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Fractional Order AGC for Distributed Energy Resources Using Robust Optimization

Indranil Pan, Saptarshi Das

arXiv:1611.09755v1eess.SYcs.AIcs.NEmath.OC

TL;DR

The paper addresses fractional-order AGC for frequency oscillation damping in a distributed-energy system with stochastic communication delays and uncertain controller implementation. It combines centralized FOPID control with PSO-based robust optimization and finds that robust designs better tolerate parameter perturbations, while FOPID outperforms PID in the reported designs. The study also shows that archive-based evaluation reduces the computational expense of robust optimization.

  • Problem

    The paper investigates how to damp frequency oscillations in a complex distributed-energy system whose renewable generation, storage devices, and communication network introduce variability and stochastic delays.

  • Method

    The study uses centralized FOPID AGC, PSO-based robust optimization, and an archive strategy to tune controllers and reduce costly objective-function evaluations.

  • Results

    Robust designs outperform optimal designs under controller and system-parameter perturbations, while FOPID outperforms PID and nominal optimal and robust performance is nearly unchanged.

  • Takeaways & Limitations

    The robust design accommodates variation in controller gains, fractional orders, and implementation realizations without significant performance loss.

  • Takeaways & Limitations

    The study does not model packet dropouts and identifies physical hardware and real-time implementation as future work.

Abstract

from arXiv · show

The applicability of fractional order (FO) automatic generation control (AGC) for power system frequency oscillation damping is investigated in this paper, employing distributed energy generation. The hybrid power system employs various autonomous generation systems like wind turbine, solar photovoltaic, diesel engine, fuel-cell and aqua electrolyzer along with other energy storage devices like the battery and flywheel. The controller is placed in a remote location while receiving and sending signals over an unreliable communication network with stochastic delay. The controller parameters are tuned using robust optimization techniques employing different variants of Particle Swarm Optimization (PSO) and are compared with the corresponding optimal solutions. An archival based strategy is used for reducing the number of function evaluations for the robust optimization methods. The solutions obtained through the robust optimization are able to handle higher variation in the controller gains and orders without significant decrease in the system performance. This is desirable from the FO controller implementation point of view, as the design is able to accommodate variations in the system parameter which may result due to the approximation of FO operators, using different realization methods and order of accuracy. Also a comparison is made between the FO and the integer order (IO) controllers to highlight the merits and demerits of each scheme.

I. INTRODUCTION

The paper develops centralized fractional-order AGC for a hybrid distributed-energy system with renewable generation, storage, nonlinear device constraints, and stochastic communication delays. It uses robust optimization to tune the controller under these operating conditions.

  • Distributed energy resources combine wind, solar, storage, and combined heat and power technologies, increasing the importance of control and communication.
  • Random delays in forward and feedback communication paths make the control loop unreliable and motivate incorporating stochastic delays into controller design.
  • The proposed scheme uses a FOPID-based centralized AGC with PSO-based robust optimization to tune controller parameters and assess parametric robustness.
  • The hybrid model includes wind-turbine, photovoltaic, fuel-cell, diesel-engine, and aqua-electrolyzer subsystems, plus flywheel and battery energy storage.
  • The storage and generation devices are actuated by controller signals, with stochastic actuator delay and rate-constraint nonlinearities representing electromechanical limits.

D. Power System Model Using Grid Frequency Deviation

The power-system model links stochastic wind behavior and renewable power conversion to grid-frequency deviation. Wind generation is represented with base and noise components, while turbine output depends on aerodynamic operating variables.

  • The system transfer function maps grid power deficit or surplus to frequency deviation using equivalent inertia and damping constants.The simulation uses M = 0.4 and D = 0.03 as typical values.
  • Wind speed is modeled as a sum of a constant base component and a stochastic noise component to capture fluctuation and turbulence.
  • The noise model uses a spectral-density formulation with random phase, variance σ^2 = 200, N = 50, and frequency step Δω = 0.5 rad/s.
  • Wind-turbine power is characterized through the power coefficient as a function of tip-speed ratio and blade pitch angle.
  • The turbine mechanical-power model depends on air density, swept blade area, wind speed, and the power coefficient.

G. Characteristics of PV Output Power and Demand Power

The photovoltaic and demand-power models represent stochastic renewable supply and load variation for evaluating frequency-control behavior. A plotted realization is independent of controller structure.

  • Photovoltaic output is modeled from solar radiation, cell conversion efficiency, array area, and ambient temperature.The specified efficiency is η = 10%, array area is S = 4084 m^2, and ambient temperature is T_a = 25 °C.
  • The PV and load models include stochastic variations that produce sudden fluctuations in renewable generation and demand power.
  • Figure 2 shows one realization of photovoltaic generation, wind-turbine generation, load demand, and net renewable power supplied to the grid.

H. Control Over Unreliable Communication Network

The controller operates across a shared network with independently random delays before and after the controller. A fractional-order PID structure is rationally approximated for numerical implementation.

  • Measurements and actuator commands communicate through a shared medium that introduces random delays in both feedback and forward control paths.The delays are sampled uniformly from [0.05, 0.15].
  • The FOPID controller has five tuning variables: proportional, integral, and derivative gains plus fractional orders λ and μ.
  • Setting λ = 1 and μ = 1 reduces the fractional-order controller to the classical parallel PID structure.
  • Each candidate pair of fractional orders is continuously rationalized using Oustaloup’s 5th-order approximation over 10^-2 to 10^2 rad/s.
  • The delayed grid-frequency deviation enters the controller, which produces the control action sent to the actuated system.

IV. OPTIMIZATION ALGORITHMS AND CONTROL OBJECTIVES

Robust optimization evaluates solutions by their sensitivity to input variation rather than nominal objective value alone. A robust solution may sacrifice nominal optimality to reduce worst-case degradation.

  • A. The Concept of Robust Optimization: An optimal solution minimizes the nominal objective, whereas a robust solution limits objective variation when design variables fluctuate.The robust solution has a higher objective value but a less severe worst-case outcome.
  • A. The Concept of Robust Optimization: Robustness is assessed by evaluating how input-variable fluctuations affect the objective function through their probability distribution.The paper formulates a fitness measure using the expected objective value over the fluctuation space.
  • A. The Concept of Robust Optimization: The optimization problem has an n-dimensional decision space, where n denotes the number of decision variables.The supplied formulation identifies n as the problem dimension.
  • A. The Concept of Robust Optimization: Fig. 3 schematically contrasts the nominal optimum with the less variation-sensitive robust solution.The figure illustrates the distinction between the two solution types.

B. Objective Function for Optimization Based Control

The controller-design objective jointly penalizes frequency deviation and incremental control effort. Equal weighting is used, while implementation differences among fractional-order realizations motivate robust parameter design.

  • B. Objective Function for Optimization Based Control: The objective function minimizes both hybrid-system frequency deviation and the incremental control signal.These correspond to weighted ISE and ISDCO terms.
  • B. Objective Function for Optimization Based Control: Including control-signal variation limits battery-capacity requirements, flywheel jerk, and diesel consumption.The objective is formulated to reduce actuator demands alongside frequency oscillations.
  • B. Objective Function for Optimization Based Control: Derivative action and the control-signal difference operator can amplify noise, motivating scaling before assigning objective weights.The paper notes that alternative weightings remain possible, while multi-objective optimization can expose trade-offs.
  • B. Objective Function for Optimization Based Control: Different analog or digital FO realization methods and approximation orders can produce different time- and frequency-domain responses.The paper uses this implementation variability as a motivation for robust controller design.
  • B. Objective Function for Optimization Based Control: Figs. 4 and 5 compare phase and impulse responses across band-limited FO realizations and approximation orders.Together, they illustrate discrepancies relevant to implementing a single FO operator.

D. Canonical PSO (CPSO) Optimizer and Its Variants

The paper uses canonical PSO to search the controller-design space and examines fully informed and charged variants. These variants differ mainly in how particle velocities incorporate neighborhood information and repulsion.

  • D. Canonical PSO (CPSO) Optimizer and Its Variants: CPSO represents candidate controller designs as particle positions and updates each particle’s position using its velocity.The objective is minimized over an n-dimensional real-valued search space.
  • D. Canonical PSO (CPSO) Optimizer and Its Variants: CPSO velocity updates combine inertia, cognitive learning from each particle’s best position, and social learning from the neighborhood best.The constriction coefficient and learning rates control the update.
  • D. Canonical PSO (CPSO) Optimizer and Its Variants: FIPS differs from CPSO by incorporating all neighboring solutions rather than only the best neighbor in its velocity update.The neighborhood contribution is summed across particle neighbors.
  • D. Canonical PSO (CPSO) Optimizer and Its Variants: CCPSO adds an acceleration or repelling term to maintain population diversity and avoid premature convergence.The repulsion force depends on particle separation and parameters including core radius and perception limit.
  • D. Canonical PSO (CPSO) Optimizer and Its Variants: The experiments use 2500 effective fitness evaluations, a population of 30, learning rates β1 = 2.8 and β2 = 1.3, and a ring topology for CPSO.For CCPSO, the perturbation of controller gains and orders sets the relevant repulsion parameters.

E. Archive Strategy in PSO Variants for Robust Optimization

The archive strategy approximates robust fitness by reusing previously evaluated points instead of repeatedly calling the expensive objective function. Sampling and archive cleanup control approximation quality and storage.

  • E. Archive Strategy in PSO Variants for Robust Optimization: The multi-evaluation model approximates expected robust fitness by sampling perturbed objective values and averaging them.The perturbations are drawn according to the input-variation probability distribution.
  • E. Archive Strategy in PSO Variants for Robust Optimization: An archive stores evaluated points and their fitness values to reduce calls to the original objective function.Increasing the number of samples improves approximation but increases computational expense.
  • E. Archive Strategy in PSO Variants for Robust Optimization: Latin Hypercube sampling identifies representative archive points for robust fitness evaluation without recomputing the expensive objective.Points are assigned to usable or resampling sets according to nearest-point relationships.
  • E. Archive Strategy in PSO Variants for Robust Optimization: Using all usable archive values provides a more accurate approximation of robust fitness.The formulation is modified to incorporate the available archive points and their weighting function.
  • E. Archive Strategy in PSO Variants for Robust Optimization: The archive is cleaned up at a maximum size of 5000 elements, and 10 samples are used for effective fitness approximation.These settings bound archive growth and sampling effort.

A. Controller Design for Distributed Energy Resources

The study tunes optimal and robust PID/FOPID controllers with PSO variants under controller-gain and fractional-order perturbations. Robust designs preserve performance under uncertainty while archive-based evaluation substantially reduces computational expense.

  • Controller Design for Distributed Energy Resources: Robust tuning perturbs controller gains and fractional orders in both positive and negative directions, while CPSO tunes optimal controllers and CPSO, CCPSO, and FIPS tune robust controllers.The perturbation set includes controller gains and orders, and the algorithms are compared across PID and FOPID structures.
  • Controller Design for Distributed Energy Resources: Optimal FOPID achieves the lowest min J, followed by optimal PID; among robust designs, CCPSO is best for FOPID and CPSO for PID.Robust solutions have higher min J because it is evaluated over gain and order variations around nominal values.
  • Controller Design for Distributed Energy Resources: After 2500 effective fitness function evaluations, min J and controller parameters become almost constant, indicating convergence to the respective minima.The convergence and parameter-evolution plots report this stabilization for optimal and robust PID/FOPID cases.
  • Controller Design for Distributed Energy Resources: Optimal controllers are slightly better under nominal conditions, whereas robust controllers maintain good performance and better tolerate controller-parameter uncertainties.The individual component powers show no appreciable difference between optimal and robust solutions.
  • Complexity of Robust Optimization Algorithms: Archive-based robust algorithms require less than 22% of the actual evaluations for FOPID and less than 9% for PID compared with algorithms without archiving.Archive interpolation reduces calls to the computationally expensive original objective function; FIPS uses the fewest evaluations but has lower performance.

C. Performance Assessment under Perturbed Condition

Perturbation tests show that robust designs preserve acceptable performance when controller or power-system parameters vary, while optimal designs can degrade severely. FOPID generally outperforms PID under these perturbations, with practical relevance for variable FO realizations.

  • Controller parameter perturbation: 100 Monte-Carlo runs perturbing controller gains and orders show robust designs maintain acceptable performance near nominal values, whereas optimal controllers frequently become unstable or infeasible.Perturbations use uniform and Gaussian sampling; infeasible or unstable cases receive J = 10^4.
  • Controller parameter perturbation: FOPID yields a better robust design than PID according to the expected objective-function value under controller-parameter perturbations.
  • Power-system parameter perturbation: FOPID maintains lower peaks in Δf than PID when power-system parameters are perturbed, especially for decreased M and increased KFESS, τSC, and τCA.Figures 13–14 show M perturbations; analogous effects for other parameters are reported in supplementary material and Table IV.
  • Overall robustness: Under complex dynamics, stochastic communication delays, renewable/load variability, and storage rate constraints, all optimal controllers stabilize the nominal system but lose comparable performance after perturbation.
  • Overall robustness: Robust and optimal controllers show hardly any nominal-performance variation, while robust designs remain satisfactory under controller-parameter perturbations.This supports robust tuning for FO implementations whose realized gains and orders can differ across approximation methods and filter orders.
  • Scope boundaries: The study considers stochastic delays but not packet dropouts, and evaluates robust optimization using PSO variants rather than newer optimizer families.The authors state that the methodology could accommodate delays and dropouts together and extend beyond PSO variants.

APPENDIX

The appendix contains supplementary figures covering the hybrid-system schematic, input powers, optimization behavior, controller performance, component powers, archive evolution, evaluation counts, and parameter perturbations.

  • System and inputs: The supplementary figures include the hybrid power-system schematic and a realization of renewable-generation and demand powers independent of controller structure.
  • Optimization behavior: Figures 6–8 show PSO convergence, PID/FOPID tuning-parameter evolution, and box-and-whisker distributions of Jmin across 10 runs.
  • Performance and computation: Figures 9–13 depict optimal/robust PID and FOPID performance, component powers, archive-size evolution, evaluation counts, and the effect of increasing M.
  • Perturbation studies: Figures 14–18 cover decreasing M, robust-FOPID frequency deviation and control signal under controller perturbation, and analogous power-system-parameter perturbations.
  • Perturbation studies: Figures 19–20 compare grid-frequency deviation and control signals under controller perturbations across optimal/robust PID/FOPID structures.
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