Mixed-integer exponential conic optimization for reliability enhancement of power distribution systems

Abstract

This paper develops an optimization model for determining the placement of switches, tie lines, and underground cables in order to enhance the reliability of an electric power distribution system. A central novelty in the model is the inclusion of nodal reliability constraints, which consider network topology and are important in practice. The model can be reformulated either as a mixed-integer exponential conic optimization problem or as a mixed-integer linear program. We demonstrate both theoretically and empirically that the judicious application of partial linearization is key to rendering a practically tractable formulation. Computational studies indicate that realistic instances can indeed be solved in a reasonable amount of time on standard hardware.

Appendices

Appendix A: Linearization of constraints (10) and (12)–(15)

In this section, we present a similar linearization to constraints (10), (12), (13), and (14), (15). Particularly, for constraint (10), we have (constraints (47)–(49) are written $\forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}$):

$$\begin{aligned} \tau _{n,l}&\ge \overline{\lambda }_l \overline{R}_l \Bigg [1-x^{RC}_{1,l}-x^{M}_{1,l} - \sum _{s=1}^2\sum _{j \in \Omega _{n,l}} \Big (x^{RC}_{s,j} + x^{M}_{s,j} \Big ) \Bigg ] \nonumber \\&\quad + (\underline{\lambda }_l \underline{R}_l-\overline{\lambda }_l \overline{R}_l)z^2_l \qquad \forall n \in \mathcal {N}^U_l, \forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}, \end{aligned}$$

(46)

$$\begin{aligned}&\qquad (1-M_f)u_l \le z^2_{l} \le u_l, \end{aligned}$$

(47)

$$\begin{aligned} z^2_{l}&\ge \Bigg [1-x^{RC}_{1,l}-x^{M}_{1,l} - \sum _{s=1}^2\sum _{j \in \Omega _{n,l}} \Big (x^{RC}_{s,j} + x^{M}_{s,j} \Big ) \Bigg ] - (1-u_l), \end{aligned}$$

(48)

$$\begin{aligned} z^2_{l}&\le \Bigg [1-x^{RC}_{1,l}-x^{M}_{1,l} - \sum _{s=1}^2\sum _{j \in \Omega _{n,l}} \Big (x^{RC}_{s,j} + x^{M}_{s,j} \Big ) \Bigg ] - \nonumber \\&\qquad (1-M_f)(1-u_l). \end{aligned}$$

(49)

For constraint (12) we have the following set of linear constraints (constraint (50) is written $\forall n \in \mathcal {N}^D_l, \forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}$ and constraints (51)–(53) are written $\forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}$):

$$\begin{aligned}&\tau _{n,l} \ge \overline{\lambda }_l T^M \Bigg [1-x^{RC}_{2,l}-\sum _{s=1}^2\sum _{j \in \Omega _{n,l}} x^{RC}_{s,j} \Bigg ] + (\underline{\lambda }_l -\overline{\lambda }_l )T^Mz^3_l, \end{aligned}$$

(50)

$$\begin{aligned}&(1-M_f)u_l \le z^3_{l} \le u_l, \end{aligned}$$

(51)

$$\begin{aligned}&z^3_{l} \ge \Bigg [1-x^{RC}_{2,l}-\sum _{s=1}^2\sum _{j \in \Omega _{n,l}} x^{RC}_{s,j} \Bigg ] - (1-u_l) , \end{aligned}$$

(52)

$$\begin{aligned}&z^3_{l} \le \Bigg [1-x^{RC}_{2,l}-\sum _{s=1}^2\sum _{j \in \Omega _{n,l}} x^{RC}_{s,j} \Bigg ]- (1-M_f)(1-u_l). \end{aligned}$$

(53)

Similarly, constraint (13) is linearized as follows (Constraint (54) is written $\forall n \in \mathcal {N}^U_l, \forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}$ and constraints (55)–(57) are written $\forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}$):

$$\begin{aligned}&\tau _{n,l} \ge \overline{\lambda }_l T^M \Bigg [1- \sum _{r \in \Gamma _f} x^{T,RC}_{r} \Bigg ] + (\underline{\lambda }_l -\overline{\lambda }_l )T^Mz^4_l, \end{aligned}$$

(54)

$$\begin{aligned}&(1-|\Gamma _f|)u_l \le z^4_{l} \le u_l, \end{aligned}$$

(55)

$$\begin{aligned}&z^4_{l} \ge \Bigg [1- \sum _{r \in \Gamma _f} x^{T,RC}_{r} \Bigg ] - (1-u_l), \end{aligned}$$

(56)

$$\begin{aligned}&z^4_{l} \le \Bigg [1- \sum _{r \in \Gamma _f} x^{T,RC}_{r} \Bigg ]- (1-|\Gamma _f|)(1-u_l). \end{aligned}$$

(57)

where $\Gamma _f$ is an upper bound for the number of potential tie switches connected to feeder f.

Finally, constraints (58)–(61) correspond to the linear reformulation of constraint (14) and constraints (62)–(65) linearize constraint (15) (constraint (58) and (62) are written $\forall n \in \mathcal {N}^D_l, \forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}$ and the rest are written $\forall l \in \mathcal {L}_f, \forall f \in \mathcal {F}$):

$$\begin{aligned} \tau _{n,l}&\ge \overline{\lambda }_l \overline{R}_l \Bigg [1- x^{RC}_{2,l}-x^{M}_{2,l} - \sum _{s=1}^2\sum _{j \in \Omega _{n,l}} (x^{RC}_{s,j}+x^{M}_{s,j}) \Bigg ] \nonumber \\&\quad + (\underline{\lambda }_l \underline{R}_l-\overline{\lambda }_l \overline{R}_l)z^5_l, \end{aligned}$$

(58)

$$\begin{aligned}&\qquad (1-M_f)u_l \le z^5_{l} \le u_l, \end{aligned}$$

(59)

$$\begin{aligned} z^5_{l}&\ge \Bigg [1- x^{RC}_{2,l}-x^{M}_{2,l} - \sum _{s=1}^2\sum _{j \in \Omega _{n,l}} (x^{RC}_{s,j}+x^{M}_{s,j}) \Bigg ] - (1-u_l), \end{aligned}$$

(60)

$$\begin{aligned} z^5_{l}&\le \Bigg [1- x^{RC}_{2,l}-x^{M}_{2,l} - \sum _{s=1}^2\sum _{j \in \Omega _{n,l}} (x^{RC}_{s,j}+x^{M}_{s,j}) \Bigg ] - \nonumber \\ {}&\qquad (1-M_f)(1-u_l), \end{aligned}$$

(61)

$$\begin{aligned} \tau _{n,l}&\ge \overline{\lambda }_l \overline{R}_l \Bigg [1 - \sum _{r \in \Gamma _f} (x^{T,RC}_{r} +x^{T,M}_{r} ) \Bigg ] + (\underline{\lambda }_l \underline{R}_l -\overline{\lambda }_l\overline{R}_l )z^6_l, \end{aligned}$$

(62)

$$\begin{aligned}&\qquad (1-|\Gamma _f|)u_l \le z^6_{l} \le u_l, \end{aligned}$$

(63)

$$\begin{aligned} z^6_{l}&\ge \Bigg [1 - \sum _{r \in \Gamma _f} (x^{T,RC}_{r} +x^{T,M}_{r} ) \Bigg ] - (1-u_l), \end{aligned}$$

(64)

$$\begin{aligned} z^6_{l}&\le \Bigg [1 - \sum _{r \in \Gamma _f} (x^{T,RC}_{r} +x^{T,M}_{r} ) \Bigg ] - (1-|\Gamma _f|)(1-u_l). \end{aligned}$$

(65)

Appendix B: Linearization of constraint (22)

Similar to the linear reformulation (30)–(32) corresponding to constraint (21), we present a linear reformulation for nodal reliability constraint (29). Let $\hat{d}_r=1-\overline{\mu }_r$ and $\hat{h}_r=1-{\underline{\mu }}_r$. Also, let $\hat{\delta }_{j,n}(i)$ be the jth subset of $\mathcal {L}^D_n \cup \mathcal {L}_{f'}$ with i members, $\hat{D}=\mathcal {L}^D_n \cup \mathcal {L}_{f'}$, and $\hat{T}_i = {\hat{D} \atopwithdelims ()i}$. Further, let variable $\hat{U}^{(i)}_{j,n} \in [0,1]$ represent the value of $(\hat{d}_r\overline{t}_r + \hat{h}_r \underline{t}_r)\prod _{l \in \delta _{j,n}(i)} u_l$ corresponding to the jth term of polynomial with degree $i+1$, where $\hat{U}^{(i)}_{j,n}$ takes 0, $\hat{d}_r$, or $\hat{h}_r$. The MILP reformulation of constraint (29) is obtained through constraints (66) to (69) (constraints (67)–(69) are written $\forall l \in \hat{\delta }_{j,n}(i), \forall j \in \{1,\ldots , \hat{T}_i\}, \forall i \in \{1,\ldots , \hat{D}\}, n \in \mathcal {N}^*_f, \forall (f,r,f') \in \mathcal {V}$):

$$\begin{aligned}&\hat{v}_0 (\hat{d}_r\overline{t}_r + \hat{h}_r \underline{t}_r) + \sum _{i=1}^{\hat{D}} \sum _{j = 1}^{\hat{T}_i} \hat{v}^{(i)}_{j,n}\hat{U}^{(i)}_{j,n} \ge \phi _n p_{n,r} \nonumber \\&\qquad \qquad \qquad \qquad \forall n \in \mathcal {N}^*_f, \forall (f,r,f') \in \mathcal {V}, \end{aligned}$$

(66)

$$\begin{aligned}&\hat{U}^{(i)}_{j,n} \le \overline{u}_l, \qquad \qquad \quad \end{aligned}$$

(67)

$$\begin{aligned}&\hat{U}^{(i)}_{j,n} \le \hat{d}_r\overline{t}_r + \hat{h}_r \underline{t}_r, \qquad \qquad \quad \end{aligned}$$

(68)

$$\begin{aligned}&\hat{U}^{(i)}_{j,n} \ge \hat{d}_r\overline{t}_r + \hat{h}_r \underline{t}_r + \sum _{\forall l \in \delta _{j,n}(i)} u_l - |\delta _{j,n}(i)|, \qquad \end{aligned}$$

(69)

where

$$\begin{aligned}&\hat{v}^{(i)}_{j,n} = \prod _{l \in \delta _n(i)} h_l \prod _{{l} \in \{\mathcal {L}^D_n \cup \mathcal {L}_{f'}\}/\delta _n(i)} d_{{l}} f\qquad \forall j \in \{1,\ldots , \hat{T}_i\}, \nonumber \\&\qquad \quad \qquad \forall i \in \{1,\ldots , \hat{D}\}, n \in \mathcal {N}^*_f, \forall (f,r,f') \in \mathcal {V}. \end{aligned}$$

(70)