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A matrixperturbationtheorybased optimal strategy for smallsignal stability analysis of largescale power grid
Protection and Control of Modern Power Systems volume 3, Article number: 34 (2018)
Abstract
In this paper, a sensitivity matrix based approach is proposed to improve the minimum damping ratio. The proposed method also avoids burdensome deviation calculations of damping ratio of largescale power grids when compared to the SmallSignalStability Constrained Optimal Power Flow (SSSCOPF) approach. This is achieved using the Matrix Perturbation Theory (MPT) to deal with the 2nd order sensitivity matrices, and the establishment of an optimal corrective control model to regulate the output power of generating units to improve the minimum damping ratio of power grids. Finally, simulation results on the IEEE 9bus, IEEE 39bus and a China 634bus systems show that the proposed approach can significantly reduce the burden of deviation calculation, while enhancing power system stability and ensuring calculation accuracy.
Introduction
Smallsignal stability of power systems is critical to the system security of power grids due to largescale connections of power grids [1, 2], rapid development of Ultra High Voltage Alternating Current (UHVAC) transmission [3,4,5], high penetration of renewable energy sources, etc [6,7,8,9].
Many reported work have shown that damping controllers can enhance smallsignal stability [10]. In practice, the redispatch of generator powers can also provide additional measure to ensure the smallsignal stability of a power grid. Therefore, SmallSignalStability Constrained Optimal Power Flow (SSSCOPF) has become a ‘hot’ research topic. This is because SSSCOPF can achieve an appropriate security level while considering economic objectives and technical constraints [11].
SSSCOPF has been reported for improvement of smallsignal stability, but the existing work with high calculation precision is timeconsuming due to large computation requirement when calculating the eigenvalue [10,11,12,13,14,15]. Eigenvalue sensitivity based Interior Point Methods (IPM) have been proposed to improve the power transfer capability with smallsignal stability constraints [10, 11], which is an SSSCOPF based Numerical Eigenvalue Sensitivity (NESSSSCOPF). However, these methods have to deal with the burden of heavy computation from the repetitive calculation of eigenvalues, and at the same time, high precision cannot be guaranteed due to the neglected highorder terms of the smallsignal stability constraints. In [12], this expectedsecuritycost optimal power flow with smallsignal stability constraints is addressed by the closedform formula with extra computational burden to guarantee the calculation accuracy. Approximatesingularvaluesensitivitybased IPM is used to coordinate oscillation control in electricity market [13]. For the two methods proposed in [12, 13], both are very timeconsuming for calculating the Hessian of smallsignal stability constraints. An optimization method based on the sequential quadratic programming algorithm with Gradient Sampling [14] can ensure the global and efficient convergence of SSSCOPF [15]. However, the computation with high precision is again timeconsuming due to the complex in formula derivation and sampling process.
Matrix Perturbation Theory plays an important role in describing the changes in eigensolution [16,17,18] and has been applied for structural dynamics [19], automation control [20], design of power parameters [21], error analysis of power systems [22, 23], etc. In a power system, the sensitivity obtained by MPT can evaluate the key parameters to smallsignal stability and can also be applied for power dispatch and parameter design [24]. In the case of small fluctuations in the operating parameters, the 1storder sensitivity of MPT can obtain the results with sufficient precision [25]. Power systems show nonlinear characteristics, while larger disturbance causes abrupt change of power system parameters [26]. The 2ndorder control strategy tolerates a larger disturbance which can be regarded as a parameter disturbance of more than 20% variation [24]. Therefore, the 2ndorder sensitivity of system parameters with high precision becomes a concern.
In this paper, Matrix Perturbation Theory based Optimal Strategy (MPTOS), a hybrid method combining IPM and the 2ndorder MPT is proposed to deal with the sensitivity of damping ratio for improving of the smallsignal stability. The optimal calibration model is established to improve system stability by IPM and the MPT is used to deal with the Hessian Matrix of the damping ratio. Since the Hessian Matrix calculation only needs the original state matrix and perturbation variables in MPT, deviation calculation of the 2ndorder sensitivities is not needed.
The remainder of this paper is organized as follows: In Section 2, the control model of smallsignal stability and the partial derivative of damping ratio in SSSCOPF are addressed. A MPT based computing method for the sensitivity of damping ratio is introduced in Section 3. In Section 4, the proposed approach is validated on three systems, and Section 5 draws the conclusion.
Optimal model of smallsignal stability
The optimal model
The corrective control model can be described as follows:

1)
The objective function.
The minimum change of the output powers of the generators is considered as the objective function as
where i ∈ S_{Gi} and S_{Gi} is the set of generators. P_{Gi} is the active power output of the generator and \( {P}_{Gi}^0 \) is its initial active power output of the generator. ΔP_{Gi} is the regulating output power of the generator given as \( \Delta {P}_{Gi}={P}_{Gi}{P}_{Gi}^0 \).

2)
The equality constraints.
The power balance equation is:
where i ∈ S_{n} and S_{n} is the set of nodes. P_{Di} is the active load, Q_{Ri} is the reactive power output of the generator and Q_{Di} is the reactive load; K_{λi} and θ_{i} are the voltage amplitude and phase angles respectively. Y_{ij} and α_{ij} are the amplitude and phase angle of the admittance matrix, respectively.

3)
Inequality constraints.

(1)
Operation constraints are:

(1)
where S_{R} is the set of reactive power supplies. \( {P}_{Gi}^{\mathrm{min}} \) and \( {P}_{Gi}^{\mathrm{max}} \) are the upper and lower limits of the active power output of the generator, respectively. \( {Q}_{Ri}^{\mathrm{min}} \) and \( {Q}_{Ri}^{\mathrm{max}} \) are the respective upper and lower limits of the reactive power output of the generator, \( {V}_i^{\mathrm{min}} \) and \( {V}_i^{\mathrm{max}} \) are the upper and lower limits of the voltage, respectively.

(2)
Stability constraint is:
where ξ is the system damping ratio and \( \underline{\xi} \) is its lower limit. The index \( \underline{\xi} \) is to guarantee the system to be smallsignal stable, and for \( \xi <\underline{\xi} \), the system has smallsignal stability problem.
After the improvement of system stability, ξ = ξ_{0} + Δξ where ξ_{0} is the initial damping ratio, and Δξ can be written as:
where K_{ξ} and H_{ξ} are the respective gradient matrix and Hessian matrix of the damping ratio and active power of the generators.
The smallsignal stability corrective control model of (1)–(5) can be described as:
where x_{0} is the initial state variable, f(x) is the objective function, h(x) and g(x) are the equality and inequality constraints, respectively. \( \overline{g} \) and \( \underline{g} \) are the upper and lower limits, respectively.
According to IPM, (8) can be written as \( g\left({x}_0\right){l}_0\underline{g}=0 \) and \( g\left({x}_0\right)+{l}_0\overline{g}=0 \), where l_{0} is the slack variables with l_{0} > 0.
Equations (6)–(8) can be written as Lagrange function as:
Firstorder optimality conditions (Karush Kuhn Tucker, KKT) are:
where y and z are the respective Lagrange multipliers, μ is the blockage parameter, L and Z are the diagonal matrix of l and z, respectively, and e is the unit column vector. The damping ratio constraints part of ∇_{x}g(x) can be written as K_{ξ}.
Using Newton’s method to solve (10), the correction equation is obtained:
where \( H={\nabla}_x^2f{\nabla}_x^2 hy{\nabla}_x^2 gz{\nabla}_x{gL}^{1}Z{\nabla}_x^Tg \), and corrections are solved by iteration in (11). The damping ratio constraints part of \( {\nabla}_x^2g(x) \) can be represent by H_{ξ}. Section 3 describes the detailed derivation process of K_{ξ} and H_{ξ}.
The eigenvalue constraint has been considered in many studies, but the damping ratio constraint has been ignored. In this paper, the models of (1)–(5) are established in a similar way by IPM. However, derivative calculations of damping ratio are complex due to the implicit relationship between the generator active power and the damping ratio, which seriously affects the practicability of SSSCOPF. The existing method [12] spends significant computation time on heavy eigenvalue computation of the QR method, which avoids derivative calculations. Therefore, it is necessary to effectively simplify computation in practical application of SSSCOPF. Compared to the existing methods [12], the K_{ξ} and H_{ξ} are calculated by MPT in the presented method, which avoids demanding derivative calculations and reduces computing time.
Corrective control process
In Fig. 1, the steps of the smallsignal stability optimal strategy are as follows.

1)
Small signal stability analysis.
The system stability is judged by smallsignal stability analysis.

2)
Damping ratio sensitivity.
The eigenvalue and eigenvector sensitivities are calculated by MPT. The damping ratio sensitivity is deduced by eigenvalue and eigenvector sensitivities.

3)
The corrective control model of power dispatch.
The corrective control model is established to improve the system stability by regulating the output power of generating units.

4)
Judgement of system stability.
The corrective control results are checked by small signal stability analysis. If the results meet the system stability requirement, the calculation is stopped, otherwise, the corrective control model is reestablished to improve the system stability.
Methods: damping sensitivity matrixes
Matrix perturbation theory
The calculation of K_{ξ} and H_{ξ} involves the change of state matrix, perturbation variable and initial eigensolution in MPT.

(1)
Initial eigensolution
The state matrix is described as follows [27, 28]:
where \( \tilde{A} \) and \( \tilde{B} \) are the coefficient matrixes of the state variables, \( \tilde{C} \) and \( \tilde{D} \) are the coefficient matrixes of the nonstate variables, A_{0} is the initial system state matrix, and Δx is the small change of the system state variables. The initial eigensolution contains the initial matrix eigenvalue λ_{0}, initial left eigenvector u_{0}, initial right eigenvector v_{0} and initial damping ratio ξ_{0}:

(2)
Perturbation variable and the change of state matrix
When the perturbation variable is the output power of generating units ΔP_{Gi}, the change of state matrix ΔA can be expressed as:
where A_{0} and A are the original and new state matrixes, respectively.
The change of state matrix ΔA is written as:
where A_{si} is the sensitivity of the state matrix and is given as A_{si} = ΔA_{i}/ΔP_{Gi}.

(3)
Eigenvalue sensitivity
The eigensolution of the uncertain relation problem is described as [21, 22]:
where S_{L} is the set of system state variables, λ_{i0} is the ith initial eigenvalue, v_{i0} is the ith initial right eigenvector, Δv_{i} is the change of the ith right eigenvector, and Δλ_{i} is the change of the ith eigenvalue. According to MPT, the eigenvalue λ_{i} and eigenvector v_{i} of (19) are represented by [16]:
where i ∈ S_{L}, λ_{i1} and v_{i1} are the 1storder perturbation values, and λ_{i2} and v_{i2} are the 2ndorder perturbation values.
Substituting (19) into (20) and ignoring the perturbation values equal or higher than the 3rdorder yield:
The 2ndorder Taylor expansion of (21) is described as
where i ∈ S_{L}, K_{λi} and K_{ui} are the gradient matrixes of eigenvalue and eigenvector, respectively, and H_{λi} is the Hessian matrix of eigenvalue. The gradient matrix and Hessian matrix can be written as:
where i ∈ S_{L}, and j ∈ S_{G}.
According to MPT and the normalized eigenvectors, the eigenfunctions are substituted into (15) and combined with (21) and (23). The 1storder perturbations can be described as [22]:
Combining (17), (25) and (26), K_{λi}, K_{vi} can be written as:
where i ∈ S_{L}, and K_{λi} can be described by the 1storder eigenvalue sensitivity ∂Δα/∂ΔP_{G} and ∂Δβ/∂ΔP_{G}.
v_{i1} is given as:
Combining (21) and (29), the 2ndorder eigenvalue can be expressed:
Equation (30) is multiplied by the left eigenvector u_{i0}:
Utilizing orthogonality relation of eigenvector, (31) is derived to:
Utilizing (17), (27) and (28), (32) can be express:
where matrix h is symmetric and the Hessian matrix of eigenvalue H_{λi} are given as:

(4)
The gradient matrix and the Hessian matrix of damping ratio
With the 2ndorder Taylor expansion of damping ratio:
where ξ_{i0} is the ith initial damping ratio. Damping ratio describes the system stability and is calculated by (18). The first and secondorder sensitivities of damping ratio are derived by (27), (28) and (33) considering i ∈ S_{L}, (k, j) ∈ S_{G} as:
To utilize (39), the expression of the Hessian matrix H_{ξi} is:
The H_{ξi} and K_{ξi} are deduced by the sensitivity of state matrix A_{s} and the initial eigenfunction. The sensitivity of state matrix A_{s} is deduced by the change of state matrix ΔA and perturbation variable ΔP_{G}. Because system stability is estimated by the state matrix A_{0}, the initial eigenfunction is known.
The existing approach [10] contains heavy eigenvalue computation used in the QR method, and thus, the computation speed of QR method increases with the size of state matrix. The QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The basic idea is to perform QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate. The sensitivity matrices A_{s} are obtained through the perturbation variable and the change of state matrix ΔA in the proposed method. The computational process of H_{ξi} and K_{ξi} only contains simple multiplications and divisions, and thus simplifies the calculation procedure. Compared with the existing methods [12, 13, 15], burdensome deviation calculations of damping ratio are avoided in the presented method, and the calculation steps are direct and explicit. Thus, the method has higher efficiency for high order state matrix calculation.
In order to obtain the eigenvalue sensitivities in (17), (25), (32), ΔA needs to be calculated by A_{0} and A, while system voltage obtained by active power perturbation can constitute A by (13) in perturbation process. In process of active power perturbation, node voltage is obtained by timeconsuming power flow calculations, whereas the system voltage is calculated by power sensitivity so as to avoid repeated power flow calculations.
The power sensitivity application
The system voltage can describe the system state and the state matrix is established by the system state. The voltage can be calculated by power sensitivity so that the iteration time is saved in power flow calculation.
Ja is the Jacobian matrix in the NewtonRaphson method [29, 30] and the derivatives of voltage can be derived by the Jacobian matrix Ja from (5). The derivatives of the voltage amplitude and phase angle can be written as:
where ∂ΔV/∂ΔP and ∂Δθ/∂ΔP are the voltage amplitude sensitivity and phase angle sensitivity, respectively. Δθ and ΔV are the changes of system voltage amplitude and phase angle, respectively. ΔS and ΔP are the respective power changes in the system and generator, and ∂ΔS/∂ΔP = [0, ....0, 1, 0, ...0]^{T}.
Utilizing the voltage sensitivities in (44) and (45), new state matrix A can be obtained by (13). Then, ΔA required in the above section can be calculated by A_{0} and A in (16). The method avoids iterative computation of power flow, which greatly reduce required calculation.
Results and discussion
Following examples are built on the MATLAB platform. The WSCC 3machine 9bus, New England 10machine 39bus and the China 124machine 634bus systems are used for illustrating the proposed technique. The convergence precision is 10^{− 4} and damping ratio limit \( \underline{\xi} \) is 0.03.
The 9bus 3machine system
In modeling the system, the generator model is a 6thorder model, the excitation model is a selfshunt static model and the load is modelled as constant resistors. Through the smallsignal stability analysis, Tables 1 and 2 show the initial operation conditions.
The oscillation frequencies of multiple modes are in the range of 0–2.4 Hz in the original system with only one damping ratio state being less than 0.03 at 0.0175. Thus, the system is prone to low frequency oscillation after disturbance.
The above system stability is improved by the proposed methodology. Tables 3, 4, and 5 show the optimal results (\( \underline{\xi} \)= 0.03). The index ξ_{err} is to evaluate the advantages of the 2ndorder damping, and is defined as the relative error of damping as:
where ξ_{t} is the minimum true value of damping after system calibration, and ξ_{r} is the result of correction strategy.
As can be seen from Table 3, the minimum damping ratio is 0.033, which is greater than ξ_{limit} of 0.03. The system can suppress oscillation in a short time after stability correction. For (46), the relative error is 3.125% and the numerical error is 0.001 in minimum damping ratio, when considering the Hessian matrix of damping ratio in the model. However, the corresponding relative error is 6.25% and the numerical error is 0.002 when considering the gradient matrix of damping ratio. Table 4 shows the test results of NESSSSCOPF and MPTOS, indicating that MPTOS is faster than NESSSSCOPF in computation time.
Table 5 shows the relationship between the damping ratio and the active power adjustment. The increase in damping ratio will increase the generator active output adjustment. The active power adjustment of the 1storder sensitivity is higher than that of the 2ndorder sensitivity. This is due to the control strategy only considering the damping ratio gradient matrix, which has lower accuracy than the Hessian matrix. Similarly, the 1storder relative error is always bigger than that of the 2ndorder in minimum damping ratio.
The 39bus 10machine system
In the system model, the generators also adopt the 6thorder model, the excitation system is a faststatic excitation model and the constant resistance model is also used.
From smallsignal stability analysis, the damping ratios below 0.03 contain 0.0281, 0.0257 and 0.0153 as shown in Table 6.
The system may become unstable due to insufficient damping. Generators have the ability to provide additional measure to the system to maintain the smallsignal stability. The 39bus system is adjusted by NESSSSCOPF and MPTOS respectively, and Tables 7, 8, 9, 10 and 11 describes the adjustment results.
The 1storder ξ_{err} is 6.7% and the 2ndorder ξ_{err} is 0% according to (46), and active power adjustment of the 2ndorder sensitivity is smaller than that of the 1storder one. Comparing the two methods, the NESSSSCOPF is more time consuming than the MPTOS shown in Table 9. The eigenfunction problem is solved by QR algorithm with significant computation.
In Fig. 2, the rotor frequencies fluctuate severely at 10s, and its maximum frequency deviation is 0.1884 Hz. This is due to the small damping ratio of the initial system and thus the system cannot quickly recover to the initial state. By contrast, Fig. 3 shows the rotor frequency deviations after a 10% load disturbance. System stability has been improved by the two redispatch plans, one has the 1storder damping sensitivity and the other 2ndorder damping sensitivity. At 20s, rotor frequency deviation with the 1storder damping sensitivity is smaller than that with the 2ndorder damping sensitivity. Compared with Fig. 2, the fluctuation of rotor frequency is smaller in Fig. 3. The maximum frequency deviation is 0.0328 Hz in Fig. 3a compared to 0.0358 Hz in Fig. 3b. The damping ratio of the 1storder sensitivity is higher than that of the 2ndorder sensitivity, resulting in reduced rotor frequency deviation of the 1storder sensitivity compared to that of the 2ndorder sensitivity.
From Tables 7, 8, 10 and 11, the active power adjustment of generators and damping ratio variable Δξ gradually decline with the increase of the order of sensitivity. The main reason for this is that high order sensitivity allows more accurate control of damping ratio.
From Tables 10 and 11, it is seen that the active power adjustment results satisfy the smallsignal stability constraints by MPTOS. In addition, the adjustment results with the 2ndorder sensitivity is close to actual adjustment results in the control model, while damping ratio of the 1storder sensitivity is higher than that of the 2ndorder sensitivity. This means that, the system can provide more damping ratio, its stability is superior to the original state, and it can suppress rotor frequency fluctuation in a shorter time.
The 634bus 124machine system
The actual power grid in China is taken as an example, which has 634 nodes, 532 lines, 124 generators, 190 reactive power compensation points and 879 transformers. In the modelled system, the generators use the 6thorder model, the load is again modelled as constant load and the excitation model is the faststatic excitation.
The initial operating condition in Table 12 shows unstable modes with damping ratio less than the threshold of 0.03. Thus, if the system is subject to small disturbance, system is prone to oscillation.
With the optimal model in Section 2, the stability of the system is improved by the output power of generating units.
Comparing Tables 4, 9 and 13, it can be seen that MPTOS is faster than NESSSSCOPF in computation time while increasing the system scale leads to increased difference of the total CPU time between the two. It can be concluded that MPTOS has higher efficiency for high order state matrix calculation than NESSSSCOPF.
After system damping ratio improvement, ξ_{r} is 0.034. As can be seen from Table 14, the minimum damping ratio is 0.0372 when the optimal strategy considers the 1storder sensitivity. However, the minimum damping ratio is 0.0350 when the 2ndorder sensitivity is considered. From (46), ξ_{err} is 8.6% with the 1storder sensitivity and is 2.8% with the 2ndorder sensitivity. The active power adjustment with 2ndorder sensitivity is smaller than that with the 1storder sensitivity. This is because high order sensitivity can allow a more accurate control of damping ratio. It can be seen from the above table that, compared to the 1storder sensitivity, the 2ndorder sensitivity more accurately characterizes damping ratio state, leading to better precision.
Conclusion
This paper has showed that optimization strategy with constraints for smallsignal stability can improve system stability by using methods from Matrix Perturbation Theory (MPT). First, a small signal analysis is used to identify the degree of influence of unstable modes on the system. The desired sensitivity matrix is then calculated through MPT, which can describe damping ratio constraints. In the optimized corrective control, a series of comprehensive restrictions are proposed to enhance the system damping ratio while meeting the normal operation requirements.
The strategy presented in this paper does not require burdensome deviation calculation to describe the 2ndorder damping ratio sensitivity. The desired sensitivity can be obtained by perturbation variable and the change of the state matrix. The solving steps are direct and explicit, and the calculation of high order sensitivity matrix is simplified. Furthermore, the active power adjustment with the 1storder sensitivity is higher than that with the 2ndorder sensitivity in the optimization strategy. Since the accuracy of the gradient matrix is lower than that of the Hessian matrix, high order sensitivity can gain higher accuracy in controlling damping ratio and active power adjustment. Compared to other methods, the proposed MPT based optimal strategy can avoid the deviation calculation and reduce computation time, while ensuring calculation accuracy.
Abbreviations
 IPM:

Interior Point Method
 MPT:

Matrix Perturbation Theory
 SSCOPF:

SmallSignalStability Constrained Optimal Power Flow
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Funding
This work was supported by the National Natural Science Foundation of China (Grant No.51577085).
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YY conceived and designed the study. YY and JZ performed the experiments and simulations. YY, JZ and HL wrote the paper. YY, JZ, HL, ZQ, JD and JQ reviewed and edited the manuscript. All authors read and approve the manuscript.
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Yang, Y., Zhao, J., Liu, H. et al. A matrixperturbationtheorybased optimal strategy for smallsignal stability analysis of largescale power grid. Prot Control Mod Power Syst 3, 34 (2018). https://doi.org/10.1186/s416010180107z
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Keywords
 Matrix perturbation theory
 2nd order sensitivity
 Optimal strategy
 Smallsignal stability