 s and 0 s, respectively. The other ranges of the unknown parameters in the proposed transformed model are the same as with the previous test system.

It can be inferred from Figure 4 that the inertia estimation accuracy with the proposed transformed model is obviously closer to the actual value than that with the traditional model. The mean values of the estimation results are 12.5809 s and 11.0644 s for the traditional and proposed transformed models, respectively, where 11.1067 s is the actual value.

Further, the performance of the SO-, PSO-, GA- and DE-based methods is also compared, whose estimation errors are 0.2025 s, 0.2962 s, 0.3164 s and 0.3268 s, respectively. The specific estimation results and errors of the SO- and PSO-based methods are summarized in

Table 3. Performance of SO and PSO algorithms under different observation time windows.

Observation Time Window	SO Algorithm	PSO Algorithm
	Result/Error	Result/Error
0.1 s	11.1811 s/0.1644 s	11.0044 s/0.0122 s
0.2 s	11.2491 s/0.2324 s	10.6033 s/0.4134 s
0.3 s	11.1360 s/0.1193 s	10.8402 s/0.1764 s
0.4 s	11.2394 s/0.2228 s	10.8879 s/0.1288 s
0.5 s	11.0946 s/0.0779 s	10.8111 s/0.2056 s
0.6 s	11.1578 s/0.1411 s	10.6317 s/0.3850 s
0.7 s	10.4590 s/0.5577 s	10.3673 s/0.6493 s
0.8 s	11.1742 s/0.1575 s	10.7420 s/0.2747 s
0.9 s	10.8005 s/0.2161 s	10.4266 s/0.5901 s
1.0 s	11.1524 s/0.1357 s	10.8906 s/0.1261 s

.

It is indicated in Table 3 that the SO algorithm has more powerful ability in global optimization than the PSO algorithm, where the estimation accuracy of the SO algorithm is superior to that of the PSO algorithm in most cases. In addition, it is also can be inferred from Figure 5 that the SO algorithm has good capability in convergence speed, whose fitness value can decrease sharply within several steps.

4.3. Test System 3: Five Generators and Fourteen Buses Test System

In Figure 6, the structure of the test system is presented. In this test system, the amount of total loads is 147 MW. The rated power of the generators at buses 1, 2, 3, 6 and 8 is 100 MVA, 60 MVA, 60 MVA, 25 MVA and 25 MVA, respectively. The inertia time constants of the generators are 6.54 s, 5.06 s, 5.06 s, 5.06 s and 5.06 s, respectively. Hence, the equivalent inertia constant of the generators is 5.6081 s, theoretically. The disturbance is assumed as a load increase at bus 2, whose amount is 80 MW.

In this test system, the upper and lower limits of the inertia constant are set as 10 s and 0 s, respectively. The other ranges of the unknown parameters in the proposed transformed model are the same as with the previous test system.

In the aspect of model difference, the performance of the SO algorithm on the proposed transformed model is better than that on the traditional model, where the mean errors of the proposed transformed model and the traditional model are 0.4189 s and 0.8334 s, respectively.

Further, the inertia estimation errors of the SO and PSO algorithms on the proposed transformed model under different observation time windows are demonstrated in Figure 7. The average absolute errors of the SO and PSO algorithms are 0.4189 s and 0.6809 s, respectively. It then can be inferred that the estimation result by the SO algorithm is more accurate than that by the PSO algorithm.

On the basis of the proposed transformed model, the fitness value variations during the optimization process under the SO and PSO algorithms are presented in Figure 8, which once again indicates the promising advantage in ability of the SO algorithm in convergence speed.

4.4. Test System 4: Ten Generators and Thirty-Nine Buses Test System

The total loads of the IEEE 39 test system reach 6.1491 GW. The inertia time constants of the rotor are 42 s, 30.3 s, 35.8 s, 28.6 s, 26 s, 34.8 s, 26.4 s, 24.3 s, 34.5 s and 500 s, respectively. The theoretical inertia value of the system then can be figured out to be 78.27 s. The disturbance power of load increase is set as 200 MW according to the scale of this test system.

In this test system, the upper and lower limits of the inertia constant are set as 150 s and 0 s, respectively. The other ranges of the unknown parameters in the proposed transformed model are the same as with the previous test system. By applying the SO algorithm in different inertia estimation models, the inertia estimation results with different observation time windows are summarized in

Table 4. Performance of SO algorithm under different models and observation time windows in test system 3.

Observation Time Window	Traditional Model	Proposed Transformed Model
	Result/Error	Result/Error
0.1 s	95.7974 s/17.5274 s	90.4395 s/12.1695 s
0.2 s	91.0843 s/12.8143 s	81.9380 s/3.6680 s
0.3 s	90.0319 s/11.7619 s	80.9137 s/2.6437 s
0.4 s	90.3709 s/12.1009 s	77.1445 s/1.1255 s
0.5 s	91.1961 s/12.9261 s	69.9666 s/8.3034 s
0.6 s	92.4855 s/14.2155 s	71.4888 s/6.7812 s
0.7 s	93.7382 s/15.4682 s	70.3299 s/7.9401 s
0.8 s	95.2051 s/16.9351 s	72.9294 s/5.3406 s
0.9 s	96.5379 s/18.2679 s	71.8028 s/6.4672 s
1.0 s	97.6417 s/19.3717 s	67.0750 s/9.1950 s

.

It can be inferred from Table 4 that the performance of the SO algorithm on the proposed transformed inertia estimation model is obviously better than that on the traditional model, where the inertia estimation error is much lower than that of the traditional model. It proves the effectiveness of considering frequency and voltage effects in inertia estimation.

In addition, the SO and PSO algorithms are implemented 20 times each for inertia estimation and their estimation error rates are presented with the box diagram in Figure 9.

In Figure 9, it is obvious that the results of the SO algorithm are more concentrated than those of the PSO algorithm because the height of the box of the SO algorithm is much lower than that of the PSO algorithm. In specific, the medians for the SO and PSO algorithms in the left figure are 2.004% and 1.684%, while the absolute mean values for the SO and PSO algorithms are 2.38% and 4.57%, respectively. In the right figure, the SO algorithm shows an obvious advantage in accuracy and solution reliability. Hence, it can be concluded that the SO algorithm has a strong ability to find optimal solutions.

5. Conclusions

To estimate the inertia provided by synchronous generators accurately, this paper proposes a transformed inertia estimation model. Different from the traditional inertia estimation model, the proposed transformed inertia estimation model considers the regulation effects of frequency and voltage variations on the power deficit. Furthermore, an SO algorithm-based inertia estimation method is put forward, which is believed to have global optimization ability and high convergence speed. According to the simulation results on four typical test systems, it is validated that the proposed transformed inertia estimation model can approximate the inertia of synchronous generators more accurately than the traditional inertia estimation model. Moreover, the SO algorithm shows great inertia estimation accuracy and convergence speed.