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Distributed Optimal Load Frequency Control Considering Nonsmooth Cost Functions

Zhaojian Wang, Feng Liua, Changhong Zhao, Zhiyuan Ma, Wei Wei

TL;DR

This work addresses the distributed frequency control problem in power systems considering controllable load with a nonsmooth cost using the Clark generalized gradient and proves the optimality of the equilibrium of the closed-loop system as well as its asymptotic stability.

Abstract

This work addresses the distributed frequency control problem in power systems considering controllable load with a nonsmooth cost. The nonsmoothness exists widely in power systems, such as tiered price, greatly challenging the design of distributed optimal controllers. In this regard, we first formulate an optimization problem that minimizes the nonsmooth regulation cost, where both capacity limits of controllable load and tie-line flow are considered. Then, a distributed controller is derived using the Clark generalized gradient. We also prove the optimality of the equilibrium of the closed-loop system as well as its asymptotic stability. Simulations carried out on the IEEE 68-bus system verifies the effectiveness of the proposed method.

Distributed Optimal Load Frequency Control Considering Nonsmooth Cost Functions

TL;DR

This work addresses the distributed frequency control problem in power systems considering controllable load with a nonsmooth cost using the Clark generalized gradient and proves the optimality of the equilibrium of the closed-loop system as well as its asymptotic stability.

Abstract

This work addresses the distributed frequency control problem in power systems considering controllable load with a nonsmooth cost. The nonsmoothness exists widely in power systems, such as tiered price, greatly challenging the design of distributed optimal controllers. In this regard, we first formulate an optimization problem that minimizes the nonsmooth regulation cost, where both capacity limits of controllable load and tie-line flow are considered. Then, a distributed controller is derived using the Clark generalized gradient. We also prove the optimality of the equilibrium of the closed-loop system as well as its asymptotic stability. Simulations carried out on the IEEE 68-bus system verifies the effectiveness of the proposed method.

Paper Structure

This paper contains 16 sections, 3 theorems, 29 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Preliminaries and notations
  4. Notations
  5. Preliminaries
  6. Network model
  7. Problem Formulation
  8. Optimization problem
  9. Controller Design
  10. Optimality and Convergence
  11. Optimality
  12. Convergence
  13. Case studies
  14. Test system
  15. Simulation results
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose Assumptions convex and Slater hold. We have

Figures (5)

  • Figure 1: IEEE 68-bus system
  • Figure 2: Frequency dynamics under AGC and OLC
  • Figure 3: Dynamics of $\mu$ and controllable load
  • Figure 4: Active power dynamics of line $1$ and controllable load
  • Figure 5: Dynamics of $\mu$ when line power congestion exists

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4: Load demand estimate
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof