Project purpose

Increasing penetration of renewable generation and demand-side penetration from DERs calls for operational planning methods that account for uncertainty taking the form of significant large and complex load fluctuations, see Figure 1. The goal of the project is to develop stochastic optimization tools that able to accommodate more accurate power flow models. These methods need to be specifically designed to account for the non-linearity of AC power equations and uncertainty at the same time while maintaining efficiency.

 

Technical approach

The first part of the approach deals with the primary challenge of finding scalable ways to integrate nonlinearity of the power flow physics and the effect of uncertainty into optimization. This involves investigation of several uncertainty quantification methods that can be incorporated into an AC optimal power flow formulation. The methods explored are (i) Partial Linearization that linearizes the effect of uncertainty while keeping the nonlinearity of the power flows for the nominal case, (ii) the Lassere’s hierarchy that approximates probabilistic constraints with polynomials, and (iii) the polynomial chaos expansion that represents the effect of uncertainty as additional constraints in the optimization. Based on observations during the first year of the project, the polynomial chaos expansion has been identified as the most promising method in term of quality and scaling potentiality, see Figure 2. Subsequent efforts have been devoted to making this approach scalable to large systems by taking advantage of network structure and the specifics of the power-flow physics.

The second technical approach attempts to accelerate real-time stochastic OPF by using machine learning tools to identify features of OPF that can be learnt offline. Specifically, this approach creates a mapping between the realization of the uncertainty and the active set of constraints at optimality. This method ultimately aims at performing near real-time stochastic OPF and preliminary tests have been successful on DC-OPF.

The third part is dedicated to formulating inner approximations a robust AC optimal power flow formulation. Special consideration is given to enforcing stability under uncertainty, which requires guaranteeing feasibility of the AC power flow equations themselves. The main technical tool is creating a convex restriction of the robust optimal power flow in the form of an explicit set of convex quadratic constraints, that guarantees feasibility of both the power flow equations and the technical limits. An algorithm consisting of solving a sequence of convex optimization problems is designed to solve the robust AC optimal power flow problem. Experiments on test cases from the PGLib suite demonstrates the efficiency of the algorithm.

Summary of Executed and Planned Results

In 2018 we developed and implemented three uncertainty quantification methods that can be incorporated in AC-OPF formulation. We assessed pros and cons of each UQ methods, and we identified polynomial chaos expansion to be the most promising method for its quality of UQ and potential of scalability. We also developed an active set learning based method and perform successful preliminary testing on DC-OPF.
In 2019 we improve the scalability of polynomial chaos expansion for AC-OPF by exploiting network structure and power flow physics. Our results show a significant reduction in computational time and the method has been scaled to thousand bus systems. We also developed an inner-approximation for robust AC-OPF formulation and validated the method on PGLib test-cases.

 

Illustrations

Figure 1: Our approach guarantees safe generation dispatch in electric transmission networks with heavily fluctuating loads, even when fluctuations are highly correlated within subregions (in orange).

 

 

 

 

Figure 2: Estimated probability densities of voltage resulting from load fluctuations. Voltages higher than Vmax are unsafe. Our sparse PCE method (in green) is as accurate as the costly exhaustive Monte-Carlo computation (in orange). For comparison, the vanilla low-degree PCE (in blue) overestimates the probability of failure.