Source-linked AI summary
Fractional Order Load-Frequency Control of Interconnected Power Systems Using Chaotic Multi-objective Optimization
Indranil Pan, Saptarshi Das
TL;DR
The paper addresses multi-objective load-frequency control for a two-area interconnected power system with competing time-domain objectives. It designs and compares PID and FOPID controllers using standard and chaotic NSGA-II variants, finding better Pareto solutions from chaotic optimization and generally stronger FOPID performance. Robustness is examined under parameter perturbations and random load changes.
Problem
The problem is designing effective load-frequency controllers for interconnected two-area systems under conflicting time-domain objectives and nonlinear operating features.
Method
The method uses multi-objective optimization of PID and FOPID controllers with standard, Logistic-map, and Henon-map NSGA-II variants, evaluated by Pareto metrics and fuzzy compromise selection.
Results
Chaotic NSGA-II produces better Pareto solutions than ordinary NSGA-II, while FOPID outperforms PID for multi-objective two-area LFC designs.
Takeaways & Limitations
Under nominal conditions and random load changes, slow FOPID provides faster settling and keeps the listed frequency, control-input, and tie-line deviations low.
Takeaways & Limitations
The study is bounded by a two-area model with generation-rate constraints, governor dead-band, specified step-load disturbances, and selected parameter perturbations.
Abstract
from arXiv · showhide
Fractional order proportional-integral-derivative (FOPID) controllers are designed for load frequency control (LFC) of two interconnected power systems. Conflicting time domain design objectives are considered in a multi objective optimization (MOO) based design framework to design the gains and the fractional differ-integral orders of the FOPID controllers in the two areas. Here, we explore the effect of augmenting two different chaotic maps along with the uniform random number generator (RNG) in the popular MOO algorithm - the Non-dominated Sorting Genetic Algorithm-II (NSGA-II). Different measures of quality for MOO e.g. hypervolume indicator, moment of inertia based diversity metric, total Pareto spread, spacing metric are adopted to select the best set of controller parameters from multiple runs of all the NSGA-II variants (i.e. nominal and chaotic versions). The chaotic versions of the NSGA-II algorithm are compared with the standard NSGA-II in terms of solution quality and computational time. In addition, the Pareto optimal fronts showing the trade-off between the two conflicting time domain design objectives are compared to show the advantage of using the FOPID controller over that with simple PID controller. The nature of fast/slow and high/low noise amplification effects of the FOPID structure or the four quadrant operation in the two inter-connected areas of the power system is also explored. A fuzzy logic based method has been adopted next to select the best compromise solution from the best Pareto fronts corresponding to each MOO comparison criteria. The time domain system responses are shown for the fuzzy best compromise solutions under nominal operating conditions. Comparative analysis on the merits and de-merits of each controller structure is reported then. A robustness analysis is also done for the PID and the FOPID controllers.
1. Introduction
The paper develops multi-objective FOPID design for two-area load-frequency control, emphasizing trade-offs, chaotic NSGA-II optimization, and robustness. It compares FOPID with PID and uses Pareto metrics and fuzzy selection to identify compromise controllers.
- Logistic and Henon chaotic maps augment NSGA-II to seek improved Pareto-optimal solutions.
- Hypervolume, spacing, Pareto spread, and diversity metrics assess chaotic multi-objective optimization performance.
- FOPID controllers are used to improve two-area load-frequency-control performance.
- The study demonstrates conflicting time-domain trade-offs between FOPID and PID performance and selects compromise solutions with fuzzy logic.
- The paper studies FOPID robustness against system-parameter uncertainty and random load-pattern changes.
2. Load frequency control of interconnected two area power system
The two-area LFC model regulates frequency deviations and tie-line power through area control errors and secondary controllers. It includes turbine, governor, reheater, dead-band, and generation-rate constraints, with multi-objective FOPID design motivated by nonlinearities not previously studied in this setting.
- Area control error combines frequency deviation and tie-line power deviation for interconnected-area regulation.
- The FOPID controller converts area control error into a signal that drives the actuator and turbine mechanical torque.
- Each area models a steam turbine, governor, reheater stages, generation-rate constraint, and governor dead-band, while tie-line dynamics depend on T12.
- The governor dead-band is set to 0.06% in each area, and the generation-rate constraint limits power-rate change to ±0.005.
- Multi-objective fractional-order control design for the modeled generation-rate and dead-zone nonlinearities is identified as the paper’s motivation.
3. Fractional calculus and Fractional order PIλDµ (FOPID) controller
Fractional calculus provides several definitions for non-integer differ-integrals, while the FOPID controller extends PID by tuning both conventional gains and fractional orders. Varying these orders enables trade-offs among response speed and noise amplification across the two areas.
- Fractional calculus: Fractional differentiation and integration are commonly defined using Grunwald-Letnikov, Riemann-Liouville, and Caputo formulations.The paper uses Caputo’s definition in fractional-order systems and control, particularly for deriving transfer-function models under zero initial conditions.
- FOPID controller: FOPID extends integer-order PID by adding non-integer differ-integral orders to the proportional, integral, and derivative gains.Its five tuning variables are K_p, K_i, K_d, λ, and µ.
- FOPID controller: For λ=1 and µ=1, the FOPID structure reduces to the classical PID controller.
- Four-quadrant operation: Increasing λ above one amplifies low-frequency components, whereas increasing µ above one increases high-frequency gain and noise amplification.
- Four-quadrant operation: Within the slow and fast FOPID families, multi-objective optimization selects area-specific noise-amplification levels that form a non-dominated Pareto front.
4. Need of multi-objective optimization and conflicting time domain control objectives
The LFC design problem contains conflicting objectives: faster damping requires stronger control action, while desirable operation limits controller output. Multi-objective optimization retains trade-off solutions for selecting a compromise controller.
- Motivation: A single controller structure cannot generally satisfy all design specifications, so controller comparisons must account for trade-offs among conflicting requirements.
- Objectives: The optimization uses ITSE, the Integral of the Time multiplied Squared Error, to evaluate area-control-error performance.The error signal is the ACE in area i.
- Objectives: The second objective is ISDCO, the Integral of the Squared Deviation in Controller Output, which measures control-signal deviation.The control signal is denoted u_i(t).
- Objective conflict: Higher controller gains can damp ACEs and frequency oscillations faster, but they also increase the control action applied to the generating system.
- Objective conflict: The tuning parameters therefore span solutions offering better load-disturbance rejection at the cost of higher control signal and vice versa.After generating trade-off solutions, a compromise solution can be selected as the controller setting.
5. Multi-objective controller design using chaotic maps
The controller design is formulated as a Pareto-based multi-objective optimization and implemented with nominal or chaotic NSGA-II variants. Chaotic maps modify the random-number generation used by evolutionary operations, while multiple Pareto metrics assess resulting fronts.
- Multi-objective formulation: The optimization seeks non-dominated objective vectors, where improving one objective is impossible without worsening another.The corresponding set of objective vectors is the Pareto front.
- NSGA-II design: NSGA-II ranks solutions by non-domination fronts and uses crowding distance to preserve sparsely populated regions of each front.Tournament selection prefers lower fronts and, within one front, larger crowding distance.
- Controller variables: FOPID optimization tunes K_p, K_i, K_d, λ, and µ, whereas integer-order PID optimization tunes only K_p, K_i, and K_d.
- Optimization settings: The study uses population and stopping settings based on the number of controller parameters, with n_var=6 for PID and n_var=10 for FOPID across both areas.The algorithm stops at a fitness tolerance of 10^-6 or after n_var×200 generations.
- Chaotic randomization: Multiplying chaotic outputs by uniform RNG produces distributions with more frequent lower values, although histograms do not capture time-domain randomness and evolution.
- Pareto-front assessment: The paper compares Pareto fronts using hypervolume, diversity, spread, and spacing-related measures across multiple stochastic runs.Thirty independent runs are used for each nominal or chaotic NSGA-II variant, alongside convergence-time comparisons.
6. Simulation and Results
The simulations compare PID and fast/slow FOPID controllers under multi-objective optimization using Pareto-quality metrics and time-domain responses. Chaotic NSGA-II variants generally produce more nondominated FOPID fronts, while slow FOPID gives the strongest nominal compromise in the reported comparisons.
- Simulation setup: The study tunes PID, slow FOPID, and fast FOPID controllers for two-area LFC using three optimization algorithms and 30 runs per case.The algorithms are standard NSGA-II, Logistic-map NSGA-II, and Henon-map NSGA-II.
- Pareto-quality metrics: FOPID controllers show better hypervolume performance than PID controllers, although traditional NSGA-II is better for FOPID median values and worse for PID.The comparison uses box plots of the hypervolume indicator across algorithms and controller structures.
- Pareto-quality metrics: Chaotic NSGA-II produces higher diversity and wider Pareto spreads for FOPID controllers, whereas standard NSGA-II performs better for PID.The additional FOPID fractional orders enlarge the search space and can yield more diverse solutions.
- Pareto-front comparison: Across the Pareto metrics, chaotic NSGA-II generally gives the best nondominated fronts, with Logistic-map NSGA-II usually outperforming the alternatives.The fronts are compared for slow FOPID, fast FOPID, and PID controller structures.
- Pareto-front comparison: Under nominal conditions, slow FOPID is more nondominated than the other controller structures across the compared Pareto criteria.The study identifies best compromise solutions for four metrics and three controller structures.
- Time-domain performance: Slow FOPID reduces oscillations, overshoot, and control effort relative to PID and fast FOPID in the reported compromise-solution responses.For maximum Pareto spread, fast FOPID has a smaller control signal but a sluggish response, while slow FOPID trades faster response for higher effort; minimum spacing also favors slow FOPID.
7. Robustness analysis of the designed solutions
Robustness tests vary the tie-line synchronizing coefficient and introduce random load changes to assess the designed controllers beyond nominal conditions. FOPID variants generally retain better robustness than PID, although the preferred fast or slow variant depends on the perturbation and Pareto criterion.
- Robustness tests: Robustness is evaluated for fuzzy best-compromise PID, fast FOPID, and slow FOPID solutions under parameter changes and random load variations.The tests include changes in T12 and random load changes in both areas.
- Synchronizing-coefficient variation: When T12 is doubled, PID may appear to have better frequency and tie-line responses, but it requires much higher control signals with sharp jumps.Those control actions may be detrimental for the governor.
- Synchronizing-coefficient variation: With gradual T12 increases, fast FOPID remains close to nominal behavior while slow FOPID and PID become unstable.This result is reported for the maximum diversity metric criterion.
- Random-load variation: Under nominal conditions and random load changes, slow FOPID gives fast settling and keeps frequency, control, and tie-line deviations low.The conclusion summarizes these responses across the reported system variables.
- Overall robustness: Overall, both PID and FOPID solutions show sufficient robustness, but FOPID variants outperform PID under parameter variations and random load changes.The paper presents this as support for FOPID use where system parameters and load patterns are uncertain.
8. Discussions and Conclusions
The paper concludes that chaotic NSGA-II improves Pareto solutions for the two-area FOPID LFC design problem, with Logistic-map assistance generally strongest. FOPID controllers outperform conventional PID in the reported multi-objective and robustness studies.
- Main findings: Chaotic NSGA-II produces more nondominated solutions than non-chaotic NSGA-II across the evaluated Pareto metrics.The conclusion states that Logistic-map assistance generally performs better than Henon-map assistance.
- Main findings: Slow FOPID performs best under nominal conditions and random load changes by settling quickly while keeping system deviations low.The summarized variables include frequency deviations, control signals, and tie-line power deviation.
- Robustness conclusions: Under perturbed system parameters, either fast or slow FOPID is preferred depending on the Pareto metric, but both outperform PID.The preferred FOPID variant is therefore criterion-dependent in the robustness tests.
- Overall conclusion: The study identifies FOPID controllers as a viable alternative to conventional integer-order PID controllers for interconnected-power-system LFC.This conclusion follows the reported multi-objective design and robustness simulations.