kriging based surrogate modeling for fractional order control of microgrids

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Kriging Based Surrogate Modeling for FractionalOrder Control of Microgrids

Indranil Pan and Saptarshi Das

Abstract—This paper investigates the use of fractional order(FO) controllers for a microgrid. The microgrid employs vari-ous autonomous generation systems like wind turbine generator,solar photovoltaic, diesel energy generator, and fuel-cells. Otherstorage devices like the battery energy storage system and theflywheel energy storage system are also present in the powernetwork. An FO control strategy is employed and the FO-proportional integral derivative (PID) controller parameters aretuned with a global optimization algorithm to meet system per-formance specifications. A kriging based surrogate modelingtechnique is employed to alleviate the issue of expensive objec-tive function evaluation for the optimization based controllertuning. Numerical simulations are reported to prove the valid-ity of the proposed methods. The results for both the FO andthe integer order controllers are compared with standard evo-lutionary optimization techniques, and the relative merits anddemerits of the kriging based surrogate modeling are discussed.This kind of optimization technique is not only limited to thisspecific case of microgrid control, but also can be ported to othercomputationally expensive power system optimization problems.

Index Terms—Fractional order proportional integral deriva-tive (FOPID) controller, global optimization, kriging, microgridcontrol, surrogate modeling.

I. INTRODUCTION

PENETRATION of renewable energy technologies in elec-trical power systems, in recent years, have increased

the system complexity and requires efficient monitoring andcontrol methods to ensure smooth operation of the whole sys-tem [1]. The distributed generation (DG) model [2] which usessmall capacity generators, typically in the order of 5 kW to10 MW, can easily include renewable energy storage systemsand other fuel based generators. The DG model offers theadvantage of localized generation with consequent reductionin transmission costs and losses. They offer increased relia-bility and ease of maintenance. Retrofitting other units to theDG is also simple and helps in easier capacity planning andimprovement in later stages of operation. Smart grid refers tothe integration of these DGs into the grid where there is signifi-cant interplay of information and communication technologies

Manuscript received November 22, 2013; revised May 12, 2014; acceptedJuly 4, 2014. Paper no. TSG-00871-2013.

I. Pan is with the Energy, Environment, Modelling, and Minerals ResearchSection, Department of Earth Science and Engineering, Imperial CollegeLondon, London SW7 2AZ, U.K. (e-mail: [email protected]).

S. Das is with the Communications, Signal Processing, and Control Group,School of Electronics and Computer Science, University of Southampton,Southampton SO17 1BJ, U.K. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSG.2014.2336771

for increasing the grid flexibility and system reliability [3].These smart grids may be composed of different smaller unitsknown as microgrids (MG) [4]. These MGs are comprised ofsmall generation and load units connected at multiple pointsand can either operate autonomously as an isolated system orcan work in a grid connected mode. Since the generating unitsare small in capacity, their inertia is smaller. This results insevere fluctuations in system parameters like frequency andvoltage in cases where the input is stochastic (wind or solar)or there are outages in the generating units. To improve thesystem stability and performance, energy storage devices likeflywheels, batteries, and ultracapacitors are often used [4].These serve as backup devices and store excess power whenthe generation is more than the demand and release power tothe grid when the demand is more than the generation. Thisessentially helps in maintaining a steady flow of power irre-spective of the generation and load power level fluctuationsand consequently keeps the deviations in system frequency atacceptable levels.

For efficient operation of these interconnected generatorsand energy storage devices, the control of microgrids hasreceived increased attention in recent years [5]. Proper man-agement and control of domestic smart grid technology canbe achieved by good prediction, advanced planning, and realtime control [6]. This helps in a better matching of demandand supply. There are different levels of control schemes inthe grid viz. the local controls, centralized controls, and decen-tralized controls [5]. Recently intelligent frequency controltechniques using particle swarm optimization (PSO) and fuzzylogic have been used in the context of microgrids [7] withencouraging results. Multiagent system for microgrid controlhas been investigated in [8]. Genetic algorithms have alsobeen employed for frequency control design in hybrid energygeneration/storage system [9].

Application of fractional calculus based control systemdesigns have gained impetus in recent times due to the flex-ibility and effectiveness that can be gained through suchmethodologies [10], [11]. Merging computational intelligencetechniques with fractional order controller designs are alsobeing recently explored [12]. However, applicability of frac-tional order (FO) controllers for electrical power systems isstill largely unexplored. A few studies have been done for theapplication of the FO proportional integral derivative (FOPID)controller to the design of the automatic voltage regulator(AVR) and load-frequency control (LFC) in a power systemusing single objective [13] and multiobjective formalisms infrequency domain [14] and in time domain [15], which have

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2 IEEE TRANSACTIONS ON SMART GRID

been shown to give better results over the traditional PID con-troller. The FOPID controller has also been applied to theproblem of two area load frequency control in a deregulatedenvironment [16]. A fractional calculus based maximum pho-tovoltaic power tracking (MPPT) controller is also proposedfor microgrid system in [17].

Most of these optimization based controller design prob-lems involve multiple calls to computationally expensivetime domain simulations for power system’s dynamics withdifferent guess values of controller parameters which arerefined iteratively. This becomes computationally prohibitiveand therefore algorithms which can arrive at optimal solutionsin less number of iterations are necessary. Meta-models orsurrogate modeling methodology for evolutionary algorithmsare expedient in such circumstances [18]. These meta-modelsapproximate the fitness landscape of the expensive optimiza-tion function and take much less computational time to simu-late. Surrogate based modeling methodologies have also beenexpedient in problems involving dynamic optimization [19],constrained optimization, combinatorial optimization [20] etc.amongst many others. Various techniques can be used for theconstruction of the surrogate fitness function like radial basisfunctions [21], neural networks [22], kriging methods [20],response surface models [23], etc. However, surrogate mod-eling techniques have not been popular for optimization anddesign of expensive dynamic power system models.

In this paper, a kriging assisted surrogate modeling method-ology is outlined and embedded within a global optimizationframework for the design of FOPID controllers for microgridfrequency control. The FO differ-integral operators are inher-ently infinite dimensional linear filters [10], but for practicalrealization of such systems, band-limited higher order linearsystem approximations are commonly used [24]. Thereforetime domain simulation of such coupled very high orderapproximated FO systems with several power system com-ponents, is computationally expensive. The surrogate mod-eling methodology is hence suitable in such circumstancesto obtain the controller parameters in less number of itera-tions to pave the path of online tuning of this type of FOcontrollers.

II. THEORETICAL BACKGROUND

A. Fractional Calculus Basics

Fractional calculus extends the common notion of integerorder integration/differentiation to any arbitrary real number.It can be represented by aDα

t where α ∈ � is the order ofthe differentiation/integration and a and t are the bounds ofthe operation. There are many definitions of fractional cal-culus like the Grünwald–Letnikov (GL), Riemann–Liouville(RL), and Caputo definitions [11]. In control system studies,the Caputo definition is mostly used for realizing the frac-tional integro-differential operators of the FOPID controller.According to Caputo’s definition, the αth order derivative ofa function fx (t) with respect to time is given by

0Dαt fx (t) = 1

� (m − α)

∫ t

0

Dmfx (t)

(t − τ)α+1−mdτ (1)

α ∈ �+, m ∈ Z+, m − 1 ≤ α < m.

Fig. 1. Schematic of a microgrid with different connected energy sources.

Fig. 2. Block diagram schematic of the microgrid used in the study.

B. Microgrid System and the Controller Structure

Fig. 1 shows the schematic of the microgrid [7] used inthe present study with various power generating units likethe wind turbine, photovoltaic cell, fuel cells, and dieselenergy generator etc. There are also a battery and a fly-wheel energy storage system in the microgrid. The FOPIDcontroller gives the control signal to the fuel cell (FC) andthe diesel energy generator (DEG) based on the frequencydeviation in the microgrid and tries to minimize the stochas-tic fluctuations in the grid frequency. Fig. 2 shows the blockdiagram schematic of the microgrid with the transfer func-tions of the individual components. The parameters of differentcomponents of the microgrid system are adopted from [7]as follows: KWTG = KFESS = KBESS = 1, D = 0.015 pu/Hz,2H = 0.1667 pu s, TFESS = 0.1 s, TBESS = 0.1 s, TFC =0.26 s, TWTG = 1.5 s, Tg = 0.08 s, Tt = 0.4 s, TI/C = 0.004 s,TIN = 0.04 s, R = 3 Hz/pu.

The FOPID controller is used to minimize the system fre-quency fluctuations in the microgrid and ensure better powerquality. The transfer function representation of a FOPIDcontroller is given by

C (s) = Kp + Ki/

sλ + Kdsμ. (2)


PAN AND DAS: KRIGING BASED SURROGATE MODELING FOR FRACTIONAL ORDER CONTROL OF MICROGRIDS 3

This typical controller structure has five independent tun-ing knobs i.e., the three controller gains

{Kp, Ki, Kd

}and

two fractional order operators {λ, μ}. For {λ, μ} = 1 thecontroller structure (2) reduces to the classical PID con-troller in parallel structure. Few recent research results showthat band-limited implementation of FOPID controllers usinghigher order rational transfer function approximation of theintegro-differential operators gives satisfactory performance inindustrial automation [25]. The Oustaloup’s recursive approx-imation (ORA), which has been used to implement theintegro-differential operators in frequency domain is given by(3), representing a higher order analog filter [10]

sα ≈ Kf

N∏k=−N

s + ω′k

s + ωk(3)

where, the poles, zeros, and gain of the filter can be recursivelyevaluated as

ωk = ωb

(ωh

ωb

) k+N+ 12 (1+α)

2N+1

, ω′k = ωb

(ωh

ωb

) k+N+ 12 (1−α)

2N+1

Kf = ωαh . (4)

The frequency deviation signal �f (t) is passed through thefilter (3) and the output of the filter can be regarded as anapproximation to the fractionally differentiated or integratedsignal Dα

[�f (t)

]which are then linearly combined with PID

gains to obtain the final control signal for controller struc-ture (2). In (3)–(4), α is the order of the differ-integration,(2N + 1) is the order of the analog filter and (ωb, ωh) is theexpected fitting range. Here, 5th order ORA is adopted for theFO operators within a chosen frequency band ω ∈ {

10−2, 102}

rad/sec for the constant phase elements (CPEs) [10]–[12].The controller output actuates the FC and DEG to control

these units as they require expensive fuels. Depending on thegrid frequency fluctuation, the controller sends a signal to therespective actuators i.e., like the hydrogen flow rate in the FCand mass flow rate of oil in DEG. Such a scheme can regulatethe amount of power delivered by these devices to the micro-grid to minimize the operating costs. The motor speed to runthe flywheel and battery input current are directly taken fromthe grid frequency oscillation signal without the interventionof the controller as these devices does not need sophisticatedcontrol [7].

Here, the rated electrical specifications are adopted from [7]as the nominal case (without stochastic fluctuations), where allthe powers are represented in pu with respect to the total baseload demand of 410 kW (1 pu). The electrical load specifica-tions of [7] are incorporated in the present model in the form ofoutput saturation in each of the power producing components.The rated wind and solar powers are considered as 100 kW(0.244 pu) and 30 kW (0.073 pu) respectively with total renew-able generation of 130 kW (0.317 pu). Similarly, the maximumpower generated/absorbed by battery energy storage system(BESS) and flywheel energy storage system (FESS) are con-sidered ±0.11 pu. In the case of the rapid power producingelements (FC/DEG), the lower bound of the saturation is zerosince they cannot absorb power and the upper bound is 0.48 pu

and 0.45 pu respectively. In [7], a sudden 0.1 pu (≈ 41 kW)of load disturbance has been introduced to study the perfor-mance of the control system. This has been improved in thispaper with a step-wise increase in the demand power havingsmall stochastic fluctuations.

An upper and lower saturation limit is imposed on eachenergy storage element along with rate constraint nonlinear-ity to avoid any possible mechanical shock due to suddenlarge frequency fluctuation. The dynamical models in Fig. 1represents small signal linearized transfer functions whichcaptures the dynamic characteristics at a specific operatingpoint [9]. The upper and lower limits of the output satu-ration will restrict the extraction/storage of large amount ofpower from/to a particular element than its rated values. Theoutput saturations (in pu) and rate constraints for differentelements are

|PFESS| < 0.11, |PBESS| < 0.11, 0 < PFC

< 0.48, 0 < PDEG < 0.45∣∣PFESS∣∣ < 0.05,

∣∣PBESS∣∣ < 0.05,

∣∣PFC∣∣ < 1,

∣∣PDEG∣∣ < 0.5.

The microgrid has different control hierarchy like the localcontrol, secondary control, centralized control, etc. [26], andhere, the secondary control scheme is adopted. In case ofemergency where one subsystem needs to be disconnected,the centralized control loop overrides the functioning of thesecondary control loop and disconnects the subsystem.

C. Characteristic Changes of the Wind and Solar PowerGeneration and the Demand Load

Large deterministic drift and small stochastic power fluc-tuations [9] are considered for the wind generation, solargeneration, and load demand and are modeled in a generaltemplate here. This kind of template gives rise to a time-serieswith small stochastic fluctuations about the mean generated ordemand power. In addition, the models take into account a sud-den change in the mean value to represent real case scenarioswhere there are significant variations of these parameters. Thegeneral template for these is chosen as

P =((

φη√

β (1 − G (s)) + β) /

β)

� = χ · � (5)

where, P represents the power output of the solar, wind orthe load model, φ is the stochastic component of the power,β contributes to the mean value of the power, G (s) is a lowpass filter, η is a constant in order to normalize the generatedor demand power (χ ) to match the per unit (pu) level, � isa time dependent switching signal with a gain which dictatesthe sudden fluctuation in mean value for the stochastic poweroutput. Due to the sudden change in base value along withstochastic fluctuations, the source of such uncertain behaviorin the power generation and demand can be modeled in a sametemplate, having different parameters as studied in [9].

For the wind power generation the parameters of (5) are

φ ∼ U (−1, 1) , η = 0.8, β = 10, G (s) = 1/ (

104s + 1)

and

� = 0.24h (t) − 0.04h (t − 140) (6)

where, h (t) is the Heaviside step function.



For the solar power generation, the parameters of (5) are

φ ∼ U (−1, 1) , η = 0.1, β = 10, G (s) = 1/ (

104s + 1)

and

� = 0.05h (t) + 0.02h (t − 180). (7)

For the demand load, the parameters of (5) are: φ ∼U (−1, 1), G (s) = (

300/

(300s + 1)) + (

1/

(1800s + 1)),

η = 0.9, β = 10 and

� = (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).

(8)

D. Objective Function for Optimization

For effective functioning of the microgrid system, the con-troller gains, and fractional integro-differential orders need tobe tuned. For the controller design problem, the objective func-tion in (9) is considered. It consists of the integral of twoweighted terms, which try to minimize the frequency devia-tion in the microgrid (�f ), as well as the incremental controlsignal (�u)

J =∫ Tmax=220

Tmin=100

[w (�f )2 + (

(1 − w)/

Kn)(�u)2

]dt (9)

where, w dictates the relative importance of the two objectives(i.e., Integral of Squared Error—ISE and Integral of squaredDeviation of Control Output—ISDCO) [15] and its value istaken as 0.7. Kn = 104 is the normalizing constant to scaleISE and ISDCO in uniform scale. For any fixed structurecontroller there is always a trade-off between the conflictingobjectives of load disturbance rejection (reducing �f to zerovery quickly) and the amount of control effort (�u) required.Previous investigations [15] have shown that the problem isinherently multiobjective and to have a fast suppression of loaddisturbance, a higher amount of controller effort is required.The choice of w as 0.7 in the present case indicates that thedesign gives more importance to the fast suppression of themicrogrid frequency oscillations in comparison to the highervalue of control signal.

E. Kriging Based Global Optimization

Kriging models have been shown to be very expedient inaccurate global approximations of the design space [27]. Thesemodels can approximate both linear and nonlinear trends inthe design space. Their flexibility arises from the fact thatdifferent spatial correlation functions can be employed forbuilding the approximation. Additionally, the kriging modelscan be built either to give more importance to the trainingdataset, by providing an exact interpolation, or they may bebuilt to have a smooth inexact interpolation [28]. For the pur-pose of constructing a surrogate model, consider a set of kdesign sites S = [

s1 . . . sk]

with si ∈ �n and corresponding

model responses Y = [y1 . . . yk

]T with yi ∈ �q. The data isnormalized to satisfy the following conditions in (10):

m[S:,j

] = 0, σ[S:,j, S:,j

] = 1, ∀ j ∈ {1, 2, . . . , n}m

[Y:,j

] = 0, σ[Y:,j, Y:,j

] = 1, ∀ j ∈ {1, 2, . . . , q} (10)

where X:,j is the vector represented by the jth column in matrixX, and m [·] and σ [·, ·] denote the mean and the covariancerespectively.

A kriging model combines a global model with localized

departures [29]. The model y that gives the deterministic

response y (x) ∈ �q, for an n dimensional input x ∈ �z ⊆ �n

as a realization of a regression model � and a stochasticprocess

yl (x) = �(ζ:,l, x

) + zl (x) , l = 1, . . . , q. (11)

The regression model � is taken as a linear combinationof p chosen functions fj:�n �→ �, ∀ j ∈ {1, 2, . . . , p} and canbe expressed as

�(ζ:,l, x

) = ζ1,lf1 (x) + · · · + ζp,lfp (x)

= [f1 (x) . . . fp (x)

]ζ:,l = f (x)T ζ:,l (12)

where, the coefficients{ζk,l

}are the regression parameters.

The random process z is assumed to have zero mean andcovariance between z (w) and z (x) is given by

E [zl (w) zl (x)] = σ 2l �(θ, w, x) , ∀l ∈ {1, 2, . . . , q} (13)

where σ 2l is the process covariance for the lth component

of the response and �(θ, w, x) is the correlation model withparameters θ .

For calculating the kriging predictor for the set S of designsites, the design matrix F ∈ �k×p with Fij = fj (si) can beexpressed as

F = [f (s1) . . . f (sk)

]T (14)

where, f (x) is defined in (12). Additionally the matrix Qis defined to be that of the stochastic process correlationsbetween z’s at the design sites

Qij = �(θ, si, sj

) ∀i, j ∈ {1, 2, . . . , k}. (15)

At a previously unsampled location x, let

q (x) = [�(θ, s1, x) . . . � (θ, sk, x)

]T (16)

represent the vector of correlations between z’s at the designsites and x.

For the regression problem of the form (17)

Fζ ≈ Y (17)

the generalized least square solution (with respect to Q) is

ζ ∗ =(

FTQ−1F)−1

FTQ−1Y (18)

and the corresponding kriging predictor can be expressed as

y (x) = qTQ−1Y +(

FTQ−1q − f)T

ζ ∗

= f (x)T ζ ∗ + q (x)T Q−1 (Y − Fζ ∗). (19)

The mean squared error estimate ϕ (x) of the predictor canbe obtained as

ϕ (x) = σ 2(

1 + bT(

FTQ−1F)−1

b − qTQ−1q

)(20)



TABLE IDIFFERENT CORRELATION FUNCTIONS FOR THE KRIGING MODEL

Fig. 3. Schematic of the kriging based optimization method.

where, b = FTQ−1q − f and σ 2 is the maximum likelihoodestimate of the variance.

The correlation models considered in this paper are of theform (21) [29]

�(θ, w, x) =n∏

j=1

�j(θ, wj − xj

)(21)

which are essentially products of stationary 1-D correlations.Table I shows the different correlation functions used in thepresent study.

Fig. 3 shows a schematic of the kriging based surrogatemodeling and optimization process. Initially a symmetric Latinhypercube sampling scheme is constructed in the design spaceof the fractional order controller (i.e., the three gains and thetwo fractional orders). This kind of a sampling scheme is atrade-off [30] between the simple Latin hypercube sampling(which might not cover the entire design space uniformly) andthe computationally expensive ‘space-filling’ Latin hypercubesampling [31] methods. The bounds of the design variables{Kp, Ki, Kd, λ, μ} are kept between {0, 0, 0, 0, 0} and{5, 5, 5, 2, 2} respectively and 50 initial sampling points aregenerated using this scheme. Using these sampling points, thekriging model is constructed with one of the correlation func-tions as in Table I. The candidate point approach [32] is usedon the kriging model, for selecting the best location to sam-ple in the next iteration. It works by creating two groupsof candidates, one which is obtained by perturbing the bestpoint obtained till the present iteration and the other which

TABLE IISTATISTICAL RESULTS OF 30 INDEPENDENT RUNS OF DIFFERENT

KRIGING BASED OPTIMIZATION WITH PID/FOPID CONTROLLER

is generated by uniformly selecting points from the wholedecision space. The kriging based response surface is usedto predict the objective function values at the candidate pointsand is used to calculate the response surface criterion. Thedistance criterion is calculated which measures the distanceof each candidate point to the existing set of sampled points.The weighted sum of these two criteria is used to determinethe best candidate point. This best candidate point is thenused to sample the computationally expensive objective func-tion (by running the microgrid control model) and the actualvalue of the objective function is obtained. The kriging modelis updated with this new value of the objective function andthe process is iterated until a specified number of functionevaluations are completed.

III. SIMULATION AND RESULTS

A. Performance Evaluation of the Kriging Based Optimizersand Different Controller Structures

Since the objective function is stochastic in nature (i.e., thesame value of controller parameters gives slightly differentobjective function values in each instance), the function isevaluated multiple times (10 in this case) and the expectedvalue of the objective function is considered for optimization.Calling the objective function multiple times in this fashionis counted as one expensive function evaluation and the totalnumber of these expensive function evaluations is limited to150. The kriging based optimization is run with the differentcorrelation models and the statistical results for 30 indepen-dent runs along with the best found expected minima (sincethe model contains stochastic components) of the objective(9)—Jmin are reported in Table II. A comparison is done witha standard genetic algorithm (GA) which is run with 10 popu-lations for 15 generations (so that the total number of functionevaluations is 150) and the statistical results for the sameare also reported in Table II. The number of elite individ-uals is taken as 2 and the crossover and mutation fraction



TABLE IIIBEST PID/FOPID CONTROLLER PARAMETERS

are taken as 0.8 and 0.2 respectively. The corresponding bestfound parameters for the PID and the FOPID controller arereported in Table III. During the optimization four differentcontroller structures are called depending on ranges of val-ues for FOPID orders i.e., {λ, μ} greater and less than one,since the ORA can only approximate FO operators less thanunity. For {λ, μ} > 1 the nominal FO part is rationalizedusing ORA and an additional integrator/differentiator is addedwithin the controller structure in series with the higher orderrational approximation [10], [12].

The results in Table II indicate that the FOPID controllerconsistently outperforms the PID controller with all of theoptimization algorithms. Also, the kriging based optimizationalgorithm with the spline correlation function gives the bestresult for both the PID/FOPID controllers. In both cases, thekriging based surrogate method outperforms the GA. Also themean and standard deviation of the kriging based optimizationalgorithms in Table II are very small as compared to the GA.This occurs since the GA is not able to find stable solutions inthe many of the runs. Therefore the kriging based method notonly gives better accuracy, but also consistently gives stablenear optimal solutions every time. Table II also shows thatthe spline correlation model gives the best average solutionfor both PID/FOPID controller amongst the five correlationmodels in Table I.

It is known in FO control that the integral order of FOPIDhigher than unity i.e., λ > 1 makes the overall system faster,whereas λ < 1 makes it slower [12]. On contrary the controleffort for λ > 1 increases drastically to ensure faster timeresponse and λ < 1 produces smaller control effort. Since thecost function (9) has got two parts balancing the impact of fasttracking and control effort as a weighted sum, under differentcircumstances integro-differential orders may be less than orgreater than one, although in Table III both the optimum FOorders are <1.

Fig. 4 shows a single realization of the stochastic pro-cess model for power generation by the wind turbine

Fig. 4. One realization of the stochastic generated and demand powersindependent of the controller structure.

Fig. 5. Frequency deviation, control signal and power deviation of themicrogrid with best obtained PID/FOPID controllers.

generator (WTG), solar photovoltaic (PV) and the load (5)which is used in the simulation study. It is observed that thereare significant fluctuations which would be reflected in thesystem frequency and the controller needs to take appropriateaction to damp out these oscillations considering all possi-ble realization of the stochastic objective function (9), therebylocating an expected minima [33], [34] based on multiple runsof the same model. Therefore, the role of the kriging modelused here is to approximate a dynamically evolving noisy func-tion taking into account the correlations among the differentcontroller parameters to find out the expected global minimain less number of iterations. Such reduction in number of itera-tions would facilitate the online implementation of tuning suchcontrollers in future [7]. Significant non-stationary nature ofthe stochastic fluctuation with drift in renewable generationand demand load are evident in Fig. 4. In the present designframework, it is considered that the microgrid was operatingat 1 pu load during 0 < t < 100 sec and the control systemperformance has been evaluated then for a finite time horizonof 100 < t < 220 sec considering change in both the demandload and renewable generations (Fig. 4).

The microgrid frequency deviation (�f ), control signal (u),and the deficit/excess power (�P) corresponding to the bestobtained PID and the FOPID controller are shown in Fig. 5.It is evident that the FOPID controller outperforms the PIDcontroller since it results in less frequency fluctuation �f(hence better power quality), faster damping of the deficit grid



Fig. 6. Individual powers of the different components of the microgrid withthe best obtained PID/FOPID controllers.

Fig. 7. Convergence of five kriging methods for PID controller.

Fig. 8. Convergence of five kriging methods for FOPID controller.

power �P with less control signal (hence less actuator sizerequirements for the FC and DEG).

The corresponding powers produced by the FESS, BESS,FC, and DEG are reported in Fig. 6. It can be observed thatthere are less power fluctuations with the FOPID controllerthan the PID controller. This implies that if the FOPID con-troller is used, the sizing of these energy supply and storagesystems can be made smaller. Also there are less require-ments of supplying and storing power to supress the microgridfrequency fluctuations. This makes the overall system moreenergy efficient.

Figs. 7 and 8 show four standard statistical measures (mean,median, best, and worst case) of the convergence curves insemi-log scale, for 30 independent runs of the kriging based

Fig. 9. Convergence of GA based tuning of PID/FOPID controllers.

optimization employing different correlation functions for thePID and the FOPID controller respectively. The correspond-ing curves for the GA are plotted in Fig. 9. It is clear thatto find the best solutions, the GA takes more number of iter-ations whereas all the kriging based optimizers converges tothe best solutions very quickly. Therefore, the kriging solutionswould significantly outperform the GA even if less number offunction evaluations (than 150) are used. Also, if there aresignificant stochastic deviations in the objective function, thencalling the function only 10 times (as done in this case) wouldnot be suitable for approximating an expected value and morenumber of samples need to be taken. In such cases, the krig-ing would significantly outperform the GA as well. As evidentfrom Figs. 7 and 8, for both the PID and FOPID controllerthe spline correlation function is capable of locating a lowervalue of the objective functions in all the four cases of statisti-cal measures which is in agreement with the findings reportedin Table II.

B. Parametric Robustness of the Optimum Solutions

The best solutions for the PID/FOPID controllers are nowobtained from Tables II and III (as the nominal case) andtheir robustness is tested for perturbed case of differentmicrogrid parameters. The expected values of the objectivefunction (9) are noted in Table IV considering multiple runsof the stochastic model. The corresponding microgrid fre-quency deviation for both the case of parameter increase anddecrease (in D, 2H, R, TFC, Tg, Tt, TI/C, TIN) are shownin Figs. 10 and 11, respectively. It can be observed fromTable IV and also Figs. 10 and 11 that the PID controller isless robust than the FOPID controller in terms of increasedvalue of J under perturbed condition and increase in �f.This establishes the superiority of using FOPID controllerin such a microgrid frequency control problem with largepositive/negative uncertainty in the system parameters. It isalso evident from Figs. 10 and 11 that the increase in sys-tem parameters is much detrimental than a decrease in equalamount in most cases, although the ill-effects are significantlysmaller with an FOPID. Among different parameters the inter-connection device, FC, and inverter time constants (TI/C, TFC,TIN) are found to be gradually most susceptible ones whichcause performance deterioration much faster than perturbingother microgrid parameters.



TABLE IVROBUSTNESS FOR PERTURBATION IN MICROGRID PARAMETER

Fig. 10. Robustness for an increase in system parameters.

C. Effect of Actuator On-Off Switching Logic

Next, it is assumed that the FC and DEG does not supplyany power if the grid frequency deviation is within a speci-fied limit of |�f | < 0.05 and only turns on when the FESSand BESS cannot supply/absorb enough power within a shortperiod to damp the frequency deviation. It turns out that thescheme suffers from chattering which is common for suchswitched systems and sliding mode control. Chattering is detri-mental for different components of the microgrid and henceis not desirable. To alleviate this issue to some extent a mod-ification of the scheme is done such that it remains on for atleast a minimum of 10 sec until the frequency deviations arewithin acceptable limits. However, as soon as this happens andboth the DEG and FC are cut out of the microgrid, the systemfrequency starts to deviate and the scheme again triggers theDEG and FC on, after a few milliseconds. Therefore, there aresignificant switching transients after almost every 10 secondsas shown in Fig. 12 which is undesirable. The actuator on-offswitching logic is implemented in Stateflow while modelingof the microgrid with FOPID is done using Simulink which

Fig. 11. Robustness for a decrease in system parameters.

Fig. 12. Transients due to on-off switching of the actuators.

Fig. 13. Significant nonlinear operation of the energy storage/supplyingelements with rate constraint nonlinearity.

is invoked by the optimization procedure coded in MATLABscripts. The whole model (with system nonlinearity in the formof rate constraint and output saturation, FOPID controller,stochastic forcing with jumps) is numerically integrated withthe 3rd order accurate Bogacki-Shampine formula with a fixedstep-size of 0.01 sec.

Operation in significant nonlinear zone has also been shownin Fig. 13 along with the actuator on-off logic in Fig. 12 forthe present microgrid system with PID/FOPID controller. It isevident from Fig. 13 that the rate of change of power in eachelement above or below the respective thresholds have beencut-off and a constant rate is maintained. Also the DEG andFC in Fig. 12 produces a constant amount of power once they



reach their maximum limits due to the presence of the outputnonlinearity in the system.

IV. CONCLUSION

The paper proposes the use of fractional order controllerfor supressing the system frequency deviation in a nonlin-ear and stochastic model of a microgrid. Simulation resultsshow that the FOPID controller is better than the standardPID controller under nominal operating condition and givesbetter robustness for large parametric uncertainty of the micro-grid. A kriging based surrogate modeling and optimizationtechnique is proposed which reduces the time taken for opti-mizing the controller parameters for the microgrid system.This can facilitate the online tuning of such controllers infuture. Simulation results show that the kriging based opti-mization outperforms the standard genetic algorithm in termsof the quality of solutions and faster convergence.

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kriging based surrogate modeling for fractional order control of microgrids

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