A robust and informative method for solving large-scale power flow problems

Abstract

Solving power flow problems is essential for the reliable and efficient operation of an electric power network. However, current software for solving these problems have questionable robustness due to the limitations of the solution methods used. These methods are typically based on the Newton–Raphson method combined with switching heuristics for handling generator reactive power limits and voltage regulation. Among the limitations are the requirement of a good initial solution estimate, the inability to handle near rank-deficient Jacobian matrices, and the convergence issues that may arise due to conflicts between the switching heuristics and the Newton–Raphson process. These limitations are addressed by reformulating the power flow problem and using robust optimization techniques. In particular, the problem is formulated as a constrained optimization problem in which the objective function incorporates prior knowledge about power flow solutions, and solved using an augmented Lagrangian algorithm. The prior information included in the objective adds convexity to the problem, guiding iterates towards physically meaningful solutions, and helps the algorithm handle near rank-deficient Jacobian matrices as well as poor initial solution estimates. To eliminate the negative effects of using switching heuristics, generator reactive power limits and voltage regulation are modeled with complementarity constraints, and these are handled using smooth approximations of the Fischer–Burmeister function. Furthermore, when no solution exists, the new method is able to provide sensitivity information that aids an operator on how best to alter the system. The proposed method has been extensively tested on real power flow networks of up to 58k buses.

Appendices

Appendix 1: Global convergence proof

Under the assumption that bounded power mismatches implies bounded voltage magnitudes, it is shown that the sequence of iterates $\{x_k\}$ of the algorithm described in Sect. 5.2 has a feasible point (according to the required tolerance) or a limit point that is a stationary point of the function $c^Tc$ (restricted to the set of $x$ such that $Ax=b$). This result is obtained without assuming that the Jacobian of the power flow equations is full rank. In the following proofs, the subscript $i$ is used for buses or terms inside a sum or product series, the subscript $j$ is used to associate an object with the $j$-th inner iteration of subproblem (60), and the subscript $k$ is used to associate an object with the $k$-th outer iteration. The continuity of the functions involved is also used throughout. To simplify the notation, derivatives are assumed to be with respect to $x$ and hence a subscript $x$ is not used for derivatives of functions that include other variables.

The reader is reminded that $x$ is a vector of voltage magnitudes $\{v_i\}_{i \in [n] \setminus \{s\}}$, voltage angles $\{\theta _i\}_{i \in [n] \setminus \{s\}}$, generator reactive powers $\{Q^g_i\}_{i \in \fancyscript{R}}$, and positive and negative regulated voltage magnitude deviations $\{y_i\}_{i \in \fancyscript{R}}$ and $\{z_i\}_{i \in \fancyscript{R}}$, respectively. The function $\varPhi$ is the smooth vector-valued function whose scalar function entries are given by (44) and (45). The objective function $\varphi$ is a strongly convex non-negative quadratic function, and the function $f$ is the vector-valued function of power mismatches (46) and (47). The function $c$ consists of both $f$ and $\varPhi$, as defined in (58). Throughout this section, $J$ is used to denote the Jacobian of $c$, and $Z$ is used to denote the matrix whose columns span the null space of $A$, the Jacobian of the linear equality constraints.

When dealing with outer iterations, $\varphi _k$, $f_k$, $\varPhi _k$, $c_k$ and $J_k$ are used to denote $\varphi (x_k)$, $f(x_k)$, $\varPhi (x_k)$, $c(x_k)$ and $J(x_k)$, respectively. Similar shorthand notation is used when dealing with inner iterations.

Assumption 1

For all $M > 0$, the set $\{ x \ | \ f(x)^Tf(x) \le M\}$ has bounded voltage magnitudes $\{v_i\}_{i \in [n] \setminus \{s\}}$.

Lemma 9.1

For any given penalty parameter $\mu > 0$ and vector $\lambda$, the search directions $\{p_j\}_{j \in \mathbb {Z}_+}$ used during the solution of subproblem (60) are descent directions. That is, they satisfy

$$\begin{aligned} \nabla L_{\mu }(x_j,\lambda )^Tp_j < 0 \end{aligned}$$

(95)

for each $j \in \mathbb {Z}_+$ such that $Z^T\nabla L_{\mu }(x_j,\lambda ) \ne 0$.

Proof

Suppose not. Then, for some $j$ such that $Z^Tg_j \ne 0$, where $g_j := \nabla L_{\mu }(x_j,\lambda )$, $g_j^Tp_j \ge 0$. Hence, $p_j$ does not satisfy (72) and must have been obtained from

$$\begin{aligned} Z^T\tilde{H}_{\mu }(x_j,\lambda )Zq_j = - Z^Tg_j \end{aligned}$$

(96)

and $p_j = Zq_j$. In that case, since $\tilde{H}_{\mu }(x_j,\lambda ) \succ 0$, $Z$ is full rank and $Z^Tg_j \ne 0$, it follows that $q_j \ne 0$ and

$$\begin{aligned} g_j^Tp_j = g_j^TZq_j = -q_j^TZ^T\tilde{H}_{\mu }(x_j,\lambda )Zq_j < 0, \end{aligned}$$

(97)

which is a contradiction. $\square$

Since $\varphi (x)$ is non-negative for all $x$, it is clear that for any given $\mu$ and $\lambda$, the function $L_{\mu }(\cdot ,\lambda )$ is bounded below. From this, Lemma 9.1, and the continuity of the functions involved, it can be shown that there exists a step length that satisfies the strong Wolfe conditions (34) and (35), provided that $c_1$ and $c_2$ satisfy $0 < c_1 < c_2 < 1$ [27]. From this, the following lemma is obtained.

Lemma 9.2

For any given $\mu > 0$ and vector $\lambda$, the sequence $\{L_{\mu }(x_j,\lambda )\}_{j \in \mathbb {Z}_+}$ is monotonically non-increasing and convergent, where $\{x_j\}_{j \in \mathbb {Z}_+}$ are the inner iterates generated during the solution of subproblem (60).

Proof

Let $h_j$ and $g_j$ denote $L_{\mu }(x_j,\lambda )$ and $\nabla L_{\mu }(x_j,\lambda )$, respectively, for each $j$. From Lemma 9.1, for each $j$ such that $Z^Tg_j \ne 0$, it holds that $g_j^Tp_j < 0$. Hence, for each such $j$, the line search procedure gives $\alpha _j > 0$ such that

$$\begin{aligned} h_{j+1} \le h_j + c_1 \alpha _j g_j^Tp_j. \end{aligned}$$

(98)

This implies that $h_{j+1} \le h_j$ for all $j$ and hence that the sequence $\{h_j\}$ is monotonically non-increasing. Since $L_{\mu }(\cdot ,\lambda )$ is bounded below, the sequence $\{h_j\}$ is bounded and hence also convergent. If $Z^Tg_j = 0$ for some $j$, the process clearly stops and there is nothing more to check. $\square$

Using Lemma 9.2 and exploiting the properties of the objective function $\varphi$, a compactness result can be obtained for the inner iterates.

Lemma 9.3

For any $\mu > 0$ and vector $\lambda$, the sequence of inner iterates $\{x_j\}_{j \in \mathbb {Z}_+}$ generated during the solution of subproblem (60) lies in a compact set.

Proof

From Lemma 9.2, there exist $N_1$, $N_2 > 0$ such that

$$\begin{aligned} N_1 \le \mu \varphi _j - \mu \lambda ^Tc_j + \frac{1}{2}\Vert c_j\Vert _2^2 \le N_2 \end{aligned}$$

(99)

for all $j$. This implies that the sequence $\{\varphi _j\}$ is bounded. Since $\varphi$ is non-negative, quadratic and strongly convex, it can be expressed as

$$\begin{aligned} \varphi (x) = (x-a)^TA(x-a) + b, \end{aligned}$$

(100)

where $a$ is some vector, $A$ is a positive definite matrix, and $b$ is a non-negative scalar. Since

$$\begin{aligned} \Vert x_j\Vert _2&\le \Vert x_j - a\Vert _2 + \Vert a\Vert _2 \end{aligned}$$

(101)

$$\begin{aligned}&= \Vert A^{-1/2}A^{1/2}(x_j-a)\Vert _2 + \Vert a\Vert _2 \end{aligned}$$

(102)

$$\begin{aligned}&\le \Vert A^{-1/2}\Vert _2 \Vert A^{1/2}(x_j-a)\Vert _2 + \Vert a\Vert _2 \end{aligned}$$

(103)

$$\begin{aligned}&\le \Vert A^{-1/2}\Vert _2 \varphi _j^{1/2} + \Vert a\Vert _2 \end{aligned}$$

(104)

for all $j$, it follows that $\{x_j\}$ is a bounded sequence and hence that it lies in a compact set. $\square$

With Lemmas 9.2 and 9.3, the following theorem is obtained.

Theorem 9.1

For any given $\mu > 0$ and vector $\lambda$, if $\ Z^T\nabla L_{\mu }(x_j,\lambda ) \ne 0$ for all $j \in \mathbb {Z}_+$, the iterates $\{x_j\}_{j \in \mathbb {Z}_+}$ generated during the solution of subproblem (60) satisfy

$$\begin{aligned} \lim _{j \rightarrow \infty } \frac{\nabla L_{\mu }(x_j,\lambda )^Tp_j}{\Vert q_j\Vert _2} = 0, \end{aligned}$$

(105)

where $p_j = Zq_j$.

Proof

Let $h_j$ and $g_j$ denote $L_{\mu }(x_j,\lambda )$ and $\nabla L_{\mu }(x_j,\lambda )$, respectively, for all $j$, and suppose that (105) does not hold. Then, there exists an $\epsilon > 0$ and a countably infinite set $\fancyscript{I} \subset \mathbb {Z}_+$ such that for all $j \in \fancyscript{I}$,

$$\begin{aligned} -\frac{g_j^Tp_j}{\Vert q_j\Vert _2} \ge \epsilon . \end{aligned}$$

(106)

From condition (34) of the line search, for all such $j \in \fancyscript{I}$,

$$\begin{aligned} h_j - h_{j+1}&\ge - c_1 \alpha _j g_j^Tp_j \end{aligned}$$

(107)

$$\begin{aligned}&= -c_1 \alpha _j \Vert q_j\Vert _2 \Bigg (\frac{g_j^Tp_j}{\Vert q_j\Vert _2}\Bigg ) \end{aligned}$$

(108)

$$\begin{aligned}&\ge c_1 \alpha _j \Vert q_j\Vert _2 \epsilon . \end{aligned}$$

(109)

From Lemma 9.2, $h_j - h_{j+1} \rightarrow 0$ as $j \rightarrow \infty$ so (109) implies that the sequence $\{\alpha _j \Vert q_j\Vert _2\}_{j \in \fancyscript{I}}$ converges to 0. From condition (35) of the line search, for all $j \in \fancyscript{I}$,

$$\begin{aligned} \big (g_{j+1} - g_j\big )^TZq_j \ge - (1-c_2) g_j^Tp_j. \end{aligned}$$

(110)

From this, the Cauchy–Schwarz inequality and (106), it follows that for all $j \in \fancyscript{I}$,

$$\begin{aligned} \frac{\Vert Z^Tg_{j+1} - Z^Tg_j\Vert _2}{1-c_2} \ge - \frac{g_j^Tp_j}{\Vert q_j\Vert _2} \ge \epsilon . \end{aligned}$$

(111)

From Lemma 9.3, the sequence $\{x_j\}$ lies in some compact set $\fancyscript{K}$. Hence, the continuous function $Z^T\nabla L_{\mu }(\cdot ,\lambda )$ is actually uniformly continuous in $\fancyscript{K}$. This implies that there exists a $\delta$ such that for all $y$ and $z \in \fancyscript{K}$ such that $\Vert y-z\Vert _2 < \delta$,

$$\begin{aligned} \Vert Z^T\nabla L_{\mu }(y,\lambda ) - Z^T\nabla L_{\mu }(z,\lambda )\Vert _2 < \frac{\epsilon (1-c_2)}{2}. \end{aligned}$$

(112)

Now, since $\{\alpha _j\Vert q_j\Vert _2\}_{j \in \fancyscript{I}}$ converges to zero, so does $\{\alpha _j\Vert p_j\Vert _2\}_{j \in \fancyscript{I}}$ and hence there exists an $l \in \fancyscript{I}$ such that

$$\begin{aligned} \alpha _l\Vert p_l\Vert _2 < \delta . \end{aligned}$$

(113)

For such $l$,

$$\begin{aligned} \Vert x_{l+1} - x_l\Vert _2 = \alpha _l\Vert p_l\Vert _2 < \delta \end{aligned}$$

(114)

so

$$\begin{aligned} \frac{\Vert Z^Tg_{l+1} - Z^Tg_l\Vert _2}{1-c_2} < \frac{\epsilon }{2}, \end{aligned}$$

(115)

which contradicts (111). Therefore, (105) must hold. $\square$

An important property of the Hessian approximation used for computing the search directions is now shown. This property allows showing that the search directions used by the algorithm are not only descent directions, as shown in Lemma 9.1, but sufficient descent directions.

Lemma 9.4

For any given $\mu > 0$ and vector $\lambda$, there exists a $\eta > 0$ such that $\tilde{H}_{\mu }(x_j,\lambda )$, as defined in (66), satisfies

$$\begin{aligned} \kappa \Big (Z^T\tilde{H}_{\mu }(x_j)Z\Big ) < \eta \end{aligned}$$

(116)

for all inner iterations $j \in \mathbb {Z}_+$, where $\kappa (\cdot )$ gives the condition number of its argument.

Proof

From Lemma 9.3, $\{x_j\}$ lies in some compact set $\fancyscript{K}$. Since, $\Vert \cdot \Vert _2$ and $Z^TJ^TJZ$ are continuous functions, the image of $\fancyscript{K}$ under their composition is compact. Hence, there exists a $\sigma > 0$ such that for each $j$,

$$\begin{aligned} \lambda _{\max }\Big (Z^TJ_j^TJ_jZ\Big ) = \big \Vert Z^TJ_j^TJ_jZ\big \Vert _2 < \sigma , \end{aligned}$$

(117)

where $\lambda _{\max }(\cdot )$ gives the largest eigenvalue of its argument. Letting $A = Z^T\nabla ^2 \varphi _jZ$, which is positive definite and independent of $x_j$, and $B_j = Z^TJ_j^TJ_jZ$, which is positive semi-definite, the following inequalities hold:

$$\begin{aligned} \lambda _{\min }\big (\mu A + B_j\big )&= \underset{\Vert v\Vert _2 = 1}{\inf } v^T\big (\mu A + B_j\big )v \end{aligned}$$

(118)

$$\begin{aligned}&\ge \mu \underset{\Vert v\Vert _2 = 1}{\inf } v^TAv + \underset{\Vert v\Vert _2 = 1}{\inf } v^TB_jv \end{aligned}$$

(119)

$$\begin{aligned}&= \mu \lambda _{\min }(A) + \lambda _{\min }(B_j) \end{aligned}$$

(120)

$$\begin{aligned}&\ge \mu \lambda _{\min }(A) > 0, \end{aligned}$$

(121)

where $\lambda _{\min }(\cdot )$ gives the smallest eigenvalue of its argument. Similarly,

$$\begin{aligned} \lambda _{\max }\big (\mu A + B_j\big )&= \underset{\Vert v\Vert _2 = 1}{\sup } v^T\big (\mu A + B_j\big )v \end{aligned}$$

(122)

$$\begin{aligned}&\le \mu \underset{\Vert v\Vert _2 = 1}{\sup } v^TAv + \underset{\Vert v\Vert _2 = 1}{\sup } v^TB_jv \end{aligned}$$

(123)

$$\begin{aligned}&= \mu \lambda _{\max }(A) + \lambda _{\max }(B_j) \end{aligned}$$

(124)

$$\begin{aligned}&< \mu \lambda _{\max }(A) + \sigma . \end{aligned}$$

(125)

Letting $\tilde{H}_j$ denote $\tilde{H}_{\mu }(x_j,\lambda )$, these inequalities imply that for all $j$,

$$\begin{aligned} \kappa (Z^T\tilde{H}_jZ) = \frac{\lambda _{\max }\big (Z^T\tilde{H}_jZ\big )}{\lambda _{\min }\big (Z^T\tilde{H}_jZ\big )} = \frac{\lambda _{\max }(\mu A + B_j)}{\lambda _{\min }(\mu A + B_j)} < \frac{\mu \lambda _{\max }(A) + \sigma }{\mu \lambda _{\min }(A)}. \end{aligned}$$

(126)

$\square$

Lemma 9.5

For any given $\mu > 0$ and vector $\lambda$, there exists a $\rho > 0$ such that the search directions $\{p_j\}_{j \in \mathbb {Z}_+}$ computed during the solution of subproblem (60) satisfy

$$\begin{aligned} -\frac{\nabla L_{\mu }(x_j,\lambda )^Tp_j}{\Vert Z^T \nabla L_{\mu }(x_j,\lambda )\Vert _2\Vert q_j\Vert _2} > \varrho , \end{aligned}$$

(127)

for all $j \in \mathbb {Z}_+$ such that $Z^T\nabla L_{\mu }(x_j,\lambda ) \ne 0$, where $p_j = Zq_j$.

Proof

From Lemma 9.4, there exists an $\eta > 0$ such that for all $j$,

$$\begin{aligned} \kappa \Big (Z^T\tilde{H}_jZ\Big ) < \eta , \end{aligned}$$

(128)

where $\tilde{H}_j$ denotes $\tilde{H}_{\mu }(x_j,\lambda )$. Letting $g_j$ and $C_j$ denote $\nabla L_{\mu }(x_j,\lambda )$ and $Z^T\tilde{H}_jZ$, respectively, it follows that for all $j$ such that $Z^Tg_j \ne 0$ and $q_j$ was computed using $C_j$,

$$\begin{aligned} -\frac{g_j^Tp_j}{\Vert Z^Tg_j\Vert _2\Vert q_j\Vert _2}&= \frac{g_j^TZC_j^{-1}Z^Tg_j}{\Vert Z^Tg_j\Vert _2 \Vert C_j^{-1}Z^Tg_j\Vert _2} \end{aligned}$$

(129)

$$\begin{aligned}&\ge \frac{\lambda _{\min }(C_j^{-1})}{\lambda _{\max }(C_j^{-1})} \end{aligned}$$

(130)

$$\begin{aligned}&= \frac{\lambda _{\min }(C_j)}{\lambda _{\max }(C_j)} \end{aligned}$$

(131)

$$\begin{aligned}&= \frac{1}{\kappa (C_j)} > \frac{1}{\eta }. \end{aligned}$$

(132)

For $j$ such that $Z^Tg_j \ne 0$ and $q_j$ was not computed using $C_j$, the exact Hessian was used and $p_j$ must have passed condition (72). Hence, letting $\varrho = \min \{\xi ,1/\eta \}$ completes the proof, where $\xi$ is the predefined small positive scalar described in Sect. 5.2.4. $\square$

With Lemmas 9.5 and Theorem 9.1, it can be proved that the vPF algorithm always solves each subproblem (60) to the required accuracy.

Theorem 9.2

For any given $\mu > 0$ and vector $\lambda$, the iterates $\{x_j\}_{j \in \mathbb {Z}_+}$ generated during the solution of subproblem (60) satisfy either

$$\begin{aligned} Z^T\nabla L_{\mu }(x_l,\lambda ) = 0 \end{aligned}$$

(133)

for some $l \in \mathbb {Z}_+$, or

$$\begin{aligned} \lim _{j \rightarrow \infty } Z^T\nabla L_{\mu }(x_j,\lambda ) = 0. \end{aligned}$$

(134)

Proof

If $l$ exists such that $Z^T\nabla L_{\mu }(x_l,\lambda ) = 0$, the theorem is proved. Otherwise, Lemma 9.5 gives that

$$\begin{aligned} -\frac{g_j^Tp_j}{\Vert q_j\Vert _2} > \varrho \Vert Z^Tg_j\Vert _2 \end{aligned}$$

(135)

for all $j$, where $g_j$ denotes $\nabla L_{\mu }(x_j,\lambda )$. From Theorem 9.1, the left-hand side of this inequality goes to zero as $j$ goes to infinity. Hence,

$$\begin{aligned} \lim _{j \rightarrow \infty } Z^Tg_j = 0. \end{aligned}$$

(136)

$\square$

Corollary 9.1

Given a penalty parameter $\mu > 0$, vector $\lambda$ and any subproblem optimality tolerance $\delta > 0$, the inner level of the vPF algorithm always finds $\bar{x}$ such that

$$\begin{aligned} \Big \Vert Z^T\nabla L_{\mu }(\bar{x},\lambda )\Big \Vert _{\infty } < \delta \end{aligned}$$

(137)

in a finite number of iterations.

Proof

This results follows immediately from Theorem 9.2. $\square$

Properties of the outer iterates generated by the vPF algorithm are now proved. First, it is shown that the sequence of values of the Augmented Lagrangian function associated with the outer iterates is bounded.

Lemma 9.6

Given any initial point $x_0$, initial penalty parameter $\mu _0 > 0$, and initial vector of Lagrange multiplier estimates $\lambda _0$, if $\Vert c(x_k)\Vert _{\infty } \ge \epsilon _f$ for all $k \in \mathbb {Z}_+$, the sequence $\{L_{\mu _k}(x_k,\lambda _k)\}_{k \in \mathbb {Z}_+}$ is bounded, where $\{x_k\}_{k \in \mathbb {Z}_+}$ are the outer iterates generated by the vPF algorithm.

Proof

Suppose $\Vert c_k\Vert _{\infty } \ge \epsilon _f$ for all $k$. Hence, the vPF algorithm never terminates and generates an infinite sequence of outer iterates $x_k$. By the equivalency of norms in finite dimensional spaces, there exists some $\epsilon$ such that $\Vert c_k\Vert _2 \ge \epsilon$ for all $k$. During outer iteration $k$, the inner level of the algorithm takes $x_k$, $\mu _k$ and $\lambda _k$ and generates the next outer iterate $x_{k+1}$. By Lemma 9.2, it must be that

$$\begin{aligned} L_{\mu _k}\big (x_{k+1},\lambda _k\big ) \le L_{\mu _k}\big (x_k,\lambda _k\big ), \end{aligned}$$

(138)

for all $k$. Equivalently,

$$\begin{aligned} \mu _k \varphi _{k+1} - \mu _k\lambda _k^Tc_{k+1} + \frac{1}{2}\Vert c_{k+1}\Vert _2^2 \le \mu _k \varphi _k - \mu _k\lambda _k^Tc_k + \frac{1}{2}\Vert c_k\Vert _2^2. \end{aligned}$$

(139)

By construction (last paragraph of Sect. 5.2.5), there exists some $M > 0$ such that $\Vert \lambda _k\Vert _2 \le M$ for all $k$. Hence, using the Cauchy-Schwarz inequality, the fact that $\mu _k$ is non-increasing, and the non-negativity of $\varphi$, it follows that

$$\begin{aligned} \mu _k \varphi _{k+1} - \mu _k M\Vert c_{k+1}\Vert _2 + \frac{1}{2}\Vert c_{k+1}\Vert _2^2&\le \mu _k \varphi _k + \mu _k M\Vert c_k\Vert _2 + \frac{1}{2}\Vert c_k\Vert _2^2 \end{aligned}$$

(140)

$$\begin{aligned} \mu _k \varphi _{k+1} + \frac{1}{2}\Vert c_{k+1}\Vert _2^2 \Bigg (1 - \frac{2 \mu _k M}{\Vert c_{k+1}\Vert _2} \Bigg )&\le \mu _k \varphi _k + \frac{1}{2}\Vert c_k\Vert _2^2 \Bigg (1 + \frac{2 \mu _k M}{\Vert c_k\Vert _2} \Bigg ) \end{aligned}$$

(141)

$$\begin{aligned} \mu _{k+1} \varphi _{k+1} + \frac{1}{2}\Vert c_{k+1}\Vert _2^2 \Bigg (1 - \frac{2 \mu _k M}{\epsilon } \Bigg )&\le \mu _k \varphi _k + \frac{1}{2}\Vert c_k\Vert _2^2 \Bigg (1 + \frac{2 \mu _k M}{\epsilon } \Bigg ) \end{aligned}$$

(142)

for all $k$. Since $\mu _k \downarrow 0$ as $k \rightarrow \infty$, there is some $K \in \mathbb {N}$ such that for all $k \ge K$,

$$\begin{aligned} 1/2 < 1- 2\mu _kM/\epsilon < 1. \end{aligned}$$

(143)

For such $k$,

$$\begin{aligned} \Bigg (1 - \frac{2 \mu _k M}{\epsilon } \Bigg ) \Bigg ( \mu _{k+1} \varphi _{k+1} + \frac{1}{2}\Vert c_{k+1}\Vert _2^2 \Bigg )&\le \Bigg (1 + \frac{2 \mu _k M}{\epsilon } \Bigg ) \Bigg ( \mu _k \varphi _k + \frac{1}{2}\Vert c_k\Vert _2^2 \Bigg ), \end{aligned}$$

(144)

or

$$\begin{aligned} \mu _{k+1} \varphi _{k+1} + \frac{1}{2}\Vert c_{k+1}\Vert _2^2 \le \eta _k \Bigg ( \mu _k \varphi _k + \frac{1}{2}\Vert c_k\Vert _2^2 \Bigg ), \end{aligned}$$

(145)

where

$$\begin{aligned} \eta _k := \frac{1 + \frac{2 \mu _k M}{\epsilon }}{1 - \frac{2 \mu _k M}{\epsilon }}. \end{aligned}$$

(146)

It follows that for $k > K$,

$$\begin{aligned} \mu _k \varphi _k + \frac{1}{2}\Vert c_k\Vert _2^2 \le \Bigg ( \mu _K \varphi _K + \frac{1}{2}\Vert c_K\Vert _2^2 \Bigg ) \prod _{i=K}^{k-1} \eta _i. \end{aligned}$$

(147)

Now, for $k \ge K$, it holds that

$$\begin{aligned} \log \eta _k = \log \Bigg (1 + \frac{2 \mu _kM}{\epsilon } \Bigg ) - \log \Bigg (1 - \frac{2 \mu _kM}{\epsilon } \Bigg ). \end{aligned}$$

(148)

Hence, the bounds

$$\begin{aligned} 1-\frac{1}{y} \le \log y \le y - 1, \end{aligned}$$

(149)

for all $y \in \mathbb {R}_{++}$, imply that

$$\begin{aligned} \log \eta _k \le \mu _k\frac{2M}{\epsilon } + \mu _K\frac{4M}{\epsilon } = \mu _k\frac{6M}{\epsilon }. \end{aligned}$$

(150)

Since, $\mu _k \le \beta _l^k\mu _0$, where $\beta _l \in (0,1)$, there exists some $N > 0$ such that for all $k > K$,

$$\begin{aligned} \log \prod _{i=N}^{k-1}\eta _i = \sum _{i=N}^{k-1}\log \eta _i \le \frac{6M\mu _0}{\epsilon } \sum _{i=N}^{\infty } \beta _l^i < N. \end{aligned}$$

(151)

This implies that for all $k > K$,

$$\begin{aligned} \mu _k \varphi _k + \frac{1}{2}\Vert c_k\Vert _2^2 \le \Bigg ( \mu _K \varphi _K + \frac{1}{2}\Vert c_K\Vert _2^2 \Bigg ) e^N \end{aligned}$$

(152)

and hence that $\{L_{\mu _k}(x_k,\lambda _k)\}$ is a bounded sequence. $\square$

With the boundedness result of Lemma 9.6, the boundedness of other important quantities can be shown.

Lemma 9.7

If the vPF algorithm never terminates and Assumption 1 holds, the voltage magnitudes $\{v_i\}_{i \in [n] \setminus \{s\}}$, generator reactive powers $\{Q_i^g\}_{i \in \fancyscript{R}}$, and regulated voltage magnitude deviations $\{y_i\}_{i \in \fancyscript{R}}$ and $\{z_i\}_{i \in \fancyscript{R}}$ associated with each of the outer iterates $\{x_k\}_{k \in \mathbb {Z}_+}$ are uniformly bounded. Moreover, the sequence $\{J_k\}_{k \in \mathbb {Z}_+}$ of Jacobian matrices is bounded.

Proof

If the vPF algorithm never terminates, $\Vert c_k\Vert _{\infty } \ge \epsilon _f$ for all $k$. Then, by Lemma 9.6, the sequence $\{L_{\mu _k}(x_k,\lambda _k)\}$ is bounded. From the definition of $L_{\mu }$, the sequence $\{\Vert c_k\Vert _2\}$ is bounded. Then, from the definition of $c$ (Sect. 5.2), $\{\Vert f_k\Vert _2\}$ must be bounded. Hence, Assumption 1 gives that the voltage magnitudes $\{v_i\}$ associated with the outer iterates are uniformly bounded. The uniform boundedness of the generator reactive powers $\{Q_i^g\}$ then follows from the boundedness of reactive power mismatches and voltage magnitudes, and from (47). Similarly, from the definition of $c$, $\{\Vert \varPhi _k\Vert _2\}$ is also bounded. The uniform boundedness of $\{y_i\}$ and $\{z_i\}$ then follows from the uniform boundedness of $\{Q_i^g\}$, the fact that $\{\varPhi _k\}$ is a bounded sequence, and from (44) and (45). Lastly, the boundedness of $\{J_k\}$ is implied from the uniform boundedness of $\{v_i\}$, $\{Q^g_i\}$, $\{y_i\}$ and $\{z_i\}$, and the fact that angles only appear inside sine and cosine functions. $\square$

Lemma 9.8

If the vPF algorithm never terminates, the sequence $\{\mu _k \nabla \varphi _k\}_{k \in \mathbb {Z}_+}$ satisfies $\mu _k \nabla \varphi _k \rightarrow 0$ as $k \rightarrow \infty$, where $\{x_k\}_{k \in \mathbb {Z}_+}$ are the outer iterates.

Proof

If the vPF algorithm never terminates, $\Vert c_k\Vert _{\infty } \ge \epsilon _f$ for all $k$. Then, by Lemma 9.6, the sequence $\{L_{\mu _k}(x_k,\lambda _k)\}$ is bounded. From the definition of $L_{\mu }$, the sequence $\{\mu _k \varphi _k\}$ is bounded. Since $\varphi$ is non-negative, quadratic and strongly convex, it can be expressed as

$$\begin{aligned} \varphi (x) = (x-a)^TA(x-a) + b, \end{aligned}$$

(153)

where $a$ is some vector, $A$ is a positive definite matrix, and $b$ is a non-negative scalar. Hence, $\{\mu _k \varphi _k\}$ bounded implies that there exists some $M > 0$ such that for all $k$,

$$\begin{aligned} \big \Vert A^{1/2}(x-a)\big \Vert _2 \le \sqrt{\frac{M}{\mu _k}}. \end{aligned}$$

(154)

It follows that

$$\begin{aligned} \mu _k \Vert \nabla \varphi _k\Vert _2&= \mu _k \Vert 2A(x-a)\Vert _2 \end{aligned}$$

(155)

$$\begin{aligned}&\le \mu _k 2 \big \Vert A^{1/2}\big \Vert _2 \big \Vert A^{1/2}(x-a)\big \Vert _2 \end{aligned}$$

(156)

$$\begin{aligned}&\le 2 \sqrt{\mu _kM} \big \Vert A^{1/2}\big \Vert _2. \end{aligned}$$

(157)

The result then follows from the fact that $\mu _k \rightarrow 0$ as $k \rightarrow \infty$. $\square$

The main result about the convergence of the vPF algorithm can now be proved.

Theorem 9.3

Under Assumption 1, the vPF algorithm either finds during some iteration $l \in \mathbb {Z}_+$ an outer iterate $x_l$ that is feasible according to the required tolerance, i.e., that satisfies $\Vert c_l\Vert _{\infty } < \epsilon _f$, or it generates a sequence of outer iterates $\{x_k\}_{k \in \mathbb {Z}_+}$ that satisfies

$$\begin{aligned} \lim _{k \rightarrow \infty } Z^TJ_k^Tc_k = 0. \end{aligned}$$

(158)

From $\{x_k\}_{k \in \mathbb {Z}_+}$, a bounded sequence $\{\tilde{x}_k\}_{k \in \mathbb {Z}_+}$ can be constructed such that

$$\begin{aligned} J(\tilde{x}_k)^Tc(\tilde{x}_k) = J_k^Tc_k, \end{aligned}$$

(159)

for all $k$, by translating the voltage angles of $\{x_k\}_{k \in \mathbb {Z}_+}$ to $[-\pi ,\pi ]$. This new sequence is guaranteed to have a limit point $x^*$ that satisfies

$$\begin{aligned} Z^TJ(x^*)^Tc(x^*) = 0. \end{aligned}$$

(160)

Proof

Suppose that Assumption 1 holds. If for some $l \in \mathbb {Z}_+$, the outer iterate $x_l$ satisfies $\Vert c(x_l)\Vert _{\infty } < \epsilon _f$, the result holds. Otherwise the vPF algorithm does not terminate and it generates a sequence of outer iterates $\{x_k\}$ such that

$$\begin{aligned} \big \Vert Z^T\nabla L_{\mu _k}(x_{k+1},\lambda _k)\big \Vert _{\infty } < \delta _k, \end{aligned}$$

(161)

where $\delta _k \downarrow 0$. Hence,

$$\begin{aligned} \nu _k := \big \Vert Z^T\nabla L_{\mu _k}(x_{k+1},\lambda _k)\big \Vert _2 \rightarrow 0 \end{aligned}$$

(162)

as $k \rightarrow \infty$. Now, for all $k$,

$$\begin{aligned} \Big \Vert Z^TJ_{k+1}^Tc_{k+1}\Big \Vert _2&= \Big \Vert Z^T \nabla L_{\mu _k}(x_{k+1},\lambda _k) - \mu _kZ^T\nabla \varphi _{k+1} + \mu _kZ^TJ_{k+1}^T\lambda _k\Big \Vert _2\end{aligned}$$

(163)

$$\begin{aligned}&\le \nu _k + \mu _k \Vert Z\Vert _2 \Vert \nabla \varphi _{k+1}\Vert _2 + \mu _k \Vert Z\Vert _2 \Vert J_{k+1}\Vert _2 \Vert \lambda _k\Vert _2 \end{aligned}$$

(164)

$$\begin{aligned}&\le \nu _k + \frac{\mu _{k+1}}{\beta _s} \Vert Z\Vert _2 \Vert \nabla \varphi _{k+1}\Vert _2 + \mu _k \Vert Z\Vert _2 \Vert J_{k+1}\Vert _2 \Vert \lambda _k\Vert _2, \end{aligned}$$

(165)

where the last inequality follow from $\mu _{k+1} \ge \beta _s \mu _k$. Since $\{\lambda _k\}$ is bounded by construction and $\{J_k\}$ is bounded from Lemma 9.7, the last term on the right-hand side of (165) goes to zero as $k\rightarrow 0$. From Lemma 9.8, the middle term goes to zero as $k \rightarrow 0$. Hence, the entire right-hand side of (165) goes to zero and (158) holds.

Since the voltage angles appear in $c$ and $J$ only inside sine and cosine functions, they can be translated to lie inside $[-\pi ,\pi ]$ by adding or subtracting multiplies of $2\pi$ to create a sequence $\{\tilde{x}_k\}$ with uniformly bounded angles that satisfies

$$\begin{aligned} J(\tilde{x}_k)^Tc(\tilde{x}_k) = J_k^Tc_k, \end{aligned}$$

(166)

for all $k$. The sequence $\{\tilde{x}_k\}$ hence lies in a compact set and has a limit point $x^*$ that satisfies (160). $\square$

This completes the proof of the convergence of the vPF algorithm. It has been shown that this algorithm either terminates with a feasible point according to the required tolerance, or that it generates a sequence of iterates that has a limit point that is a stationary point of the function $c^Tc$ (restricted to the set of $x$ such that $Ax=b$).

Appendix 2: Lossy networks

In Sect. 5.3, lossy networks were introduced as power networks for which active power losses are positive for any set of complex bus voltages that are not all zero. In this section, a simple characterization of these networks based on the nodal admittance matrix is proved, and also that for these networks, bounded power mismatches implies bounded voltage magnitudes.

Lemma 10.1

A power network is a lossy network if an only if the Hermitian matrix $\tilde{G}$, as defined in (20), is positive definite.

Proof

Let $\{P_k^g\}_{k \in [n]}$ be the active powers injected by generators and $\{P_k^l\}_{k \in [n]}$ the active powers consumed by loads at each bus of the network. It is known that the total active power injected into the system must equal the total active power consumed by loads plus the total active power lost in the system. Hence

$$\begin{aligned} \sum _{k \in [n]} P_k^g - \sum _{k \in [n]}P^l_k = L, \end{aligned}$$

(167)

where $L$ denotes the total active power losses. For each $k \in [n]$, let $w_k$ be the complex voltage at bus $k$, i.e.,

$$\begin{aligned} w_k := v_k e^{j\theta _k}. \end{aligned}$$

(168)

Then, from (23),

$$\begin{aligned} \sum _{k \in [n]} P_k^g - \sum _{k \in [n]}P^l_k&= \mathfrak {R}\Bigg \{ \sum _{k \in [n]}\sum _{m \in [n]} v_k v_m Y^*_{km} e^{j(\theta _k-\theta _m-\phi _{km})} \Bigg \}\end{aligned}$$

(169)

$$\begin{aligned}&= \mathfrak {R}\Bigg \{ \sum _{k \in [n]}\sum _{m \in [n]} w_k w^*_m \tilde{Y}^*_{km} \Bigg \}\end{aligned}$$

(170)

$$\begin{aligned}&= \mathfrak {R}\big \{w^* \tilde{Y}^* w \big \}, \end{aligned}$$

(171)

where $\tilde{Y}$ is as defined in (19), $\mathfrak {R}\{\cdot \}$ gives the real part of its argument, and $*$ denotes conjugate transpose. From (20) and (21), it follow that

$$\begin{aligned} w^* \tilde{Y}^* w&= w^*\big (\tilde{G} + j \tilde{B}\big )^* w \end{aligned}$$

(172)

$$\begin{aligned}&= w^*\tilde{G}^*w - j w^*\tilde{B}^*w. \end{aligned}$$

(173)

Since both $\tilde{G}$ and $\tilde{B}$ are Hermitian, $\tilde{G} = \tilde{G}^*$, $\tilde{B} = \tilde{B}^*$, and hence that $w^*\tilde{G}^*w$ and $w^*\tilde{B}^*w$ are real. Therefore,

$$\begin{aligned} \sum _{k \in [n]} P_k^g - \sum _{k \in [n]}P^l_k = w^*\tilde{G}w, \end{aligned}$$

(174)

so $L = w^*\tilde{G}w$ and that for any $w \ne 0$, $L > 0$ if and only if $\tilde{G} \succ 0$. $\square$

Lemma 10.2

For all lossy networks and for all $M$, the set $\{x \ | \ f(x)^Tf(x) \le M \}$ has bounded voltage magnitudes, where $x$ and $f$ are the vector of power flow variables and the vector-valued function of power mismatches, respectively, as defined in Sect. 5.1.

Proof

Suppose the network is lossy and let $M > 0$. Let $x$ be such that $f(x)^Tf(x) \le M$ and $w$ the corresponding vector of complex bus voltages. Also, let $\fancyscript{I}$ denote the set $[n] \setminus \{s\}$, i.e., the set of all buses except the slack. Then, using results derived in Lemma 10.1,

$$\begin{aligned} \mathfrak {R}\Bigg \{ \sum _{k \in \fancyscript{I}} \sum _{m \in [n]} w_k w^*_m \tilde{Y}^*_{km} \Bigg \}&= \mathfrak {R}\Bigg \{ \sum _{k \in [n]} \sum _{m \in [n]} w_k w^*_m \tilde{Y}^*_{km} \Bigg \} - \mathfrak {R}\Bigg \{ \sum _{m \in [n]} w_s w^*_m \tilde{Y}^*_{sm}\Bigg \}\end{aligned}$$

(175)

$$\begin{aligned}&= w^* \tilde{G}w - \mathfrak {R}\{w_s w^*\eta \}, \end{aligned}$$

(176)

where $\eta ^*$ is the $s$-th row of $\tilde{Y}$, and $*$ denotes conjugate transpose. Since $w_s$ is a complex constant in the power flow problem, let $\nu$ to denote the constant complex vector $w_s\eta$. From (176) and the definition of $f$,

$$\begin{aligned} w^* \tilde{G}w - \mathfrak {R}\{w^*\nu \} = \sum _{k \in \fancyscript{I}} \Big (P_k^g - P_k^l\Big ) - \sum _{i \in \fancyscript{A}} f_i(x), \end{aligned}$$

(177)

where $\fancyscript{A}$ is the set of indices of the vector $f(x)$ that correspond to active power mismatches at buses $k \in [n] \setminus \{s\}$. Hence,

$$\begin{aligned} \lambda _{\min }(\tilde{G})\Vert w\Vert ^2 - \Vert w\Vert \Vert \nu \Vert \le \sum _{k \in \fancyscript{I}} \big |P_k^g - P_k^l\big | + \Vert f(x)\Vert _1, \end{aligned}$$

(178)

where $\Vert \cdot \Vert$ is the norm induced by the inner product $\langle a,b \rangle := b^*a$, for any complex vectors $a$ and $b$. Since $P_k^g$ and $P_k^l$ are constants (independent of $x$) for $k \in \fancyscript{I}$, the quantity

$$\begin{aligned} N := \sum _{k \in \fancyscript{I}} \big |P_k^g - P_k^l\big | \end{aligned}$$

(179)

is also a constant. Then, using the inequalities

$$\begin{aligned} \Vert f\Vert _1 \le m \Vert f\Vert _{\infty } \le m \sqrt{f^Tf} \le m\sqrt{M}, \end{aligned}$$

(180)

where $m$ is the dimension of $f(x)$, it follows that

$$\begin{aligned} \lambda _{\min }(\tilde{G})\Vert w\Vert ^2 - \Vert w\Vert \Vert \nu \Vert \le N + m\sqrt{M}. \end{aligned}$$

(181)

From Lemma 10.1, $\lambda _{\min }(\tilde{G}) > 0$. From this and the fact that $N$ and $M$ are independent of $x$, it can be concluded that the set

$$\begin{aligned} \{w \ | \ f(x)^Tf(x) \le M \} \end{aligned}$$

(182)

is bounded. $\square$