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Kriging based Surrogate Modeling for Fractional Order Control of Microgrids
Indranil Pan, Saptarshi Das
TL;DR
Microgrid frequency control becomes difficult with low-inertia, stochastic generation and expensive fractional-order simulations. The paper combines FOPID control with kriging-assisted global optimization, reporting better FOPID performance than PID and better kriging optimization than GA in solution quality and convergence. It also discusses the method’s computational trade-offs and potential for online tuning.
Problem
Low-inertia microgrids with stochastic renewable generation need frequency control, while repeated time-domain evaluations make optimization-based controller tuning computationally expensive.
Method
The paper embeds kriging surrogate modeling in a global optimization framework to tune FOPID controllers for microgrid frequency control.
Results
FOPID outperforms PID, while kriging-based optimization outperforms GA in solution quality, stability, and convergence.
Takeaways & Limitations
Kriging-assisted optimization reduces controller-parameter optimization time and may facilitate future online tuning of fractional-order controllers.
Abstract
from arXiv · showhide
This paper investigates the use of fractional order (FO) controllers for a microgrid. The microgrid employs various autonomous generation systems like wind turbine generator (WTG), solar photovoltaic (PV), diesel energy generator (DEG) and fuel-cells (FC). Other storage devices like the battery energy storage system (BESS) and the flywheel energy storage system (FESS) are also present in the power network. An FO control strategy is employed and the FO-PID controller parameters are tuned with a global optimization algorithm to meet system performance specifications. A kriging based surrogate modeling technique is employed to alleviate the issue of expensive objective function evaluation for the optimization based controller tuning. Numerical simulations are reported to prove the validity of the proposed methods. The results for both the FO and the integer order (IO) controllers are compared with standard evolutionary optimization techniques and the relative merits and demerits of the kriging based surrogate modeling are discussed. This kind of optimization technique is not only limited to this specific case of microgrid control but also can be ported to other computationally expensive power system optimization problems.
I. INTRODUCTION
Renewable-rich microgrids require effective control because low inertia and stochastic generation can cause severe frequency and voltage fluctuations. The paper addresses expensive FO-controller tuning with kriging-assisted surrogate optimization.
- Renewable integration increases power-system complexity and requires efficient monitoring and control for smooth operation.
- Microgrids can operate autonomously or grid-connected, but their small generators provide low inertia.
- Stochastic wind and solar inputs or generation outages can produce severe frequency and voltage fluctuations in microgrids.
- Optimization-based controller design is computationally prohibitive because it repeatedly evaluates expensive time-domain power-system simulations.
- The paper embeds kriging-assisted surrogate modeling within global optimization to design FOPID controllers for microgrid frequency control.The approach targets computationally expensive simulations of coupled, high-order approximations of fractional-order systems.
A. Fractional Calculus Basics
Fractional calculus generalizes integration and differentiation from integer orders to arbitrary real orders. The paper introduces this basis for fractional-order control.
- Fractional calculus extends integration and differentiation to arbitrary real orders.
- The paper identifies Grünwald-Letnikov, Riemann-Liouville, and Caputo as definitions of fractional calculus.
- Caputo’s definition is commonly used in control studies to realize the fractional integro-differential operators of FOPID controllers.
B. Microgrid System and the Controller Structure
The studied microgrid combines renewable generators, fuel-based generators, and storage, while an FOPID controller regulates frequency through FC and DEG actuation. Its fractional operators are implemented by band-limited rational approximations.
- The microgrid includes wind, photovoltaic, fuel-cell, diesel, battery, and flywheel energy systems.
- The FOPID controller uses frequency deviation to control the fuel cell and diesel generator and reduce stochastic grid-frequency fluctuations.
- The FOPID structure has five tuning variables: gains K_p, K_i, K_d and fractional orders λ, μ.
- Setting λ=μ=1 reduces the FOPID structure to the classical parallel PID controller.
- Higher-order rational transfer-function approximations provide practical band-limited implementations of fractional operators.
- The model imposes saturation and rate constraints on generation and storage components to limit power extraction, storage, and mechanical shock.
- The secondary control scheme is used, with centralized control able to disconnect a subsystem during emergencies.
C. Characteristic Changes of the Wind and Solar Power Generation and the Demand Load
Wind generation, solar generation, and demand are modeled with a shared stochastic template that combines mean behavior, filtering, normalization, and sudden switching changes. Different parameters represent each source or load.
- Wind, solar, and load profiles include deterministic drift, stochastic fluctuations, and sudden changes in their mean values.
- The template models power output P using a stochastic component, a mean-value contribution, a low-pass filter, normalization, and a switching signal.
- The same template represents uncertain generation and demand behavior while allowing source-specific parameters.
- Demand load: Demand-load parameters use η=0.9 and a transfer function distinct from the wind and solar models.
D. Objective Function for Optimization
The objective function balances frequency-deviation suppression against incremental control effort. Its weighting prioritizes rapid reduction of microgrid frequency oscillations while retaining a penalty on control-signal variation.
- The objective integrates weighted terms for frequency deviation and incremental control signal.These correspond to Integral of Squared Error (ISE) and Integral of Squared Deviation of Control Output (ISDCO).
- The weighting factor w is set to 0.7, giving greater importance to suppressing frequency oscillations quickly.The choice favors disturbance rejection over minimizing control-signal magnitude.
- The fixed-structure controller faces a trade-off between rapid load-disturbance rejection and the control effort required.Previous investigations characterize the design problem as inherently multi-objective.
- The normalizing constant K_n = 10^4 scales ISE and ISDCO to a uniform scale.
E. Kriging Based Global Optimization
The kriging-based optimization constructs a surrogate of the controller design space and uses it to select promising candidates for expensive objective-function evaluations. Its workflow combines initial sampling, response prediction, distance-based exploration, and iterative updating.
- Kriging approximates linear and nonlinear design-space trends by combining a global model with localized departures.Different spatial correlation functions provide flexibility in constructing the approximation.
- The kriging predictor uses sampled design sites, model responses, regression structure, and correlation matrices to estimate unsampled objective values and prediction error.The formulation includes generalized least squares and a mean squared error estimate.
- The surrogate models the response using regression terms plus a stochastic process with spatially correlated deviations.The process covariance is specified through a correlation model, while sampled responses and a design matrix support prediction.
- The workflow begins with symmetric Latin hypercube sampling over five fractional-order controller variables: Kp, Ki, Kd, λ, and μ.This sampling scheme trades off uniform coverage against the cost of more expensive space-filling sampling.
- Candidate selection combines predicted objective values from the kriging response surface with each candidate’s distance from existing sampled points.The best candidate is then evaluated using the computationally expensive objective function.
A. Performance Evaluation of the Kriging Based Optimizers and Different Controller Structures
The study evaluates kriging-assisted optimization for PID and FOPID microgrid controllers under stochastic simulation conditions. FOPID controllers and kriging optimization show stronger performance than PID and GA-based alternatives, while robustness and computational sampling remain relevant considerations.
- Optimization setup: The stochastic objective is evaluated 10 times per controller setting, with those repetitions counted as one expensive evaluation and a total limit of 150 evaluations.This estimates the expected objective value before optimization.
- Controller comparison: FOPID controllers consistently outperform PID controllers across all optimization algorithms.The paper reports better controller performance for FOPID in the compared optimization settings.
- Kriging models: The spline correlation function gives the best average solution among the five kriging correlation models for both controller types.The spline model also locates lower objective-function values across the reported statistical measures.
- Controller parameters: Both optimum fractional orders in Table 3 are below one, although orders above one can make the system faster while substantially increasing control effort.The cost function balances tracking speed and control effort.
- Dynamic response: FOPID produces less frequency fluctuation, faster damping of power deviation, and less control signal than PID.The reported implications include better power quality and smaller actuator requirements for the fuel cell and diesel generator.
- Optimizer comparison: Kriging optimization converges to the best solutions more quickly than GA-based tuning for both PID and FOPID controllers.Convergence is compared using mean, median, best, and worst cases over 30 independent runs.
B. Parametric Robustness of the Optimum Solutions
The optimum PID and FOPID solutions are tested under positive and negative perturbations of microgrid parameters. FOPID is more robust, while increases in parameters and selected time constants produce stronger deterioration.
- Robustness comparison: FOPID is more robust than PID under parameter perturbations, showing smaller increases in objective value and frequency deviation.The comparison concerns robustness of the optimum controllers under perturbed microgrid parameters.
- Perturbation direction: Increasing system parameters is generally more detrimental than decreasing them by an equal amount.The reported adverse effects are significantly smaller with FOPID control.
- Sensitive parameters: The interconnection-device, fuel-cell, and inverter time constants are the most susceptible parameters to performance deterioration.The paper identifies TI/C, TFC, and TIN as degrading performance faster than the other perturbed parameters.
C. Effect of Actuator On-Off Switching Logic
The actuator logic switches DEG and FC support on when storage cannot quickly damp frequency deviations, but this produces recurring switching transients. Rate constraints and output saturation also create significant nonlinear operation in the microgrid elements.
- Actuator switching logic: The FC and DEG activate when FESS and BESS cannot sufficiently damp frequency deviations within a short period.They are assumed inactive when the frequency deviation remains within 0.05 f, and activate outside that limit.
- Actuator switching logic: 10 sec minimum activation reduces immediate cycling but still leads to significant switching transients after nearly every interval.The frequency deviation can retrigger DEG and FC within milliseconds after they are cut out.
- Actuator switching logic: Chattering from the switched actuator scheme is detrimental to microgrid components and motivates the minimum-on-time modification.The switching logic is implemented in Stateflow while modeling the system.
- Nonlinear operation: Rate constraints cut off power-rate changes beyond thresholds, maintaining constant rates during operation in the significant nonlinear zone.The nonlinear operation is illustrated for the present microgrid with PID/FOPID control.
- Nonlinear operation: Output saturation causes DEG and FC power to remain constant after their maximum limits are reached.The simulations use a nonlinear microgrid model with rate constraints and output saturation.
IV. CONCLUSION
The paper concludes that fractional-order control improves microgrid frequency suppression and robustness, while kriging-assisted optimization accelerates controller tuning and improves solution quality relative to a genetic algorithm.
- FOPID outperforms standard PID under nominal operation and provides better robustness under large microgrid parametric uncertainty.
- Kriging-based surrogate modeling reduces the time required to optimize microgrid controller parameters.The authors suggest this could facilitate online tuning in future.
- Kriging-based optimization achieves higher solution quality and faster convergence than the standard genetic algorithm.