Synchronized states of power grids and oscillator networks by convex optimization

Abstract

Synchronization is essential for the operation of AC power systems: All generators in the power grid must rotate with fixed relative phases to enable a steady flow of electric power. Understanding the conditions for and the limitations of synchronization is of utmost practical importance. In this article, we propose a novel approach to compute and analyze the stable stationary states of a power grid or an oscillator network in terms of a convex optimization problem. This approach allows to systematically compute \emph{all} stable states where the phase difference across an edge does not exceed $π/2$.Furthermore, the optimization formulation allows to rigorously establish certain properties of synchronized states and to bound the error in the widely used linear power flow approximation.

Figures (7)

• Figure 1: Assessment of the improved power flow approximation \ref{['eq:improved_lin_approx']}. We compute the real power flows ${\boldsymbol{f}}$ for an adapted Matpower 30-Bus test case, cf. Appendix \ref{['sec:app:test_case']}, and compare numerically exact values ${\boldsymbol{f}}^{\rm (rp)}$ to the linear (DC) approximation ${\boldsymbol{f}}^{\rm (lin)}$ and the improved approximation ${\boldsymbol{f}}^{\rm (approx)}$ given by Eq. \ref{['eq:improved_lin_approx']}. (a) The left panel shows the loading $|f^{\rm (rp)}_e | / K_e$ for each edge $e$ for the exact solution. The other two panels depict the errors $|f^{\rm (rp)}_e - f^{\rm (lin, approx)}_e| / K_e$ of the linear (middle) and the improved approximation (right). Errors are hardly visible for the improved approximation \ref{['eq:improved_lin_approx']}, even on a logarithmic scale (b) For a systematic assessment we increase the overall grid load by multiplying all power injections by a scaling factor $p_f$. The figure shows the error on all edges as a function of $p_f$ in a scatter plot for the linear (left) and the improved (right) approximation. The improved approximation \ref{['eq:improved_lin_approx']} reduces the error on most edges by at least two orders of magnitudes for $p_f = 20$ and up to nine orders for $p_f=1$. For $p_f=1$, the errors of the improved approximation are below $10^{-14}$ approaching the numerical precision.
• Figure 2: Assessment of bounds of theorem \ref{['thm:david1']} for the error of the linear power flow approximation. We solve the real power flow and the linear power flow equations for the adapted Matpower 30-bus test case, cf. appendix \ref{['sec:app:test_case']}, and $10^6$ randomly sampled and valid power injection vectors ${\boldsymbol{p}}$. In each case we compute the norm of error $\| {\boldsymbol{\xi}} \|_K$ and compare it to the upper bounds given by $\| {\boldsymbol{\zeta}} \|_K$ and $\| {\boldsymbol{\Pi}}_{\rm cycle} {\boldsymbol{\zeta}} \|_K$, respectively. The figure shows a histogram of the ratio $\rm {bound}/{ \| {\boldsymbol{\xi}} \|_K}$ which serves as a measure for the tightness of the bound. The improved upper bound $\| {\boldsymbol{\Pi}}_{\rm cycle} {\boldsymbol{\zeta}} \|_K$ is significantly better and appears to be tight.
• Figure 3: Geometric interpretation of the cycle conditions. The optimal solution ${\boldsymbol{f}}^{\star}$ of the optimization problems \ref{['opt:realpower1']} and \ref{['opt:dcapprox1']} are given by the point where contour lines of the respective objective function $\mathcal{F} ({\boldsymbol{f}})$ and the linear subspace spanned by the equality constraints ${\boldsymbol{E}} {\boldsymbol{f}} = {\boldsymbol{p}}$ are tangential. Equivalently, ${\boldsymbol{f}}^\star$ is the point where the gradient $\nabla \mathcal{F} ({\boldsymbol{f}})$ is orthogonal to the linear subspace. Since any flow ${\boldsymbol{f}}$ can be decomposed into directed and a cycle flow ${\boldsymbol{f}}^{(c)} = {\boldsymbol{C}} {\boldsymbol{\ell}} \in \ker ({\boldsymbol{E}})$, the linear subspace is spanned by all points of the form ${\boldsymbol{f}}^{\star} + {\boldsymbol{C}} {\boldsymbol{\ell}}$.
• Figure 4: Improving the linear power flow approximation using a projected gradient descent. We compute the real power flows ${\boldsymbol{f}}$ for an adapted Matpower 30-Bus test case, cf. appendix \ref{['sec:app:test_case']}, and compare numerically exact values ${\boldsymbol{f}}^{\rm (rp)}$ to the improved approximation given by Eq. \ref{['eq:gradient-step']}. (a) We show the error of the approximation as a function of the step size $\gamma$ for different values of the scaling factor $p_f$ that controls the grid load. The optimal step size $\gamma^\star$, drawn as a black dotted line, increases with $p_f$. The red hatched area shows where the gradient descent step does not improve the approximation. (b) The maps depict the error of the approximation $|f^{\rm (rp)}_e - f^{\rm (lin, approx)}_e| / K_e$ for each individual line $e$, comparing the linear approximation and the improved approximation given by Eq. \ref{['eq:gradient-step']} for the optimal step size $\gamma^\star$ and $p_f = 1.028$.
• Figure 5: Geometry of the optimization problems associated with the real power flow and the linear power flow approximation. (a) We consider an elementary 4-node network with a high degree of symmetry and only two degrees of freedom $f_1$ and $f_2$. (b) The contour lines of the objective function $\mathcal{F} _{lin}({\boldsymbol{f}})$ are ellipses. In comparison, the contour lines of $\mathcal{F} _{rp}({\boldsymbol{f}})$ are slightly contracted along the axes. The respective optimizers ${\boldsymbol{f}}^{(rp)}$ and ${\boldsymbol{f}}^{(lin)}$ of the constrained optimization problems (dots) are the points where the contour lines are tangent to the affine subspace defined by ${\boldsymbol{E}} {\boldsymbol{f}} = {\boldsymbol{p}}$, cf. Fig. \ref{['fig:orthogonality']}. Due to the contraction along the axes, the optimizer ${\boldsymbol{f}}^{(rp)}$ is further from the coordinate axis. Physically, this corresponds to a more balanced flow and, thus, a smaller maximum line loading.

Theorems & Definitions (20)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Lemma 3
• proof
• Theorem 2
• proof