Convex quadratic relaxations for mixed-integer nonlinear programs in power systems

Abstract

This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semidefinite programming relaxations. Three case studies in optimal power flow, optimal transmission switching and capacitor placement demonstrate the benefits of the new relaxations.

Appendices

Appendix A: The QC formulations

This appendix describes a variety of extensions to the QC relaxation to key decision problems in power systems, including those featuring discrete variables.

1.1 A.1 Optimal power flow

The OPF problem is a simple extension of the QC relaxation. It only requires integration of the operational constraints and an appropriate objective function as follows,

$$\begin{aligned} \min ~&(11) \\ \text {s.t. }&(3)-(4), (5)-(9), \\&(12)-(19), (21), (23) \end{aligned}$$

(QC-OPF)

1.2 A.2 Optimal transmission switching (OTS)

Before developing the QC-OTS formulations we first develop an on/off version of the McCormick envelopes from [37] to support the model. The on/off extension of the McCormick envelopes is given by the following disjunction:

$$\begin{aligned} \varGamma ^0_m= & {} \left\{ (w,x,y,z) \in {\mathbb {R}}^{4}: w=0,~ x=0,~ y=0,~ z=0 \right\} \\ \varGamma ^1_m= & {} \left\{ \begin{array}{ll} &{}\left( w,x,y,z\right) \in {\mathbb {R}}^{4}:\\ &{}w \ge {\varvec{x}^l}y + {\varvec{y}^l}x - {\varvec{x}^l}{\varvec{y}^l}\\ &{}w \ge {\varvec{x}^u}y + {\varvec{y}^u}x - {\varvec{x}^u}{\varvec{y}^u}\\ &{}w \le {\varvec{x}^l}y + {\varvec{y}^u}x - {\varvec{x}^l}{\varvec{y}^u}\\ &{}w \le {\varvec{x}^u}y + {\varvec{y}^l}x - {\varvec{x}^u}{\varvec{y}^l} \\ &{} z = 1 \end{array}\right\} \end{aligned}$$

Based on (24) the convex-hull formulation is given by:

$$\begin{aligned} \varGamma ^{*}_m =&\left\{ \begin{array}{ll} &{}\left( w,x,y,z\right) \in {\mathbb {R}}^{4}:\\ &{}w \ge {\varvec{x}^l}y + {\varvec{y}^l}x - z{\varvec{x}^l}{\varvec{y}^l}\\ &{}w \ge {\varvec{x}^u}y + {\varvec{y}^u}x - z{\varvec{x}^u}{\varvec{y}^u}\\ &{}w \le {\varvec{x}^l}y + {\varvec{y}^u}x - z{\varvec{x}^l}{\varvec{y}^u}\\ &{}w \le {\varvec{x}^u}y + {\varvec{y}^l}x - z{\varvec{x}^u}{\varvec{y}^l} \\ &{}z \varvec{w}^l \le w \le z \varvec{w}^u, \\ &{} z \varvec{x}^l \le x \le x \varvec{x}^u, \\ &{} z \varvec{y}^l \le y \le z \varvec{y}^u\\ &{}0 \le z \le 1 \end{array}\right\} \end{aligned}$$

(35)

We will use the notation $\langle xy,z \rangle ^{M01}$ as the on/off extension of the McCormick relaxation $\langle xy \rangle ^{M}$.

1.2.1 A.2.1 The QC-OTS formulations

In the OTS problem, discrete variables $z_{ij} \in \{0,1\}$ for $(i,j) \in E$ are used to remove lines from the network. Let $\varvec{\theta }^M$ be the largest possible phase angle difference between two disconnected buses, then a natural big-M extension of the (4.2) relaxation is as follows,

While a stronger formulation can be developed using the results from Sect. 5,

See Sect. 6.2 for an in depth study of these OTS relaxations.

1.3 A.3 Capacitor placement

All of the variable and network parameters of this model are defined in Sect. 6.3 and used in the (AC-CAP) formulation. The following model is a straight-forward relaxation of the CAP model using the QC constraints,

$$\begin{aligned} \min ~&\sum \limits _{i \in N} c_i + d_i \\ \text {s.t. }&(3), (6)-(9), \\&(12)-(19), (21), (23)\\&{ \varvec{v}^l} \le v_i \le {\varvec{v}^u}\quad \forall i \in N \\&q^c_i - q^d_i + q^g_i - {\varvec{q}^d_i} = \sum \limits _{(i,j) \in E} q_{ij} + \sum \limits _{(j,i) \in E} q_{ij}\quad \forall i \in N, \\&0 \le q^c_{i} \le c_i{ \varvec{q}^{cu}}\quad \forall i \in N,\\&0 \le q^d_{i} \le d_i{ \varvec{q}^{du}}\quad \forall i \in N,\\&c_i \in {\mathbb {Z}}\quad \forall i \in N, \\&d_i \in {\mathbb {Z}}\quad \forall i \in N.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \hbox {(QC-CAP)} \end{aligned}$$

See Sect. 6.3 for an in depth experimental study of this CAP relaxation.

Appendix B: The SOCP and SDP power flow relaxations

In this section we review the formulations of two state-of-the-art power flow relaxations, the SOCP [27] and SDP [5] relaxations, which are used as points of comparison for this work. See Sect. 6.1 for a detailed study of these relaxations on the OPF problem.

SOCP. The SOCP relaxation is inspired by the following equivalence,

$$\begin{aligned} v_{i}^2 v_{j}^2 = (v_i v_j \cos (\theta _i - \theta _j))^2 + (v_i v_j \sin (\theta _i - \theta _j))^2 ~\forall (i,j) \in E. \end{aligned}$$

(36)

The validity of this equivalence is easily verified using the trigonometric identity $\cos (x)^2 + \sin (x)^2 = 1$. The SOCP relaxation introduces the following W-space variables to replace the non-linear expressions,

In this lifted space, the valid equality (36) becomes,

(37)

And if relaxed to an inequality yields the following SOC constraint,

(38)

The SOCP relaxation is then given by lifting all of the power flow equations into the W-space and adding the valid SOC constraint as follows,

$$\begin{aligned} {\text {SOCP-core}} \equiv \left\{ \begin{array}{l} (3)-(4) \\ (12)-(13)\\ (38) \end{array}\right. \end{aligned}$$

Extending this formulation to the OPF problem requires the following two insights on modeling the operational constraints in the W-space,

(39)

(40)

The equivalence of the first constraint is clear. For a derivation of the second constraint see [36]. Combining these derivations, a complete SOCP-based OPF formulation is given by,

$$\begin{aligned} \min ~&(11) \\ \text {s.t. }&(3)-(4), (12)-(13), (38) \\&(6)-(8), (39)-(40) \end{aligned}$$

(SOCP-OPF)

SDP. Similar to the SOCP relaxation, the SDP relaxation begins by lifting the power flow problem into the W-space as specified above. It then defines the following matrix of complex numbers for $1..n \in N$,

The key insight of the SDP relaxation is that W is positive-semidefinite, that is,

$$\begin{aligned} W \succeq 0. \end{aligned}$$

(41)

A derivation of this property can be found in [5, 52]. The core of the SDP relaxation is then given by,

$$\begin{aligned} {\text {SDP-core}} \equiv \left\{ \begin{array}{l} (3)-(4) \\ (12)-(13) \\ (41) \end{array}\right. \end{aligned}$$

Utilizing the operational constraints in the W-space, a complete SDP-based OPF formulation is given by,

$$\begin{aligned} \min ~&(11) \\ \text {s.t. }&(3)-(4), (12)-(13), (41) \\&(6)-(8), (39)-(40) \end{aligned}$$

(SDP-OPF)

A key limitation of the SDP relaxation is scalability. This is due to the current state-of-the-art in SDP solvers as well as the number of variables required by this formulation. Observe that the matrix W requires $|N|^2$ new variables while the W-space from the SOCP relaxation only requires |E| variables. The sparsity of realistic power networks ensures that the number of variables needed in the SOCP relaxation is significantly less than the SDP relaxation. To increase the performance and scalability of the SDP relaxation [35] introduces a method where not all of the $|N|^2$ variables are required. In this work we utilize this state-of-the-art SDP formulation.

Appendix C: Extensions for AC transmission system datasets

Typical power network datasets, such as those distributed with Matpower [62] include the following additional operational parameters: (1) multiple generators connected to the same bus; (2) multiple lines between buses; (3) bus shunts; (4) voltage transformers; and (5) line charging. In the rest of this section we review how the models in this paper can be extended to incorporate these parameters.

Power Flow Extensions. Let L be the set of lines in the network and let A be the set of triples (i, j, k) where $k \in L$ is a unique identifier for each line in the network and $i,j \in N$ define the two buses connected by this line. Note that the set L enables multiple lines to be connected between two buses. Let $\varvec{b}^c_{k}$ be the line charging value for each line $k \in L$. Let $\varvec{tr}_k, \varvec{\theta }^s_k$ be the tap ratio and phase shifting transformer constants for each line $k \in L$. While these values are typically given in polar coordinates, we transform them into rectangular coordinates as follows,

$$\begin{aligned} \varvec{t}^R_k&= \varvec{tr}_k \cos (\varvec{\theta }^s_k) \\ \varvec{t}^I_k&= \varvec{tr}_k \sin (\varvec{\theta }^s_k) \end{aligned}$$

Combining these values with the line admittance parameters $\varvec{g}_k, \varvec{b}_k$, we define the following constants for each line $k \in L$,

$$\begin{aligned} \varvec{\gamma }_{1k}&= \varvec{b}_k + \varvec{b}^c_{k}/2 \\ \varvec{\gamma }_{2k}&= -\varvec{g}_k \varvec{t}^R_k + \varvec{b}_k \varvec{t}^I_k \\ \varvec{\gamma }_{3k}&= -\varvec{b}_k \varvec{t}^R_k - \varvec{g}_k \varvec{t}^I_k \\ \varvec{\gamma }_{4k}&= -\varvec{g}_k \varvec{t}^R_k - \varvec{b}_k \varvec{t}^I_k \\ \varvec{\gamma }_{5k}&= -\varvec{b}_k \varvec{t}^R_k + \varvec{g}_k \varvec{t}^I_k \end{aligned}$$

The power flow equations are then extended as follows,

$$\begin{aligned} (1)&\Rightarrow p_{ijk} = \frac{\varvec{g}_k}{\varvec{tr}_k^2} v_i^{2} + \frac{\varvec{\gamma }_{2k}}{\varvec{tr}_k^2}v_iv_j \cos (\theta _i - \theta _j) + \frac{\varvec{\gamma }_{3k}}{\varvec{tr}_k^2}v_iv_j\sin (\theta _i - \theta _j) \quad \forall (i,j,k) \in A \end{aligned}$$

(42)

$$\begin{aligned} (1)&\Rightarrow p_{jik} = {\varvec{g}_k} v_i^{2} + \frac{\varvec{\gamma }_{4k}}{\varvec{tr}_k^2}v_jv_i \cos (\theta _j - \theta _i) + \frac{\varvec{\gamma }_{5k}}{\varvec{tr}_k^2}v_jv_i\sin (\theta _j - \theta _i) \quad \forall (i,j,k) \in A \end{aligned}$$

(43)

$$\begin{aligned} (2)&\Rightarrow q_{ijk} = \frac{-\varvec{\gamma }_{1k}}{\varvec{tr}_k^2} v_i^{2} - \frac{\varvec{\gamma }_{3k}}{\varvec{tr}_k^2}v_iv_j \cos (\theta _i - \theta _j) + \frac{\varvec{\gamma }_{2k}}{\varvec{tr}_k^2}v_iv_j\sin (\theta _i - \theta _j) ~\forall (i,j,k) \in A \end{aligned}$$

(44)

$$\begin{aligned} (2)&\Rightarrow q_{jik} \!=\! -\varvec{\gamma }_{1k} v_i^{2} \!-\! \frac{\varvec{\gamma }_{5k}}{\varvec{tr}_k^2}v_jv_i \cos (\theta _j - \theta _i) + \frac{\varvec{\gamma }_{4k}}{\varvec{tr}_k^2}v_jv_i\sin (\theta _j - \theta _i) \quad \forall (i,j,k) \in A \end{aligned}$$

(45)

Observe that transformer parameters make the constants in these equations asymmetrical, a key difference to the simplified power flow Eqs. (1), (2).

Kirchhoff’s current law extensions. Let $G_i$ be the set of generators at bus $i \in N$ and let $\varvec{g}^s_i, \varvec{b}^s_i$ be the active and reactive bus shunt values at that bus. Then the Kirchhoff’s Current Law constraints are extended as follows,

$$\begin{aligned} (3)&\Rightarrow \sum _{j \in G_i} p^g_{j} - \varvec{g}^s_i v_i^2 - {\varvec{p}^d_i} = \sum \limits _{(i,j,k) \in A} p_{ijk} + \sum \limits _{(j,i,k) \in A} p_{ijk} \;\; i \in N \end{aligned}$$

(46)

$$\begin{aligned} (4)&\Rightarrow \sum _{j \in G_i} q^g_{j} + \varvec{b}^s_i v_i^2 - {\varvec{q}^d_i} = \sum \limits _{(i,j,k) \in A} q_{ijk} + \sum \limits _{(j,i,k) \in A} q_{ijk} \;\; i \in N \end{aligned}$$

(47)

Extended optimal power flow. Combining these extensions, the extended optimal power flow problem is given by the following set of constraints,

$$\begin{aligned} \min ~&(11) \\ \text {s.t. }&(42)-(47) \\&(5)-(9) \end{aligned}$$

(AC-OPF-E)

Relaxation extensions. Observe that all of these model extensions result in simple modifications to the constants in front of the following key terms,

$$\begin{aligned} v_i^2,~ v_iv_j \cos (\theta _i - \theta _j),~ v_iv_j \sin (\theta _i - \theta _j). \end{aligned}$$

Because these terms remain unchanged by the extensions, nearly all of the constraints in the relaxations presented here can be adapted to these extensions simply by modifying various constant terms.

A notable exception is the current magnitude constraints (21)–(23) used in the QC model. Let $A' \subseteq A$ contain one line for each pair of buses connected in A, then these two constraints are updated as follows $\forall (i,j,k) \in A'$,

(48)

(49)

A derivation of these constraints can be found in [14].

Appendix D: Extended results

This appendix presents comprehensive results on all of the test cases in NESTA v0.6.0 [13]. In addition to the three categories of cases presented previously (i.e. TYP, API, and SAD), this section includes cases from the radial topologies (RAD), and nonconvex optimization (NCO) categories.

In the following result tables these annotations are used:

ud.:

—undefined value

bold :

—a relaxation provided a feasible AC power flow

na.:

—the solver does not support all of the features of this test case

—:

—no solution available at solver termination

inf.:

—the solver proved the model is infeasible

err.:

—the solver had an internal error

oom.:

—the solver ran out of memory

tl.:

—the solver reached the time limit

$\dagger$ :

—an annotation that the solver reached an internal iteration limit

$\star$ :

—an annotation that the solver reported numerical accuracy warnings

1.1 D.1 Optimal power flow

See Tables 5, 6, 7, 8, 9, 10.

Table 5 Optimality gap for different relaxations of the OPF problem

Table 6 Optimality gap for different relaxations of the OPF problem

Table 7 Optimality gap for different relaxations of the OPF problem

Table 8 Runtimes for different relaxations of the OPF problem

Table 9 Runtimes for different relaxations of the OPF problem

Table 10 Runtimes for different relaxations of the OPF problem

1.2 D.2 Optimal transmission switching

See Tables 11, 12.

Table 11 Quality and runtime results of the QC vs QC-strong models on OTS

Table 12 Quality and runtime results of the QC vs QC-strong models on OTS

1.3 D.3 Capacitor placement

See Tables 13, 14.

Table 13 Quality and runtime results of the QC relaxation on CAP

Table 14 Quality and runtime results of the QC relaxation on CAP