A Pad\'e-Weierstrass technique for the rigorous enforcement of control limits in power flow studies

Abstract

A new technique is presented for solving the problem of enforcing control limits in power flow studies. As an added benefit, it greatly increases the achievable precision at nose points. The method is exemplified for the case of Mvar limits in generators regulating voltage on both local and remote buses. Based on the framework of the Holomorphic Embedding Loadflow Method (HELM), it provides a rigorous solution to this fundamental problem by framing it in terms of \emph{optimization}. A novel Lagrangian formulation of power-flow, which is exact for lossless networks, leads to a natural physics-based minimization criterion that yields the correct solution. For networks with small losses, as is the case in transmission, the AC power flow problem cannot be framed exactly in terms of optimization, but the criterion still retains its ability to select the correct solution. This foundation then provides a way to design a HELM scheme to solve for the minimizing solution. Although the use of barrier functions evokes interior point optimization, this method, like HELM, is based on the analytic continuation of a germ (of a particular branch) of the algebraic curve representing the solutions of the system. In this case, since the constraint equations given by limits result in an unavoidable singularity at $s=1$, direct analytic continuation by means of standard Pad\'e approximation is fraught with numerical instabilities. This has been overcome by means of a new analytic continuation procedure, denominated Pad\'e-Weierstrass, that exploits the covariant nature of the power flow equations under certain changes of variables. One colateral benefit of this procedure is that it can also be used when limits are not being enforced, in order to increase the achievable numerical precision in highly stressed cases.