Jacobian-Free Newton-Krylov Method for Risk Identification

Abstract

The identification of disruptive initial events on vulnerable components is crucial to the protection and reliability enhancement of critical infrastructure systems. Due to the evolving nature of system parameters and the complicated coupling relationship of different components, it has been a great challenge to identify the contingencies that could trigger cascading blackouts of power systems. This chapter aims to develop a generic approach for identifying the initial disruptive contingencies on vulnerable branches that can result in catastrophic cascading failures of power systems. Essentially, the problem of contingency identification is formulated in the mathematical framework of hybrid differential-algebraic system, and it can be solved by the Jacobian-Free Newton-Krylov (JFNK) method in order to circumvent the Jacobian matrix and relieve the computational burden. Moreover, an efficient numerical algorithm for contingency identification is developed to search for the disruptive disturbances that lead to catastrophic cascading failures with guaranteed convergence accuracy in theory. Finally, a case study is presented to demonstrate the efficacy of the proposed identification approach on the IEEE test systems.

3.6 Appendix

3.1.1 3.6.1 FACTS Devices

FACTS devices can greatly enhance the stability and transmission capability of power systems. As an effective FACTS device, TCSC has been widely installed to control the branch impedance and relieve system stresses. The dynamics of TCSC is described by a first order dynamical model [30]

$$\begin{aligned} T_{C,i}\frac{\text {d}{X}_{C,i}}{\text {d}t}=-X_{C,i}+X^{*}_{i}+u_i, \quad X_{\min ,i}\le X_{C,i}\le X_{\max ,i} \end{aligned}$$

(3.11)

where $X^{*}_{i}$ refers to its reference reactance of Branch i for the steady power flow. $X_{\min ,i}$ and $X_{\max ,i}$ are the lower and upper bounds of the branch reactance $X_{C,i}$ respectively and $u_i$ represents the supplementary control input, which is designed to stabilize the disturbed power system [31]. For simplicity, PID controller is adopted to regulate the power flow on each branch

$$\begin{aligned} u_i(t)=K_P\cdot e_i(t)+K_I\cdot \int \limits _{0}^{t}e_i(\tau )\text {d}\tau +K_D\cdot \frac{\text {d}e_i(t)}{\text {d}t} \end{aligned}$$

(3.12)

where $K_P$, $K_I$ and $K_D$ are tunable coefficients, and the error $e_i(t)$ is given by

$$ e_i(t)=\left\{ \begin{array}{ll} P^{*}_{e,i}-|P_{e,i}(t)|, &{} |P_{e,i}(t)|\ge P^{*}_{e,i}; \\ 0, &{} \hbox {otherwise.} \end{array} \right. $$

Here, $P^{*}_{e,i}$ and $P_{e,i}(t)$ denote the reference power flow and the actual power flow on Branch i, respectively. Note that TCSC fails to function when the transmission line is severed.

3.1.2 3.6.2 HVDC Links

HVDC links work as a protective barrier to prevent the propagation of cascading outages in practice, and it is normally composed of a transformer, a rectifier, a DC line and an inverter. Actually, the rectifier terminal can be regarded as a bus with real power consumption $P_{r}$, while the inverter terminal can be treated as a bus with real power generation $P_{i}$. The direct current from the rectifier to the inverter is computed as follows [32]

$$ I_d=\frac{3\sqrt{3}(\cos \alpha -\cos \gamma )}{\pi (R_{cr}+R_L-R_{ci})}, $$

where $\alpha \in [\pi /30,\pi /2]$ denotes the ignition delay angle of the rectifier, and $\gamma \in [\pi /12,\pi /9]$ represents the extinction advance angle of the inverter. $R_{cr}$ and $R_{ci}$ refer to the equivalent communicating resistances for the rectifier and inverter, respectively. Additionally, $R_L$ denotes the resistance of the DC transmission line. Thus the power consumption at the rectifier terminal is

$$\begin{aligned} P_r=\frac{3\sqrt{3}}{\pi }I_d\cos \alpha -R_{cr}I^2_d, \end{aligned}$$

(3.13)

and at the inverter terminal is

$$\begin{aligned} P_i=\frac{3\sqrt{3}}{\pi }I_d\cos \gamma -R_{ci}I^2_d=P_r-R_LI^2_d. \end{aligned}$$

(3.14)

Note that $P_r$ and $P_i$ keep unchanged when $\alpha$ and $\gamma$ are fixed.

3.1.3 3.6.3 Protective Relay

The protective relays are indispensable components in power systems protection and control. When the power flow exceeds the given threshold of the branch, the timer of circuit breaker starts to count down from the preset time [5]. Once the timer runs out of the preset time, the transmission line is severed by circuit breakers and its branch admittance becomes zero. Specifically, a step function is designed to reflect the physical characteristics of branch outage as follows

$$ g(P_{e,i},b_i)=\left\{ \begin{array}{ll} 0, &{} |P_{e,i}|>b_i\hbox { and }t_c>T; \\ 1, &{} \hbox {otherwise.} \end{array} \right. $$

where T is the preset time of the timer in protective relays, and $t_c$ denotes the counting time of the timer. In addition, $P_{e,i}$ denotes the power flow on Branch i with the threshold $b_i$.