Time-Varying Semidefinite Programming: Path Following a Burer--Monteiro Factorization

Bellon, Antonio; Dressler, Mareike; Kungurtsev, Vyacheslav; Mareček, Jakub; Uschmajew, André

doi:10.48550/arxiv.2210.08387

Cited by 1 publication

(4 citation statements)

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“…To validate our approach, we consider the application of a distribution network with high penetration of photovoltaic (PV) systems, introduced by Dall'Anese and Simonetto [10], albeit our approach is by no means limited to radial networks. The modified network is a single-phase variant of the IEEE 37-node test feeder, by replacing loads of 18 secondary transformers (at nodes 4,7,10,13,17,20,22,23,26,28,29,30,31,32,33,34,35, and 36, as highlighted in Figure 1) with real load data from Anatolia, California, sampled with 1 Hz frequency in August 2012 [47]. Furthermore, the generation at PV plants is simulated based on real solar irradiance data in [47], with rating of these inverters at 300 kVA at node 3; 350 kVA at nodes 15, 16, and 200 kVA for all other inverters.…”

Section: Resultsmentioning

confidence: 99%

“…This suggests to apply the Lasserre's hierarchy in a time-varying framework. This, however, might lead to complications that limit the ability to track the solution to timevarying SDP using path-following strategies as developed in [17]. As discussed above in this section, even in a best-case scenario, the trajectory of the solution may lose smoothness or even continuity, making it hard or impossible to use local information.…”

Section: B Path-following For Convex Approximationsmentioning

confidence: 99%

“…Moreover, let Assumption V.4 hold and ξ 0 , ξ 1 , • • • ∈ Ξ. Then the randomized coordinate-descent algorithm (16) for solving (17) with L k (ξ, µ) instead of L(ξ, µ), where i k is chosen uniformly at random from {1, 2, . .…”

Section: Iteration Complexitymentioning

confidence: 99%

“…A preliminary version of this paper has appeared as [15]. Subsequently, we have developed further insights [16] into the problem, as well as a path-following procedure to track the trajectory of solutions to time-varying semidefinite programming [17]. The present paper extends beyond this preliminary version [15] in several ways.…”

mentioning

confidence: 99%

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Optimal Power Flow Pursuit in the Alternating Current Model

Liu¹,

Bellon²,

Simonetto³

et al. 2022

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Transmission-constrained problems in power systems can be cast as polynomial optimization problems whose coefficients vary over time. We consider the complications therein and suggest several approaches. On the example of the alternatingcurrent optimal power flows (ACOPFs), we illustrate one of the approaches in detail. For the time-varying ACOPF, we provide an upper bound for the difference between the optimal cost for a relaxation using the most recent data and the current approximate optimal cost generated by our algorithm. This bound is a function of the properties of the instance and the rate of change of the coefficients over time. Moreover, we also bound the number of floating-point operations to perform between two subsequent updates to ensure a bounded error.Index Terms-Numerical analysis (Mathematical programming), optimization, Power system analysis computing I. INTRODUCTIONRenewable energy sources (RESs) have created several new challenges for power system analysis and control. In particular, power quality and reliability can be undermined when RESs are used widely and when all the available power is injected. Furthermore, in distribution systems, overvoltages might become frequent. There, as well as in transmission systems, rapid changes in power output can even cause power flow reversals, as well as unexpected losses and transients that current systems cannot handle. Therefore, real-time control mechanisms must be designed, for example, to limit real power at RESs inverters, taking into account transmission constraints. As a main complication, the transmission-constrained problems coming from the alternating-current model are non-convex and nonlinear. In both theory [1] and practice, linearizations tend to produce infeasible solutions. Approaches applying Newton method [2] to the non-convex problem in a rolling-horizon framework often perform well in practice, as long as the changes are limited. However, in general, they provide little to none theoretical guarantees on their performance. In contrast, solutions to specific relaxations (see, for example, [3], [4]) coincide under mild assumptions with those to non-convex problems, for all initial points. As a main complication, it can actually take a long time to solve the relaxation, so that meanwhile the inputs can change significantly, so that when the solution is available, it might already be outdated. This

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