Large-scale stochastic power grid islanding operations by line switching and controlled load shedding

Abstract

Recently, there has been an increasing concern regarding the security and reliability of power systems due to the onerous consequences of cascading failures. Among many emergency control operations, controlled power grid islanding is a last resort yet powerful method to prevent large-scale blackouts. Islanding operations split the power grid into self-sufficient operational subnetworks and avoid cascading failures by isolating the failed elements of the power system into a non-operational island. In this paper, we consider a two-stage stochastic mixed-integer program to seek the optimal islanding operations under severe contingency states. Line switching and controlled load shedding are the main tools for the islanding operations and load shedding is considered as a measurement to gauge system’s inability to respond to disruption. The number of possible extreme contingencies grows exponentially as the size of the grid increases, and this results in a large-scale mixed-integer program, which is a computationally challenging problem to solve. We present an efficient decomposition method to solve this problem for large-scale power systems.

Appendix: Proof of Theorem 2

There are some quadratic and cubic terms in the constraints (3), (4a), and (4g) that need to be linearized. First, we replace (4a) by linear constraints (13a) and (13b).

$$\begin{aligned}&f_e^{(c)} - B_e(\theta _{i_e}^{(c)}-\theta _{j_e}^{(c)}) + F_ez_e^{(c)} \le F_e (1+d_e^{(c)} + d_{i_e}^{(c)} + d_{j_e}^{(c)}),\quad \forall e \end{aligned}$$

(13a)

$$\begin{aligned} -&f_e^{(c)} + B_e(\theta _{i_e}^{(c)}-\theta _{j_e}^{(c)}) + F_ez_e^{(c)} \le F_e (1+d_e^{(c)} + d_{i_e}^{(c)} + d_{j_e}^{(c)}),\quad \forall e \end{aligned}$$

(13b)

In order to linearize (3), define

$$\begin{aligned} u_{e,k}^{(c) l}&= (1- u_k^{(c)}) x_{i_ek},\\ u_{e,k'}^{(c) r}&= (1- u_{k'}^{(c)}) x_{j_ek'},\\ \bar{u}_{e,k,k'}^{(c) l, r}&= u_{e,k}^{(c) l} u_{e,k'}^{(c) r}. \end{aligned}$$

Then one can rewrite (3) as

$$\begin{aligned}&z_e^{(c)}=\sum _k \sum _{k'} \bar{u}_{e,k,k'}^{(c) l, r}. \end{aligned}$$

Similarly, to linearize (4g) we introduce following variables

$$\begin{aligned}&y_{ik}^{(c)} = x_{ik}(1 - u_k^{(c)})\\&v_{ik}^{(c)} = s_iy_{ik}^{(c)}. \end{aligned}$$

Then (4g) can be re-written as

$$\begin{aligned}&\sum _k \sum _{i\in V}v_{ik}^{(c)} \le \epsilon ^{(c)} \sum _k \sum _{i\in V}D_i y_{ik}^{(c)}. \end{aligned}$$

Using above realization, linear programming formulation of subproblem corresponding to decision x at contingency state c is presented as follows.

$$\begin{aligned} \begin{array}{lll} &{}Q(x,c)=\min _{f,p,s,u,z}\ \sum _{i\in V}C_i s_i^{(c)} \\ &{}\quad \quad s.t.~~ f_e^{(c)} - B_e(\theta _{i_e}^{(c)}-\theta _{j_e}^{(c)}) + F_ez_e^{(c)} \le F_e (1+d_e^{(c)} + d_{i_e}^{(c)} + d_{j_e}^{(c)}), ~\forall e ~~~~~~~~ &{}\mu _e^{(c) +} \\ &{}\quad \quad \quad -f_e^{(c)} + B_e(\theta _{i_e}^{(c)}-\theta _{j_e}^{(c)}) + F_ez_e^{(c)} \le F_e (1+d_e^{(c)} + d_{i_e}^{(c)} + d_{j_e}^{(c)}), ~\forall e ~~~~~~~~ &{}\mu _e^{(c) -}\\ &{}\quad \quad \quad \quad f_e^{(c)} -F_e z_e^{(c)}(1-d_e^{(c)})(1-d_{i_e}^{(c)})(1-d_{j_e}^{(c)}) \le 0, ~\forall e ~~~~~~~~~~~~~~~~~~~~~~~~ &{}\nu _e^{(c) +} \\ &{}\quad \quad \quad -f_e^{(c)} -F_e z_e^{(c)}(1-d_e^{(c)})(1-d_{i_e}^{(c)})(1-d_{j_e}^{(c)}) \le 0, ~\forall e ~~~~~~~~~~~~~~~~~~~~~~~~ &{}\nu _e^{(c) -} \\ &{}\quad \quad \quad \sum \limits _{g: i_g=i}p_g^{(c)}+\sum \limits _{e: j_e=i}f_e^{(c)} - \sum \limits _{e:i_e=i}f_e^{(c)} + s_i^{(c)} = D_i, \forall i ~~~~~~~~~~~~~~~~~~~~~~~ &{}\phi _i^{(c)} \\ &{}\quad \quad \quad 0\le p_g^{(c)}\le \overline{P_g}(1-d_{i_g}^{(c)})(1-d_g^{(c)}),~ \forall g ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\tau _g^{(c)}\\ &{}\quad \quad \quad 0\le s_i^{(c)}\le D_i,~ \forall i ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\kappa _i^{(c)}\\ &{}\quad \quad \quad z_e^{(c)} - \sum \limits _k \sum \limits _{k'} \bar{u}_{e,k,k'}^{(c) l, r} = 0, ~ \forall e ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\gamma _e^{(c)}\\ &{}\quad \quad \quad u_{e,k}^{(c) l} + u_k^{(c)} \le 1, ~\forall e,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\omega _{1,e,k}^{(c)}\\ &{}\quad \quad \quad u_{e,k}^{(c) l} \le x_{i_ek}, ~\forall e,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\omega _{2,e,k}^{(c)}\\ &{}\quad \quad \quad u_{e,k}^{(c) l} + u_k^{(c)} \ge x_{i_ek}, ~\forall e,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\omega _{3,e,k}^{(c)}\\ &{}\quad \quad \quad u_{e,k'}^{(c) r} + u_{k'}^{(c)} \le 1, ~\forall e,k' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\beta _{1,e,k'}^{(c)}\\ &{}\quad \quad \quad u_{e,k'}^{(c) r} \le x_{j_e{k'}}, ~\forall e,k' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\beta _{2,e,k'}^{(c)}\\ &{}\quad \quad \quad u_{e,k'}^{(c) r} + u_{k'}^{(c)} \ge x_{j_ek'}, ~\forall e,k' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\beta _{3,e,k'}^{(c)}\\ &{}\quad \quad \quad \bar{u}_{e,k,k'}^{(c) l, r} - u_{e,k}^{(c) l} \le 0, ~\forall e,k,k' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\zeta _{1,e,k,k'}^{(c)}\\ &{}\quad \quad \quad \bar{u}_{e,k,k'}^{(c) l, r} - u_{e,k'}^{(c) r} \le 0, ~\forall e,k,k' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\zeta _{2,e,k,k'}^{(c)}\\ &{}\quad \quad \quad u_{e,k}^{(c) l} + u_{e,k'}^{(c) r} -\bar{u}_{e,k,k'}^{(c) l, r} \le 1, ~\forall e,k,k' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\zeta _{3,e,k,k'}^{(c)}\\ &{}\quad \quad \quad \sum \limits _k \sum \limits _{i\in V} v_{ik}^{(c)} - \epsilon ^{(c)} \sum \limits _k \sum \limits _{i\in V} D_i y_{ik}^{(c)} \le 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\chi ^{(c)}\\ \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{lll} &{}\quad \quad \quad y_{ik}^{(c)} \le x_{ik}, ~\forall i,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\sigma _{1,i,k}^{(c)}\\ &{}\quad \quad \quad y_{ik}^{(c)} + u_k^{(c)}\le 1, ~\forall i,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\sigma _{2,i,k}^{(c)}\\ &{}\quad \quad \quad y_{ik}^{(c)} + u_k^{(c)}\ge x_{ik}, ~\forall i,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\sigma _{3,i,k}^{(c)}\\ &{}\quad \quad \quad v_{ik}^{(c)} - D_i y_{ik}^{(c)} \le 0, ~\forall i,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\rho _{1,i,k}^{(c)}\\ &{}\quad \quad \quad v_{ik}^{(c)} - s_i^{(c)} \le 0, ~\forall i,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\rho _{2,i,k}^{(c)}\\ &{}\quad \quad \quad s_i^{(c)} - v_{ik}^{(c)} + D_i y_{ik}^{(c)} \le D_i, ~\forall i,k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\rho _{3,i,k}^{(c)}\\ &{}\quad \quad \quad u_k^{(c)}d_e^{(c)} \le x_{i_e k} + x_{j_e k},\ \forall e,k~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\psi _{e,k}^{(c)}\\ &{}\quad \quad \quad u_k^{(c)}d_i^{(c)} \le x_{ik},\ \forall i,k~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\lambda _{i,k}^{(c)}\\ &{}\quad \quad \quad u_k^{(c)}d_{g}^{(c)} \le x_{i_g k},\ \forall g,k~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\delta _{g,k}^{(c)}\\ &{}\quad \quad \quad \sum \limits _{k=1}^K u_k^{(c)}=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &{}\eta ^{(c)} \end{array} \end{aligned}$$

The Greek letters on the right of each constraint show the corresponding dual variable. Therefore, the dual can be obtained.