Optimal Allocation Strategy of Virtual Inertia and Damping for Improving the Small-signal Stability

The application of the virtual synchronous generator (VSG) control technology to engage the inertia response of renewable energy generation systems is an effective means of coping with the problem of weak inertia in power systems. To deal with the problem of small-signal stability in the weak inertia power system, this paper studies how the strength of virtual inertia and damping affect the small-signal stability, and proposes an optimal allocation strategy of virtual inertia and damping for improving the small-signal stability. The optimal allocation model of virtual inertia is constructed to minimize the energy imbalance of the system under small-signal by selecting the virtual inertia and damping as optimization variables. Regarding the aforementioned model, the evolutionary algorithm is used to obtain the optimal allocation of virtual inertia and damping. Simulations on a virtual synchronized microgrid system show the effectiveness of the proposed strategy in improving small-signal stability.


Introduction
In recent years, numerous scholars have concentrated on the development and utilization of renewable energy due to the increasingly critical issues of environmental degradation and energy depletion.In China, the power system is also steadily increasing the share of new energy units, and renewable energy units will gradually replace conventional thermal power units in response to the need for a national "carbon peaking and carbon neutrality" policy.In Zhou et al.'s work [1], coal will make up the remaining 30%-50% of the electricity generated in the future power system, with clean energy sources such as hydropower, wind, solar, and nuclear power making up the remaining 50%-70%.However, a significant amount of renewable energy generation alters the structure of the power system, which may impact its operation, inertia, and damping.Therefore, the dynamic stability of the power system is harmed by the loss of inertia, and the reduction of damping will have an impact on the system small-signal stability, making it more susceptible to issues like low-frequency oscillation [2].It is urgent to research the small-signal stability problem of the renewable energy power system under the condition of weak inertia to ensure the safe and stable operation of the power grid.
To address the weak inertia issue of the power system, the industry and academia have proposed several inertia compensation techniques, such as a virtual synchronous generator (VSG), which works in conjunction with some traditional rotational inertia resources to support the system inertia [3].In [4] and [5], it is contended that virtual inertia will alter the small-signal stability in power systems.However, both of them have overlooked the impact of the geographical distribution of inertia.As a result, an important study on the best distribution of virtual inertia has been done by academics.The virtual inertia optimization at the moment typically entails the resolution of the inertia safety border, the optimization resolution of nested frequency safety restrictions, and the spatial layout of inertia in three aspects [6].The distribution of virtual inertia at the weakest point of the system inertia will increase system stability and reduce frequency oscillation.In [7], non-modal tools are employed to analyze and control inter-area oscillations.In [8], while the mode damping ratio and the analytical formulation of the system parameters are challenging to analyze, the relationship between the system state space equation and the system parameters may be described analytically.When virtual inertia is set up in the system, the optimization is challenging since the issue is non-convex.To convert the nonconvex optimization issue into a convex optimization problem, this work employs the simplified model suggested in [9].
The electricity system has a complicated structure and a lot of nodes.The inertia of the generator node far from the node where the small signal occurs contributes less to the small-signal stability of the system when a small signal occurs in the system, which is influenced by the structure and size of the power system.The position of the VSG in the network becomes more crucial because if the generator node position is fixed, more inertia is needed to sustain the stability of the system when a small signal occurs.The virtual inertia of the system is optimized based on the simplified model [9], with the smallest energy imbalance following a small signal as the objective function.However, it only optimizes the virtual inertia and ignores the impact of damping on the power system small-signal stability problem, and the optimization model constraints are straightforward.This study provides an optimization model based on the Lyapunov direct technique [10] based on the simplified model in [9].The objective function of this model is the minimal energy imbalance of the system under the small signal [11], and the decision variables are virtual inertia and damping, and the damping ratio constraint is added to the state space of the system.We investigate how the size of the virtual inertia and the damping affect the power system's small-signal stability.The three synchronous generators in the system are kept in this study based on the three-machine, nine-node model in [12], and VSG is configured at the other nodes to optimize the virtual inertia and the damping.This paper focuses on how adding VSG to the system alters stability, virtual inertia, and damping.The major contributions of this article are summarized as follows.
1) For weak inertia systems, an optimal allocation strategy of virtual inertia and damping is proposed based on the energy imbalance function, which can maximize the small-signal stability of the system within a certain constraint.
2) The influence of virtual inertia and damping allocation on the small-signal stability of the system is investigated by simulation, and it is found that the more uniform the allocation of virtual inertia and damping, the more it can improve the small-signal stability of the system.
The rest of this paper is organized as follows.Section I defines inertia and provides the secondorder model of the VSG.Section II introduces the indicators for assessing the power system smallsignal stability problem.And Section III justifies the small-signal stability model and the simplified model and provides the virtual allocation optimization technique and builds, based on a streamlined model, an optimal allocation model of virtual inertia and damping.Section IV examines the optimization model as an illustration.Finally, Section V brings this essay to a close.

VSG and inertia model
To give the power supply using the converter the characteristics of the synchronous generator, such as inertia, damping, primary frequency regulation, reactive power voltage regulation, and other characteristics outside the grid-connected operation, a technology known as VSG technology simulates the electromechanical transient characteristics of the synchronous generator [13].Additionally, it is both practicable and cost-effective for the virtual synchronous machine to not require a significant transformation of the power grid.The mechanical property equation for the VSG second-order model of the generator is as follows: where  ,  , and  represent the mechanical, electromagnetic and damping torques of synchronous generators, respectively;  stands for the synchronous generator moment of inertia;  for its electrical angular velocity;  represents the synchronous angular velocity of the power system; and  is the damping coefficient.
When the rotor rotates at its rated speed  , according to the rules of physics for spinning objects, their kinetic energy is: This leads to the following equation for calculating inertia: where  is the rotor kinetic energy;  is its mechanical angular velocity;  is the power reference value; and  is the number of pole pairs.

The small-signal stability model
Currently, the state of the power system can be described by  linear independent system variables; these variables are referred to as state variables.The state of the power system is typically described in state space.State variables can be used in conjunction with the system input signal to describe the behavior of the system and determine any necessary system parameters.Vector-matrix images can be used to depict the power system: ,  5 where  represents the state variable;  represents the input variable; and  represents the outcome variable.The differential equation for each component of the electricity system is found in Equation ( 4).The electricity system output equation is Equation (5).Since the interference is small enough, it is possible to use the stability of the linearized system to study the stability of the nonlinear system when analyzing the small-signal stability problem of the power system.This is because we always assume that the normal state experiences instantaneous small-signal at the time of   .Equations ( 4) and ( 5) above, are linearized as below: where  is the   order state matrix;  is the   order input matrix; C is the   output order matrix; and  is the feed-forward matrix; ∆ is the  -dimensional state vector; ∆ is the dimensional output vector; and ∆ is the -dimensional input vector.The Lyapunov method is still the most popular in the present research for examining the smallsignal stability problem in power systems.According to Lyapunov's first method, when all the real parts of the eigenvalues of the state matrix  are negative, the actual nonlinear system is gradually stable at the equilibrium point, which determines the small-signal stability of the power system.The typical equation has the following equation: The eigenvalues of matrix  are obtained by expanding and solving the feature equation.The damping is associated with the real component of the eigenvalues, whereas the oscillation frequency is associated with the imaginary part.The oscillation amplitude decays according to the damping ratio, and the bigger the damping ratio, the faster the oscillation amplitude decays in the system.The expression for the damping ratio for a pair of complex eigenvalues    is: where  is the damping ratio;  is the eigenvalue of the power system;  and  are the real and imaginary parts of the eigenvalue  , respectively.Each damping ratio should be positive to guarantee the stability of the system even in the presence of a small signal.Consequently, it is important to make sure that during optimization, the damping ratio of each feature root of the system state matrix is bigger than 0.

The simplified model
As was previously mentioned, since the optimization of virtual inertia using a complex power system will result in a non-convex optimization problem, it is essential to carry out the required processing on the power system to simplify the issue to convert the inertia optimization problem into a convex optimization problem that can be solved.
As the line resistance in the power system is substantially lower than the line reactance, the resistance  of the network lines is disregarded for the sake of simplification, leaving just the network electrical susceptance in place.Reactive power is configured to each system node so that it does not affect the system voltage when there are minor interferences.Based on the foregoing, the VSG node  produces the following electromagnetic power in the case of a small signal: where  is the electromagnetic power output by node ;  is the electrical susceptance between node  and node ; and  and  are the phase angles of node  and node , respectively.We consider each node  to have virtual inertia and damping.Equations ( 4), ( 5), ( 6), ( 7) and ( 10) are combined to provide the matrix form of the system simplified equation for the state, which is: where ∆ is the change of angular velocity of the synchronous generator; ∆ is the change of power angle of the synchronous generator;  is a unit matrix; 0 is a zero matrix;  stands for the electrical susceptance between system nodes; ℎ stands in for virtual inertia and  stands for damping 1 ,  , respectively.The system consists of VSG nodes that may be freely designed with damping and inertia at non-synchronous generator nodes, as well as physical synchronous generator nodes.

Virtual inertia optimization allocation strategy
The power system will undergo a series of oscillations in response to a small signal before it can once again reach the original or new equilibrium state.As a result, the objective function of decreasing the energy imbalance of the system following a small signal may be used to design an optimization model.
The second technique proposed by Lyapunov uses state space functions to determine a system stability directly.Its fundamental idea derives from a reliable assessment of the mechanical system equilibrium condition [10].
If there is a positive definite energy function   whose total differential is negatively definite for a linear, constant, continuous system, the system is increasingly stable.The scalar function of the quadratic form is frequently employed when applying Lyapunov's second method:

𝑉 𝑥 𝑥 𝑃𝑥. 12
At this point, the rate of change of the energy function   is: In general, in the calculation,  is the unit matrix, and then the positive definite matrix  is derived by the formula    ; when the system is stable, there must be a positive definite matrix .If     , when  is positive, the system becomes increasingly stable.

The objective function
The objective function of minimizing the energy imbalance of the system following a small signal can be represented in this paper [9].

𝑏 𝜃 𝑡 𝜃 𝑡 𝑚 𝜔 𝑡 𝑑𝑡 14
The above equation is converted to the integral form concerning the  matrix    ,  is: where  is a diagonal matrix with the system inertia ℎ, and  is the node susceptance matrix of the system.
The system linearized state matrix  is derived from Equation (11).
The matrix with definite positive coefficients is  : where  1 is the unit diagonal matrix and  is the system node susceptance matrix.Unit pulses are used to to represent the small signal, and the input matrix  is: where  is the system perturbation coefficient and  is a diagonal matrix with the system inertia ℎ.
The second equation of Lyapunov has the following solution from the equations above:

𝐴 𝑃 𝑃𝐴 𝑄 𝑄 𝑉 𝐵 𝑃𝐵 . 19
After a small signal, the system is most steady when the  matrix two-norm is minimal.

Constraints
VSG node virtual inertia ℎ and damping  are adjusted individually, and their adjustment range is as follows: where ℎ is the virtual inertia of the VSG  ;  is the damping of the VSG ; ℎ and  are the lowest values of the virtual inertia and damping offered by the VSG , respectively; ℎ ,  are the highest amounts of virtual inertia and damping offered by the VSG , respectively; ℎ is the smallest amount of virtual inertia that the system needs; and  is the damping ratio.

The optimization model
The minimum energy imbalance after a small signal of the system can be obtained by combining the aforementioned objective function and constraints, leading to an optimization model with the virtual inertia ℎ and damping  of the VSG as the decision variables.

The optimizing processes
This paper adopts the above optimization model in Equation (24).The system optimization model is then solved after it has been obtained.In Figure 1, the precise optimization procedure is displayed: The node susceptance matrix is generated after first calculating the node admittance matrix for the power system using the input network-related data.The evolutionary algorithm is used to to optimize and solve the matrix to minimize the two-norm of the matrix  after entering the beginning value and constraint conditions of the decision variable to be optimized.

Example analysis
In this paper, as shown in Figure 2 the model of three machines and nine nodes composed of three synchronous generators, three loads, and nine branches is established in MATLAB.The VSG is linked to other nodes, and the virtual inertia and damping values that the VSG should supply are optimized.Nodes 1, 2, and 3 are all synchronous generators, and their inertia and damping have fixed values.At each system node, perturbations are applied.Wang et al. display branch and other data for three machines, nine nodes, and synchronous generator data in [13].
The four cases are compared to demonstrate the impact of virtual inertia and damping optimization on the stability of the power system when small-signal occur at each node of the system.Figure 2. Three machines, nine nodes model.

Case 1: Only the synchronous generator provides inertia support for the system
System inertia support is provided solely by the synchronous generator, and the other nodes do not connect to the VSG.Assume that the nodes without access to the VSG have zero inertia and damping, these nodes are not involved in the system inertia optimization.Table 1 shows the virtual inertia and damping values of system nodes as well as the two-norm of the goal function.To verify the effect of different numbers of VSG on the inertia of the system, the node linked to the synchronous generator is supposed to have constant inertia and zero damping, and it is also presumed that it is not involved in system inertia optimization.Table 2 shows the virtual inertia and damping values of system nodes as well as the two-norm of the goal function.To verify the effect of different numbers of VSG on the inertia of the system, the node linked to the synchronous generator is supposed to have constant inertia and zero damping, and it is also presumed that it is not involved in system inertia optimization.Table 3 shows the virtual inertia and damping values of system nodes as well as the two-norm of the goal function.To verify the effect of different numbers of VSG on the inertia of the system, the node linked to the synchronous generator is supposed to have constant inertia and zero damping, and it is also presumed that it is not involved in system inertia optimization.Just the virtual inertia is optimized to demonstrate the advantages of doing so for both virtual inertia and damping.The damping of nodes 5, 6 and 8 is fixed and does not take part in optimization, and merely the virtual inertia is improved.Table 4 shows the virtual inertia and damping values of system nodes as well as the two-norm of the goal function.Table 5 that follows displays the objective function in the four cases.If only the synchronous generators at nodes 1, 2 and 3 provide inertia for the system when the small signal occurs at every node of the system, the two-norm of the state space matrix of the system is large, and the stability of the system is poor.This can be seen by analyzing and comparing the distribution of the virtual inertia and the damping of the VSG in four cases.When virtual inertia and damping are set up on more system nodes, as opposed to the other three cases under the same small signal, it is evident that the system stability will increase and the two-norm of its state space matrix will be less.Also, the system stability will improve and each VSG virtual inertia will decrease when more VSG are linked to the system.

Conclusion
In this paper, the influence of virtual inertia and damping allocation on the small-signal stability in the system is studied.To deal with the problem of small-signal stability in the weak inertia power system, an optimal allocation strategy of virtual inertia and damping is proposed to improve the small-signal stability.The objective of the proposed optimal strategy is to minimize the energy imbalance of the system under the small signal while optimizing the virtual inertia and damping with the constraint of the small-signal stability.The simulation results show that the proposed optimal allocation strategy of virtual inertia and damping can maximize the small-signal stability of the system within a certain constraint.
There is a coupling relationship between the renewable energy unit with VSG and the dynamics, the virtual inertia maybe reduces the damping ratio of the system under certain circumstances.Therefore, in future work, the optimal position matching of virtual inertia and damping in the system will be carried out.

Figure 3 .
Figure 3. Virtual inertia optimization results for all cases.

Figure 4 .
Figure 4. Damping optimization results for all cases.

Table 1 .
The optimized result of Case 1 All nodes except generator nodes provide are optimized Other system nodes are each connected to a VSG, while Nodes 1, 2 and 3 use synchronous generators.

Table 2
Only the virtual inertia of nodes 5, 6 and 8 are optimized System nodes 5, 6 and 8 are linked to VSG, while nodes 1, 2, and 3 use synchronous generators.

Table 3 .
The optimized result of Case 3 5.4.Case 4:Only the virtual inertia of nodes 5, 6 and 8 are optimized, and the damping set a constant System nodes 5, 6 and 8 are linked to VSG, while nodes 1, 2 and 3 use synchronous generators.

Table 4 .
The optimized result of Case 4