Solving the Hamilton-Jacobi-Bellman equation for a stochastic system with state constraints

Rutquist, Per; Wik, Torsten; Breitholtz, Claes

doi:10.1109/cdc.2014.7039666

Cited by 6 publications

(6 citation statements)

References 13 publications

(19 reference statements)

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“…First and foremost, obtaining exact numerical solution of HJB function is still an academic issue, most of the solving mechanisms existing seek the approximation to viscosity solution instead [29]. Rutquist [30] proposed an approach to seek the approximate numerical solution of HJB equation with state constraints, which demands much less collocation points to give a good approximation, reduces the computing power requirements.…”

Section: B Public Information Setupmentioning

confidence: 99%

“…Although it is not common, for (28), the solving process can demand finer mesh or large number of collocation points if it has a large feasible domain together with a large initial condition (x i t0 − x i t ). Large amount of collocation points inevitably leads to knotty computational complexity even if it gets relieved by [30]. If it takes significantly long time to solve one PDE, a user may have no sufficient time to implement the control or come up with considerable error.…”

Section: B Public Information Setupmentioning

confidence: 99%

“…To tackle this, the optimization problem is segmented into several small continuous problems if the original one is too large. A numerical test is stated by the Table I in [30], which suggests a proper amount below 64 for collocation points. Therefore, an empirical limit of 5 is given to (x i t0 − x i t ).…”

Section: B Public Information Setupmentioning

confidence: 99%

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A Mean-Field Voltage Control Approach for Active Distribution Networks With Uncertainties

Wei

Qiu

2021

IEEE Trans. Smart Grid

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A model-free distributed control scheme that implements active voltage control in low voltage distribution network (LVDN) is proposed. By solving an individual Hamilton-Jacobi-Bellman-Flemming function with public information, users can compute a good approximation to their optimal control trajectory and take uncertainties into account in a distributed manner. A detailed mathematical framework is given, accompanied by a discussion on the different entries of uncertainties. The proposed control scheme uses a broadcast signal to indicate the probability distribution in mean field theory, and to streamline the demand on Fokker-Planck-Kolmogorov PDE or Mc-Kean Vlasov SDE, which relieves the computational burden. A realistic semiurban distribution network is modified as the study case, with a benchmark of centralized ACOPF to study the performance of the proposed approach. Moreover, two special cases including communication latency and packet loss are given as well, in order to test the robustness of the proposed approach. The results prove that the proposed approach is able to deliver good approximation to the optimal control with uncertainty in a model-free and distributed manner. Index Terms-Active distribution networks, Distributed optimization, Mean field theory, Hamilton-Jacobi-Bellman equation, Model-free, Voltage control. I. INTRODUCTIONW ITH the development of distributed energy resources (DERs), renewable penetration increases not only in medium voltage distribution network, but also low voltage distribution network (LVDN). Along with this trend, conventional passive distribution networks are being transformed into active distribution network (ADN) [1]. Meanwhile, shapeable/deferrable loads spring up in LVDNs, which enlarges the applicability of ADN. Nevertheless, the implementation of ADN is the synonym of a dramatic increase in the amount of smart meters, sensors, and control devices, via bidirectional communication networks. On one hand, high burdens on both communication and computation makes the infrastructure and operation expensive; on the other hand, the rising demand in data collection is being more and more discordant with the awakening public concerns on the privacy protection of data. In this paper, the term users denotes all the DERs connected to LVDNs, which includes but is not limited to photovoltaics (PV), residential wind turbines, smart households and so on. Therefore, the main challenge to implement active controls B. Wei and G. Deconinck are with the

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Section: B Public Information Setupmentioning

confidence: 99%

Section: B Public Information Setupmentioning

confidence: 99%

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A Mean-Field Voltage Control Approach for Active Distribution Networks With Uncertainties

Wei

Qiu

2021

IEEE Trans. Smart Grid

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“…Therefore, the faster the PDE is solved, the better approximation to the optimum is yielded. A typical PDE is soved within 1 second by using the approach from [16] on a Intel I7-6600U processor. Although users cannot be expected to have such capability, three PDEs per minute are enough to give an admissible control, which will be illustrated in the case study.…”

Section: Control Implementationmentioning

confidence: 99%

Distributed Online Optimization for Voltage Control in Low Voltage Distribution Networks

Wei

2020

2020 IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe)

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In order to reduce the communication burden in online optimization in low voltage distribution networks (LVDNs), a distributed online optimization approach is proposed. Hamilton-Jacobi-Bellman equation and local measurements are integrated to seek the approximation to optimum. Voltage control is implemented as the embodiment of the proposed approach, which is validated in a realistic 103 nodes LVDN.

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“…Consequently, the solution Ψ * is a continuous function defined on a bounded domain. Hence, Ψ u and Ψ l can be made arbitrary close to Ψ * by the Stone-Weierstrass Theorem [21] in (12). However, this guarantee is lost when Ψ u and Ψ l are restricted to be SOS polynomials.…”

Section: B Controller Synthesismentioning

confidence: 99%

Suboptimal stabilizing controllers for linearly solvable system

Leong

Horowitz

Burdick

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-This paper presents a novel method to synthesize stochastic control Lyapunov functions for a class of nonlinear, stochastic control systems. In this work, the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation is transformed into a linear partial differential equation for a class of systems with a particular constraint on the stochastic disturbance. It is shown that this linear partial differential equation can be relaxed to a linear differential inclusion, allowing for approximating polynomial solutions to be generated using sum of squares programming. It is shown that the resulting solutions are stochastic control Lyapunov functions with a number of compelling properties. In particular, a-priori bounds on trajectory suboptimality are shown for these approximate value functions. The result is a technique whereby approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems.

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Solving the Hamilton-Jacobi-Bellman equation for a stochastic system with state constraints

Cited by 6 publications

References 13 publications

A Mean-Field Voltage Control Approach for Active Distribution Networks With Uncertainties

A Mean-Field Voltage Control Approach for Active Distribution Networks With Uncertainties

Distributed Online Optimization for Voltage Control in Low Voltage Distribution Networks

Suboptimal stabilizing controllers for linearly solvable system

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