Matrix perturbation theory
The calculation of Kξ and Hξ involves the change of state matrix, perturbation variable and initial eigensolution in MPT.
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(1)
Initial eigensolution
The state matrix is described as follows [27, 28]:
$$ {A}_0=\tilde{A}-{\tilde{B}\tilde{D}}^{-1}\tilde{C} $$
(12)
$$ \frac{d\Delta x}{dt}={A}_0\Delta x $$
(13)
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 non-state variables, A0 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 λ0, initial left eigenvector u0, initial right eigenvector v0 and initial damping ratio ξ0:
$$ \left.\begin{array}{c}{A}_0{v}_0={\lambda}_0{v}_0\\ {}{u_0}^T{A}_0={\lambda}_0{u_0}^T\end{array}\right\} $$
(14)
$$ \left\{\begin{array}{l}{\lambda}_0={\alpha}_0+j{\beta}_0\\ {}{\xi}_0=\frac{\hbox{-} {\alpha}_0}{\sqrt{{\alpha_0}^2+{\beta_0}^2}}\end{array}\right.\kern0.5em $$
(15)
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(2)
Perturbation variable and the change of state matrix
When the perturbation variable is the output power of generating units ΔPGi, the change of state matrix ΔA can be expressed as:
$$ A={A}_0+\Delta A $$
(16)
where A0 and A are the original and new state matrixes, respectively.
The change of state matrix ΔA is written as:
$$ \Delta A=\sum \limits_{i\in {S}_G}{A}_{si}\Delta {P}_{Gi} $$
(17)
$$ \left\{\begin{array}{c}\Delta {P}_G={\left[\Delta {P}_{G1},\Delta {P}_{G2},.\dots, \Delta {P}_{Gi},\dots, \right]}^T\\ {}{A}_s={\left[{A}_{s1},{A}_{s2},.\dots, {A}_{si},\dots, \right]}^T\end{array}\right. $$
(18)
where Asi is the sensitivity of the state matrix and is given as Asi = ΔAi/ΔPGi.
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(3)
Eigenvalue sensitivity
The eigensolution of the uncertain relation problem is described as [21, 22]:
$$ \kern0.1em \left({A}_0+\Delta A\right)\left({v}_{i0}+\Delta {v}_i\right)=\left({\lambda}_{i0}+\Delta {\lambda}_i\right)\left({v}_{i0}+\Delta {v}_i\right)\kern0.6em i\in {S}_L $$
(19)
where SL is the set of system state variables, λi0 is the ith initial eigenvalue, vi0 is the ith initial right eigenvector, Δvi is the change of the ith right eigenvector, and Δλi is the change of the ith eigenvalue. According to MPT, the eigenvalue λi and eigenvector vi of (19) are represented by [16]:
$$ \left\{\begin{array}{l}{\lambda}_i={\lambda}_{i0}+{\lambda}_{i1}+{\lambda}_{i2}\\ {}{v}_i={v}_{i0}+{v}_{i1}+{v}_{i2}\end{array}\right. $$
(20)
where i ∈ SL, λi1 and vi1 are the 1st-order perturbation values, and λi2 and vi2 are the 2nd-order perturbation values.
Substituting (19) into (20) and ignoring the perturbation values equal or higher than the 3rd-order yield:
$$ \left\{\begin{array}{l}{o}^0:{A}_0{v}_{i0}={\lambda}_{i0}{v}_{i0}\\ {}{o}^1:{A}_0{v}_{i1}+\Delta {Av}_{i0}={\lambda}_{i0}{v}_{i1}+{\lambda}_{i1}{v}_{i1}\\ {}{o}^2:{A}_0{v}_{i2}+\Delta {Av}_{i1}={\lambda}_{i0}{v}_{i2}+{\lambda}_{i1}{v}_{i1}+{\lambda}_{i2}{v}_{i0}\end{array}\right.\kern0.6em $$
(21)
The 2nd-order Taylor expansion of (21) is described as
$$ \left\{\begin{array}{l}{\lambda}_i={\lambda}_{i0}\left({P}_G^0\right)+{K}_{\lambda i}\Delta {P}_G+\frac{1}{2}\Delta {P_G}^T{H}_{\lambda i}\Delta {P}_G\\ {}{v}_i={v}_{i0}\left({P}_G^0\right)+{K}_{vi}\Delta {P}_G+\frac{1}{2}\Delta {P_G}^T{H}_{vi}\Delta {P}_G\end{array}\right.\kern0.85em $$
(22)
where i ∈ SL, Kλi and Kui are the gradient matrixes of eigenvalue and eigenvector, respectively, and Hλi is the Hessian matrix of eigenvalue. The gradient matrix and Hessian matrix can be written as:
$$ \left\{\begin{array}{l}{K}_{\lambda_i}=\left[\frac{\mathrm{\partial \Delta }{\lambda}_i}{\mathrm{\partial \Delta }{P}_{G1}},\frac{\mathrm{\partial \Delta }{\lambda}_i}{\mathrm{\partial \Delta }{P}_{G2}},\dots, \frac{\mathrm{\partial \Delta }{\lambda}_i}{\mathrm{\partial \Delta }{P}_{Gj}},.\dots \right]\\ {}{K}_{v_i}=\left[\frac{\mathrm{\partial \Delta }{v}_i}{\mathrm{\partial \Delta }{P}_{G1}},\frac{\mathrm{\partial \Delta }{v}_i}{\mathrm{\partial \Delta }{P}_{G2}},\dots, \frac{\mathrm{\partial \Delta }{v}_i}{\mathrm{\partial \Delta }{P}_{Gj}},.\dots \right]\end{array}\right. $$
(23)
$$ {H}_{\lambda_i}=\left[\begin{array}{ccc}\frac{\partial^2\Delta {\lambda}_i}{\mathrm{\partial \Delta }{P_{G1}}^2}& \dots & \frac{\partial^2\Delta {\lambda}_i}{\mathrm{\partial \Delta }{P}_{Gj}\mathrm{\partial \Delta }{P}_{G1}}\\ {}.\dots & .\dots & .\dots \\ {}\frac{\partial^2\Delta {\lambda}_i}{\mathrm{\partial \Delta }{P}_{G1}\mathrm{\partial \Delta }{P}_{Gj}}& \dots & \frac{\partial^2\Delta {\lambda}_i}{\mathrm{\partial \Delta }{P_{Gj}}^2}\end{array}\right] $$
(24)
where i ∈ SL, and j ∈ SG.
According to MPT and the normalized eigenvectors, the eigenfunctions are substituted into (15) and combined with (21) and (23). The 1st-order perturbations can be described as [22]:
$$ {\lambda}_{i1}={u}_{i0}^T\Delta {Av}_{i0} $$
(25)
$$ {v}_{i1}=\frac{\begin{array}{l}\sum \left(-{u}_{j0}^T\Delta {A}^T{v}_{i0}+{\lambda}_{i0}{u}_{j0}^T{v}_{i0}\right){v}_{j0}\\ {}j=1\\ {}j\ne i\end{array}}{\lambda_{j0}-{\lambda}_{i0}} $$
(26)
Combining (17), (25) and (26), Kλi, Kvi can be written as:
$$ {K}_{\lambda i}={u}_{i0}^T{A}_s{v}_{i0}=\frac{\mathrm{\partial \Delta }{\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gj}}+\mathrm{j}\frac{\mathrm{\partial \Delta }{\beta}_i}{\mathrm{\partial \Delta }{P}_{Gj}} $$
(27)
$$ {K}_{vi}=\frac{v_{i1}}{\Delta {P}_G} $$
(28)
where i ∈ SL, and Kλi can be described by the 1st-order eigenvalue sensitivity ∂Δα/∂ΔPG and ∂Δβ/∂ΔPG.
vi1 is given as:
$$ {v}_{i2}=\sum \limits_{j=1}^{S_l}{\alpha}_j{v}_{j0} $$
(29)
Combining (21) and (29), the 2nd-order eigenvalue can be expressed:
$$ {A}_0\sum \limits_{j=1}^{S_l}{\alpha_j}^{\ast }{v}_{j0}+\Delta {Av}_{i1}={\lambda}_{i0}\sum \limits_{j=1}^{S_l}{\alpha_j}^{\ast }{v}_{j0}+{\lambda}_{i0}{v}_{i1}+{\lambda}_{i2}{v}_{i0} $$
(30)
Equation (30) is multiplied by the left eigenvector ui0:
$$ \sum \limits_{j=1}^{S_l}{\alpha}_j{u}_{i0}{A}_0{v}_{j0}+{u}_{i0}\Delta {Av}_{i1}=\sum \limits_{j=1}^{S_l}{\alpha}_j{u}_{i0}{\lambda}_{i0}{v}_{j0}+{u}_{i0}{\lambda}_{i0}{v}_{i1}+{u}_{i0}{\lambda}_{i2}{v}_{i0} $$
(31)
Utilizing orthogonality relation of eigenvector, (31) is derived to:
$$ {\lambda}_{i2}={u}_{i0}^T\Delta {Av}_{i1}-{\lambda}_{i1}{u}_{i0}^T{v}_{i1} $$
(32)
Utilizing (17), (27) and (28), (32) can be express:
$$ {\displaystyle \begin{array}{l}{\lambda}_{i2}=\Delta {P_G}^T{u}_{i0}^T{A}_s{K}_{vi}\Delta {P}_G-\Delta {P_G}^T{K}_{\lambda i}{u}_{i0}^T{K}_{vi}\Delta {P}_G\\ {}\kern1em =\Delta {P_G}^T{h}_1\Delta {P}_G+\Delta {P_G}^T{h}_2\Delta {P}_G\end{array}} $$
(33)
where matrix h is symmetric and the Hessian matrix of eigenvalue Hλi are given as:
$$ {H}_{\lambda i}=h+{h}^T=\frac{\partial^2\Delta {\alpha}_i}{\mathrm{\partial \Delta }{P_G}^2}+j\frac{\partial^2\Delta {\beta}_i}{\mathrm{\partial \Delta }{P_G}^2} $$
(35)
$$ {\lambda}_{i2}=\Delta {P_G}^T{H}_{\lambda i}\Delta {P}_G $$
(36)
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(4)
The gradient matrix and the Hessian matrix of damping ratio
With the 2nd-order Taylor expansion of damping ratio:
$$ {\xi}_i={\xi}_{i0}+{K}_{\xi i}\Delta {P}_G+\frac{1}{2}\Delta {P_G}^T{H}_{\xi i}\Delta {P}_G $$
(37)
where ξi0 is the ith initial damping ratio. Damping ratio describes the system stability and is calculated by (18). The first and second-order sensitivities of damping ratio are derived by (27), (28) and (33) considering i ∈ SL, (k, j) ∈ SG as:
$$ {\displaystyle \begin{array}{l}{K}_{\xi_i}=\frac{\mathrm{\partial \Delta }{\xi}_i}{\mathrm{\partial \Delta }{P}_{Gj}}=\frac{1}{\sqrt{{\alpha_i}^2+{\beta_i}^2}}\frac{-{\beta_i}^2}{{\alpha_i}^2+{\beta_i}^2}\frac{\mathrm{\partial \Delta}\alpha }{\mathrm{\partial \Delta }{P}_{Gj}}\\ {}\kern2em +\frac{1}{\sqrt{{\alpha_i}^2+{\beta_i}^2}}\frac{\alpha_i{\beta}_i}{{\alpha_i}^2+{\beta_i}^2}\frac{\mathrm{\partial \Delta }{\beta}_i}{\mathrm{\partial \Delta }{P}_{Gj}}\kern1em \end{array}} $$
(38)
$$ {H}_{\xi_i}=\frac{\partial^2\Delta {\xi}_i}{\mathrm{\partial \Delta }{P}_{Gj}\mathrm{\partial \Delta }{P}_{Gj}}=\frac{\partial^2\Delta {\xi}_i}{\mathrm{\partial \Delta }{P}_{Gj}\mathrm{\partial \Delta }{P}_{Gj}}(1)+\frac{\partial^2\Delta {\xi}_i}{\mathrm{\partial \Delta }{P}_{Gj}\mathrm{\partial \Delta }{P}_{Gj}}(2) $$
(39)
$$ {\displaystyle \begin{array}{l}\frac{\partial^2\Delta \xi }{\mathrm{\partial \Delta }{P}_{Gk}\mathrm{\partial \Delta }{P}_{Gj}}(1)=\frac{-2{\beta_i}^{\ast }{\frac{\mathrm{\partial \Delta }{\beta}_i}{\mathrm{\partial \Delta }{P}_{Gj}}}^{\ast}\frac{\mathrm{\partial \Delta }{\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gk}}-{\beta}^{2\ast}\frac{\partial^2\Delta {\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gk}\mathrm{\partial \Delta }{P}_{Gj}}}{{\left({\alpha_i}^2+{\beta_i}^2\right)}^3}\\ {}{}^{\ast }{\left({\alpha_i}^2+{\beta_i}^2\right)}^{3/2}-\frac{-{\beta_i}^{2\ast }{\frac{\mathrm{\partial \Delta }{\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gk}}}^{\ast}\frac{3}{2}{\left({\alpha_i}^2+{\beta_i}^2\right)}^{1/2}}{{\left({\alpha_i}^2+{\beta_i}^2\right)}^3}\\ {}{}^{\ast}\frac{\left(2{\alpha}_i\frac{\mathrm{\partial \Delta }{\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gj}}+2{\beta}_i\frac{\mathrm{\partial \Delta }{\beta}_i}{\mathrm{\partial \Delta }{P}_{Gj}}\right)}{{\left({\alpha_i}^2+{\beta_i}^2\right)}^3}\kern0.75em \end{array}} $$
(40)
$$ {\displaystyle \begin{array}{l}\frac{\partial^2\Delta {\xi}_i}{\mathrm{\partial \Delta }{P}_{Gk}\mathrm{\partial \Delta }{P}_{Gj}}(2)=\frac{{\frac{\mathrm{\partial \Delta }{\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gj}}}^{\ast }{\beta_i}^{\ast}\frac{\mathrm{\partial \Delta }{\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gk}}+{\alpha}^{\ast }{\frac{\mathrm{\partial \Delta }{\beta}_i}{\mathrm{\partial \Delta }{P}_{Gj}}}^{\ast}\frac{\mathrm{\partial \Delta }{\beta}_{ii}}{\mathrm{\partial \Delta }{P}_{Gk}}}{{\left({\alpha_i}^2+{\beta_i}^2\right)}^{3/2}}\\ {}+\frac{\alpha_i{\beta}_i\frac{{\partial_i}^2\Delta {\beta}_i}{\mathrm{\partial \Delta }{P}_{Gk}\mathrm{\partial \Delta }{P}_{Gj}}}{{\left({\alpha_i}^2+{\beta_i}^2\right)}^{3/2}}-\frac{\alpha_i{\beta}_i{\frac{\mathrm{\partial \Delta }{\beta}_i}{\mathrm{\partial \Delta }{P}_{Gk}}}^{\ast}\frac{3}{2}{\left({\alpha_i}^2+{\beta_i}^2\right)}^{1/2}}{{\left({\alpha_i}^2+{\beta_i}^2\right)}^3}\\ {}{}^{\ast}\frac{\left(2{\alpha}_i\frac{\mathrm{\partial \Delta }{\alpha}_i}{\mathrm{\partial \Delta }{P}_{Gj}}+2{\beta}_i\frac{\mathrm{\partial \Delta }{\beta}_i}{\mathrm{\partial \Delta }{P}_{Gj}}\right)}{{\left({\alpha_i}^2+{\beta_i}^2\right)}^3}\ \end{array}} $$
(41)
To utilize (39), the expression of the Hessian matrix Hξi is:
$$ {H}_{\xi_i}=\left[\begin{array}{ccc}\frac{\partial^2{\xi}_i}{\partial {P_{G1}}^2}& \dots & \frac{\partial^2{\xi}_i}{\partial {P}_{Gj}\partial {P}_{G1}}\\ {}.\dots & .\dots & .\dots \\ {}\frac{\partial^2{\xi}_i}{\partial {P}_{G1}\partial {P}_{Gj}}& \dots & \frac{\partial^2{\xi}_i}{\partial {P_{Gj}}^2}\end{array}\right] $$
(42)
The Hξi and Kξi are deduced by the sensitivity of state matrix As and the initial eigenfunction. The sensitivity of state matrix As is deduced by the change of state matrix ΔA and perturbation variable ΔPG. Because system stability is estimated by the state matrix A0, 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 As are obtained through the perturbation variable and the change of state matrix ΔA in the proposed method. The computational process of Hξi and Kξi 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 A0 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 time-consuming 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 Newton-Raphson 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:
$$ \left[\begin{array}{c}\frac{\mathrm{\partial \Delta }V}{\mathrm{\partial \Delta }P}\\ {}\frac{\mathrm{\partial \Delta}\theta }{\mathrm{\partial \Delta }P}\end{array}\right]={Ja}^{-1}\frac{\mathrm{\partial \Delta }S}{\mathrm{\partial \Delta }P} $$
(43)
$$ {V}^{\prime }=V+\frac{\mathrm{\partial \Delta }V}{\mathrm{\partial \Delta }P}\Delta P $$
(44)
$$ {\theta}^{\prime }=\theta +\frac{\mathrm{\partial \Delta}\theta }{\mathrm{\partial \Delta }P}\Delta P $$
(45)
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 A0 and A in (16). The method avoids iterative computation of power flow, which greatly reduce required calculation.