Efficient relaxations for joint chance constrained AC optimal power flow
Introduction
Increasing penetrations of intermittent energy sources in the electric power grid, evolving faster than the corresponding infrastructure, can increase the probability that line congestions may occur, and that voltages may lay outside of desired limits. Rather than considering this randomness as a deterministic input or representing the uncertainty via computationally prohibitive scenario-based approaches, we solve a joint chance constraint problem which prevents overvoltages in the grid with a certain probability. Single chance constraints, which ensure that a single constraint is satisfied with a prescribed probability, have been considered in a variety of power systems applications, many addressing the problems of line congestions [1], [2], voltage regulation [3], [4], unit commitment [5], [6], transmission expansion planning [7], and energy storage sizing [8]. These works constrain the individual probability of constraint violations independently; however, considering these constraints separately neglects to explicitly consider simultaneous violation conditions in the overall system. For example, in the single chance constraint approach, it may be independently guaranteed that the voltage at each node is under its maximum with a high probability, but this could result in a situation where multiple nodes simultaneously have overvoltage conditions. Thus, perhaps a more relevant constraint to consider is restricting the probability of all voltages, line flows, states of charge, etc. being within prescribed limits throughout the system.
Joint chance constraints, which ensure multiple constraints are simultaneously satisfied with a given probability, have been considered in [9] for the N − 1 security problem, and the joint chance constraint problem was solved by utilizing a sample-based scenario approach. In [10], both single and joint chance constraints were considered to mitigate line congestions, and a Monte-Carlo based approach was developed to estimate the joint probability. Due to the difficulty in reformulating joint chance constraints into deterministic ones that can be utilized in an optimization problem, Boole's inequality [11] is a popular choice to provide an upper bound on the original chance constraint [12], [13], separating the joint chance constraint P(g1(x, δ) ≤ 0, …, gn(x, δ) ≤ 0) ≤ 1 − ϵ, where x is a vector of decision variables, δ is a random parameter, and ϵ is the maximum constraint violation probability, into single chance constraints P(g1(x, δ) ≤ 0) ≤ 1 − ϵ1 ⋯ P(gn(x, δ) ≤ 0) ≤ 1 − ϵn for i = 1, …, n, where . A common choice for ϵi is usually ϵ/n [12], [13]; however, this parameter can also be optimized [14]. In [15] a Monte Carlo method is used to solve a sequence of convex optimization problems and compute the joint chance constraint directly; however, it is computationally slow and can only handle relatively small problems. Ambiguous joint chance constraints were studied in [16], [17], in which the random parameters are assumed to belong to a so-called ambiguity set. In addition, distributionally robust joint chance constraints are studied in [18] using semidefinite programming based approximations, where the tightness of the approximation is tuned via a sequential convex optimization algorithm.
Our approach requires no parameter tuning, or other computationally burdensome techniques such as including ϵ as an optimization variable and performing further approximations to convexify the problem. Instead, the benefits of utilizing Boole's inequality are enhanced by a fast yet effective method that tightens the chance constraint reformulations in a simple and straightforward way. In addition, by exploiting the structure of the voltage regulation problem; i.e., assuming the probability that the system is operating within normal voltage regions is high, the improved bound decreases the cost of the otherwise overly conservative nature of using Boole-based inequalities. It is shown that the new upper bound is tighter than or equal to Boole's inequality, and intuitively amounts to using a bound on the excess probabilities of the intersection of all events using Fréchet's inequality, or estimating this intersection with a small number of samples, which Boole's inequality overestimates (see Fig. 1 for a four event example).
Finally, by utilizing a linearization of the AC power flows in a distribution network, the chance-constrained voltage regulation problem is solved to address the relevant problem of overvoltages under high penetrations of distributed generation (photovoltaic systems). Simulation results are performed using a modified IEEE-37 node test feeder with community PV systems, and the new bound is compared with a deterministic formulation and the traditional Boole's inequality used in joint chance constraint reformulations. The contributions of this paper are threefold: Firstly, to the best of the authors’ knowledge, we present the first joint chance constrained optimal power flow formulation for voltage regulation in distribution grids. Secondly, we present a computationally efficient, simple improvement over a technique that is commonly used to handle joint chance constraints with a tighter upper bound. Lastly, we not only show in simulation the improvement given by the new technique, but we formally prove the tightness of the new bound.
Section snippets
Joint chance constraint relaxation
The joint chance constraint considered here requires the probability of all voltages in the system to be under than the maximum voltage limit with a probability greater than or equal to 1 − ϵ:where g1(x, δ) < 0, …, gn(x, δ) < 0 constrain the voltage magnitude at each bus i, Vi, to be less than to the maximum voltage magnitude , x is a vector of decision variables, δ is a jointly distributed Gaussian random vector with mean μ and positive definite covariance matrix Σ, and
Distribution network
Consider a distribution feeder comprising of N + 1 nodes collected in the set , , and lines represented by the set of edges . Let and denote the phasors for the line-to-ground voltage and the current injected at node , respectively, and define the N-dimensional complex vectors and . Node 0 denotes the secondary of the distribution transformer, and it is taken to be the slack bus. Using Ohm's and Kirchhoff's laws, the
Optimization problem reformulation
The original, unrelaxed joint chance constraint optimization for voltage regulation in distribution systems shown below: for all , where denotes the kth element of va. Constraint (16b) represents a surrogate for the power balance equation; constraint (16c) is the joint chance constraint that require every voltage magnitude in the grid to be within upper and lower limits with at least 1 − ϵ
Results on IEEE-37 Node Test Feeder
The IEEE-37 node test feeder [25] was used for the following simulations. The actual five-minute load and solar irradiance data was obtained from [26] for the simulations, and shown in Fig. 4. In order to emulate a situation with high-PV penetration and risks of overvoltage, 16,200-kW rated PV systems were placed at nodes 3–18. We seek to minimize renewable curtailment; specifically,where the cost of curtailing power at each node is set to be bi = $0.10. The number of
Conclusion
In this paper, we presented a method for tractable computation of joint chance constraints in probabilistic AC OPF problems that improves upon Boole's inequality, which is often used to improve the tractability of joint chance constrained problems. The paper presented three main results: Firstly, a new application of joint chance constraints was applied to the distribution grid voltage regulation problem. Secondly, a computationally efficient technique for improving the traditional bound
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