# Design of a Fractional Order Frequency PID Controller for an Islanded Microgrid: A Multi-Objective Extremal Optimization Method

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## Abstract

**:**

## 1. Introduction

_{∞}method for the frequency control of a hybrid power system. Similar research works include robust H

_{∞}and structured singular value μ-based control synthesis approaches for microgrids [15]. Bendato et al. [16] proposed an effective two-step procedure to optimize a real-time energy management system by integrating economic aspects and power quality objectives including reactive power, voltage and frequency. Its effectiveness has been demonstrated on a microgrid system called “University of Genoa Smart Polygeneration Microgrid”.

## 2. Preliminaries

#### 2.1. FOPID Controller

^{λ}D

^{μ}controller [17]. Its transfer function model is defined as follows:

**Definition**

**1.**

_{c}(s) of a FOPID controller is described as follows:

_{p}, K

_{i}and K

_{d}are the gains of proportional, integral, and derivative, respectively. λ and μ are the order numbers of the fractional order integrator and differentiator, respectively. Generally, the domain of λ and μ are defined as: 0 ≤ λ ≤ 2 and 0 ≤ μ ≤ 2. It is clear that the traditional PID controller is one special case of a FOPID controller when λ = 1 and μ = 1.

^{λ}D

^{μ}controller is computed as the following Equation:

#### 2.2. Multi-Objective Optimization

**x**= (x

_{1}, x

_{2},…, x

_{n})∈Ω is a decision vector consisting of n decision variables x

_{1}, x

_{2},…, x

_{n}, $\mathrm{\Omega}\subseteq {\mathrm{R}}^{n}$ is the decision space, m is the number of objective functions,

**L**and

**U**represent the lower and upper bounds of vector

**x**, respectively, F: $\mathrm{\Omega}\to {\mathrm{R}}^{m}$ consists of m real-valued objective functions and ${\mathrm{R}}^{m}$ is defined as the m dimensions objective space.

**u**= (u

_{1}, u

_{2}, …, u

_{m}) ∈ ${\mathrm{R}}^{m}$ is considered to dominate another objective vector

**v**= (v

_{1}, v

_{2}, …, v

_{m})∈${\mathrm{R}}^{m}$, which is denoted as $\mathbf{u}\prec \mathbf{v}$ if and only if the following two conditions are satisfied simultaneously: (1) $\forall i\in \{1,2,\dots ,m\}$, ${u}_{i}\le {v}_{i}$, and (2) $\exists i\in \{1,2,\dots ,m\},{u}_{i}<{v}_{i}$ A decision vector

**x**∈Ω is defined to be non-dominated or Pareto optimal if and only if there does not exist another decision vector

**x**∈Ω such that $F\left({\mathbf{x}}^{\ast}\right)\prec F\left(\mathbf{x}\right)$. The Pareto-optimal set is defined as all Pareto optimal solutions in $\mathrm{\Omega}$. The set of m objective functions values corresponding to the Pareto-optimal set are called Pareto front.

^{∗}## 3. Microgrid Models Based on Small-Signal Analysis

_{sol}and P

_{W}are the input stochastic power of PV and WTG, respectively; P

_{PV}, P

_{WTG}, P

_{DEG}, P

_{FC}, P

_{BESS}and P

_{FESS}are the output power of PV, WTG, DEG, FC, BESS and FESS, respectively, and P

_{L}is the variable load power. Some intermediate variables are computed as follows: P

_{t}= P

_{PV}+ P

_{WTG}, P

_{S}= P

_{t}+ P

_{FC}+ P

_{DEG}− P

_{BESS}− P

_{FESS}, and P

_{e}= P

_{L}− P

_{S}.

## 4. Multi-Objective Extremal Optimization Based FOPID Method for the Frequency Control of Islanded Microgrids

**Definition**

**2.**

_{1}and F

_{2}subject to some given constraints are defined to evaluate the performance of a FOPID controller

**x**= (K

_{p}, K

_{i}, K

_{d}, λ, μ) for the frequency control of an islanded microgrid.

_{FESSmax}, P

_{BESSmax}, P

_{FCmax}, P

_{DEGmax}are the output saturations (in pu) of P

_{FESS}, P

_{BESS}, P

_{FC}, P

_{DEG}, respectively; P

_{FESSr}, P

_{BESSr}, P

_{FCr}, P

_{DEGr}are the rate of P

_{FESS}, P

_{BESS}, P

_{FC}, P

_{DEG}, respectively; P

_{FESSrmax}, P

_{BESSrmax}, P

_{FCrmax}, P

_{DEGrmax}are the maximum constraints of P

_{FESSr}, P

_{BESSr}, P

_{FCr}, P

_{DEGr}, respectively; and

**L**and

**U**represent the lower and upper bounds of the FOPID controller parameters, respectively.

#### MOEO-FOPID-Based Frequency Controller Optimal Design Algorithm for an Islanded Microgrid

**Input:**A microgrid system with a FOPID-based frequency controller and adjustable parameters used in the MOEO-FOPID algorithm, including the maximum number of iterations I

_{max}, the maximum size of external archive A

_{max}, and the shape parameter q used in mutation operation.

**Output:**The best non-dominated solutions for the designed FOPID-based frequency controller and the corresponding best Pareto front found so far.

**Step****1:**- Generate a real-coded solution S = (s
_{1},s_{2}, s_{3}, s_{4}, s_{5}) representing the control parameters of a FOPID-based frequency controller (K_{p}, K_{i}, K_{d}, λ, μ) in an islanded microgrid subject to the given constraints (10) randomly, and set the external archive**A**as empty and S_{C}= S. **Step****2:**- By mutating each variable s
_{i}(i = 1, 2, 3, 4, 5) of the current solution S_{C}one-by-one based on multi-non-uniform mutation (MNUM)while keeping other variables unchanged, generate five candidate solutions{S_{i}, i = 1, 2, 3, 4, 5}. The detailed process is formulated as follows:$${S}_{i}=\{\begin{array}{c}{S}_{C}+\left(U-{S}_{C}\right)\times A(t),\text{}\mathrm{if}\text{}r0.5,\\ {S}_{C}+\left({S}_{C}-L\right)\times A(t),\text{}\mathrm{if}\text{}r\ge 0.5.\end{array}$$$$A(t)={\left[{r}_{1}\left(1-\frac{{I}_{C}}{{I}_{\mathrm{max}}}\right)\right]}^{q}$$_{C}is the number of current iterations in the optimization process, both r and r_{1}are uniform random numbers between 0 and 1, and q is the shape parameter used in MNUM. **Step****3:**- Rank five solutions {S
_{i}, i = 1, 2, 3, 4, 5} based on the non-dominated sorting strategy, where the two objective functions F_{1}and F_{2}are evaluated by Definition 2. **Step****4:**- If the number of non-dominated solutions is just one, then select the only non-dominated solution S
_{nd}as the new solution S_{N}; otherwise, select one from several non-dominated solutions randomly, and set this one as the new solution S_{N}. **Step****5:****Step****6:**- Accept S
_{C}= S_{N}unconditionally. **Step****7:**- If the predefined stopping criteria, e.g., maximum number of iterations I
_{max}is met, then return to Step 2; otherwise, go to Step 8. **Step****8:**- Return external archive
**A**as the best non-dominated solutions for the FOPID controller for the frequency control of an islanded microgrid, and output the best Pareto front found so far and the corresponding control performance.

Algorithm 1 The Pseudo-Code of Algorithm “Update_Archive (S_{N}, Archive)” [37] |

1: Begin |

2: If the solution S_{N} is dominated by at least one member of the archive, then |

3: The archive keeps unchanged |

4: Else if some members of archive are dominated by S_{N}, then |

5: Remove all the dominated members from the archive and add S_{N} to the archive |

6: End if |

7: Else |

8: If the number of archive is smaller than A_{max}, i.e., the predefined maximum number of the archive, then |

9: Add S_{N} to the archive |

10: Else |

11: If S_{N} resides in the most crowded region of the archive, then |

12: The archive keeps unchanged |

13: Else |

14: Replace the member in the most crowded region of the archive by S_{N} |

15: End if |

16: End if |

17: End if |

18: End |

## 5. Simulation Results

#### 5.1. Performance Comparison in Nominal Microgrid Conditions

_{FESS}

_{max}= P

_{BESS}

_{max}= 0.11, P

_{FC}

_{max}= 0.48, P

_{DEG}

_{max}= 0.45, P

_{FESSr}

_{max}= P

_{BESSr}

_{max}= 0.05, P

_{FCr}

_{max}= 1, and P

_{DEGr}

_{max}= 0.5. For the sake of fair comparison, the lower and upper bounds of the FOPID controller parameters are set the same as in [25]:

**L**= [0, 0, 0, 0, 0] and

**U**= [5, 5, 5, 2, 2]. The parameters for MOEO and NSGA-II used in the experiments are shown in Table 3.Note that there are three main differences between MOEO-FOPID and NSGA-II-FOPID. Firstly, MOEO-FOPID adopts an individual-based iterated optimization mechanism, while NSGA-II-FOPID uses a population based optimization mechanism. Secondly, MOEO-FOPID has only selection and mutation operations while NSGA-II-FOPID has more operations including selection, crossover and mutation. Thirdly, MOEO-FOPID has fewer adjustable parameters than NSGA-II-FOPID. As a consequence, MOEO-FOPID is considered to be simpler than NSGA-II-FOPID from the perspective of algorithm design.

_{1}and F

_{2}.

_{n}is set as 10

^{4}.

#### 5.2. Robustness Tests under Perturbed System Parameters

_{FC}, T

_{g}and T

_{t}, respectively. Clearly, the frequency deviations with the FOPID controller optimized by MOEO were still smaller than those by NSGA-II in all the cases. In other words, MOEO-FOPID is superior to NSGA-II in terms of parametric robustness.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Stochastic powers of WTG (labeled P

_{WTG}), PV (labeled P

_{PV}), demand-side loads (labeled P

_{L}), and the sum of WTG and PV generation (labeled P

_{t}).

**Figure 5.**Comparison of Pareto fronts for PID/FOPID controllers obtained by NSGA-II and MOEO for the microgrid under the minimum values of HI.

**Figure 6.**Comparison of frequency deviation ∆f (

**a**), control signal u (

**b**) and deficit power deviation ∆P (

**c**) of the test microgrid with the best FOPID/PID controllers obtained by the NSGA-II and MOEO algorithms.

**Figure 7.**Comparison of individual powers in the different components of the test microgrid with the best FOPID/PID controllers obtained by the NSGA-II and MOEO algorithms.

**Figure 8.**Comparison of frequency deviation ∆f (

**a**), control signal u (

**b**) and deficit power deviation ∆P (

**c**) of the test microgrid with FOPID controllers obtained by the MOEO algorithm and single-objective optimization algorithms including KSM and RPEO.

**Figure 9.**Comparison of individual powers in different components of the test microgrid with FOPID controllers obtained by MOEO and single-objective optimization algorithms including KSM and RPEO.

**Figure 10.**Comparison of the robustness against increased system parameters obtained by MOEO-FOPID and NSGA-II-FOPID.

**Figure 11.**Comparison of the robustness against decreased system parameters obtained by MOEO-FOPID and NSGA-II-FOPID.

**Table 1.**The small-signal analysis models and parameters of the components of an islanded microgrid [25].

Component | Transfer Function | Parameters |
---|---|---|

Wind turbine generator (WTG) | ${G}_{WTG}(s)=\frac{\Delta {P}_{WTG}}{\Delta {P}_{W}}=\frac{{K}_{W}}{1+s{T}_{W}}$ | K_{W} = 1, T_{W} = 1.5 s |

Solar photovoltaic (PV) | ${G}_{STPG}(s)=\frac{\Delta {P}_{PV}}{\Delta {P}_{sol}}=\frac{1}{(1+s{T}_{IN})(1+s{T}_{IC})}$ | T_{IN} = 0.04 s, T_{IC} = 0.004 s |

Fuel cell (FC) | ${G}_{FC}(s)=\frac{\Delta {P}_{FC}}{\Delta u}=\frac{1}{(1+s{T}_{FC})(1+s{T}_{IN})(1+s{T}_{IC})}$ | K_{FC} = 1, T_{FC} = 0.26 s |

Diesel energy generator (DEG) | ${G}_{DEG}(s)=\frac{\Delta {P}_{DEG}}{\Delta u}=\frac{1}{(1+s{T}_{G})(1+s{T}_{T})}$ | T_{G} = 0.08 s, T_{T} = 0.4 s |

Microgrid system | ${G}_{S}(s)=\frac{\Delta f}{\Delta {P}_{e}}=\frac{1}{D+2Hs}$ | D = 0.015 pu/Hz, H = 1/12 pu.sec, R = 3 Hz/pu |

Flywheel energy storage system (FESS) | ${G}_{FESS}(s)=\frac{\Delta {P}_{FESS}}{\Delta f}=\frac{{K}_{FESS}}{1+s{T}_{FESS}}$ | K_{FESS} = 1, T_{FESS} = 0.1 s |

Battery energy storage system (BESS) | ${G}_{FESS}(s)=\frac{\Delta {P}_{BESS}}{\Delta f}=\frac{{K}_{BESS}}{1+s{T}_{BESS}}$ | K_{BESS} = 1, T_{BESS} = 0.1 s |

Stochastic Models | Model Parameters |
---|---|

Wind power generation | ϕ~U(−1, 1), η = 0.8, β = 10, G(s) = 1/(10^{4}s + 1), Γ = 0.24H(t) − 0.04H(t − 140) |

Solar power generation | ϕ~U(−1, 1), η = 0.1, β = 10, δ = 0.1, G(s) = 1/(10^{4}s + 1), Γ = 0.05H(t) + 0.02H(t − 180) |

Demand loads | ϕ~U(−1, 1), η = 0.9, β = 10, G(s) = (300/(300s + 1)) + (1/(1800s + 1)), Γ = (1/χ)[0.9H(t) + 0.03H(t − 110) + 0.03H(t − 130) + 0.03H(t − 150) − 0.15H(t − 170)+ 0.1H(t − 190)] + 0.02H(t) |

Algorithm | Parameters |
---|---|

NSGA-II-FOPID/PID [20,38] | I_{max} = 500, population size NP = 30, crossover probability p_{c} = 0.9, mutation probability p_{m} = 1/n, distribution indexes η_{c} = 20and η_{m} = 20 for simulated binary crossover (SBX) and PLM |

MOEO-FOPID/PID | I_{max} = 500, A_{max} = 30, q = 6 |

**Table 4.**Comparison of the statistical performance metric for Pareto fronts obtained by MOEO-FOPD/PID and NSGA-II-FOPID/PID for microgrid.

Performance Metrics | Algorithm | Minimum | Median | Maxmum | Mean | Standard Deviation |
---|---|---|---|---|---|---|

Hypervolume indicator (HI, min) | NSGA-II-PID | 1.83 × 10^{−4} | 2.07 × 10^{−4} | 2.26 × 10^{−4} | 2.06 × 10^{−4} | 1.15 × 10^{−5} |

MOEO-PID | 1.26 × 10^{−4} | 2.16 × 10^{−4} | 3.92 × 10^{−4} | 2.24 × 10^{−4} | 7.78 × 10^{−5} | |

NSGA-II-FOPID | 1.56 × 10^{−4} | 2.03 × 10^{−4} | 3.35 × 10^{−4} | 2.13 × 10^{−4} | 5.08 × 10^{−5} | |

MOEO-FOPID | 1.04 × 10^{−4} | 1.63 × 10^{−4} | 2.20 × 10^{−4} | 1.63 × 10^{−4} | 2.95 × 10^{−5} | |

Spacing metric (SP, max) | NSGA-II-PID | 5.41 × 10^{−3} | 9.93 × 10^{−3} | 1.50 × 10^{−2} | 1.00 × 10^{−2} | 2.69 × 10^{−3} |

MOEO-PID | 4.28 × 10^{−3} | 1.20 × 10^{−2} | 2.32 × 10^{−2} | 1.26 × 10^{−2} | 5.43 × 10^{−3} | |

NSGA-II-FOPID | 6.93 × 10^{−3} | 1.59 × 10^{−2} | 2.84 × 10^{−2} | 1.52 × 10^{−2} | 4.56 × 10^{−3} | |

MOEO-FOPID | 1.91 × 10^{−3} | 4.24 × 10^{−3} | 7.97 × 10^{−3} | 4.61 × 10^{−3} | 1.22 × 10^{−3} | |

Inertia-based diversity metric (I, max) | NSGA-II-PID | 7.07 × 10^{−2} | 0.107 | 0.120 | 0.103 | 1.30 × 10^{−2} |

MOEO-PID | 6.50 × 10^{−2} | 6.15 × 10^{−2} | 4.35 × 10^{−2} | 6.01 × 10^{−2} | 4.28 × 10^{−3} | |

NSGA-II-FOPID | 0.131 | 0.103 | 5.78 × 10^{−2} | 0.103 | 2.14 × 10^{−2} | |

MOEO-FOPID | 0.135 | 0.117 | 8.10 × 10^{−2} | 0.113 | 1.69 × 10^{−2} | |

Inverted generational distance (IGD, min) | NSGA-II-PID | 7.00 × 10^{−3} | 8.08 × 10^{−3} | 9.58 × 10^{−3} | 8.09 × 10^{−3} | 6.16 × 10^{−4} |

MOEO-PID | 9.52 × 10^{−3} | 1.07 × 10^{−2} | 1.38 × 10^{−2} | 1.08 × 10^{−2} | 9.43 × 10^{−4} | |

NSGA-II-FOPID | 3.43 × 10^{−3} | 5.52 × 10^{−3} | 9.22 × 10^{−3} | 5.66 × 10^{−3} | 1.17 × 10^{−3} | |

MOEO-FOPID | 3.16 × 10^{−3} | 4.39 × 10^{−3} | 1.20 × 10^{−2} | 4.68 × 10^{−3} | 1.59 × 10^{−3} |

**Table 5.**Best FOPID/PID controller parameters and performance obtained by MOEO and NSGA-II under the minimum values of HI.

Algorithm | F_{1} | F_{2} | K_{p} | K_{i} | K_{d} | λ | μ |
---|---|---|---|---|---|---|---|

NSGA-II-PID | 8.3877 × 10^{−4} | 1.4204 × 10^{−3} | 4.78192 | 4.76904 | 0.95045 | 1 | 1 |

MOEO-PID | 8.1589 × 10^{−4} | 1.4250 × 10^{−3} | 4.53357 | 4.85426 | 1.05329 | 1 | 1 |

NSGA-II-FOPID | 7.8307 × 10^{−4} | 1.4198 × 10^{−3} | 5 | 4.99475 | 0.54391 | 1.00950 | 1.20039 |

MOEO-FOPID | 7.2219 × 10^{−4} | 1.4174 × 10^{−3} | 4.90695 | 4.16141 | 0.78012 | 1.00730 | 1.13911 |

**Table 6.**Best FOPID controller parameters and performance obtained by MOEO-FOPID, RPEO-FOPID and KSM-FOPID.

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**MDPI and ACS Style**

Wang, H.; Zeng, G.; Dai, Y.; Bi, D.; Sun, J.; Xie, X. Design of a Fractional Order Frequency PID Controller for an Islanded Microgrid: A Multi-Objective Extremal Optimization Method. *Energies* **2017**, *10*, 1502.
https://doi.org/10.3390/en10101502

**AMA Style**

Wang H, Zeng G, Dai Y, Bi D, Sun J, Xie X. Design of a Fractional Order Frequency PID Controller for an Islanded Microgrid: A Multi-Objective Extremal Optimization Method. *Energies*. 2017; 10(10):1502.
https://doi.org/10.3390/en10101502

**Chicago/Turabian Style**

Wang, Huan, Guoqiang Zeng, Yuxing Dai, Daqiang Bi, Jingliao Sun, and Xiaoqing Xie. 2017. "Design of a Fractional Order Frequency PID Controller for an Islanded Microgrid: A Multi-Objective Extremal Optimization Method" *Energies* 10, no. 10: 1502.
https://doi.org/10.3390/en10101502