Load Frequency Control of an Isolated Micro Grid Using Fuzzy Adaptive Model Predictive Control

This paper presents a fuzzy adaptive model predictive approach for load frequency control of an isolated micro grid. A generalized state space model of a typical isolated micro grid having controllable and uncontrollable generating power sources is derived, and the same has been utilized to predict the future output and control inputs for the micro grid frequency control. A centralized model predictive control (MPC) is implemented with a single input multi-output system model based on the controllable distributed energy resources in the micro grid. The parameter-driven MPC is made adaptable by dynamically adjusting its parameter Rw using fuzzy controller. The proposed fuzzy MPC employs a rule-based fuzzy controller to fuzzify the tuning parameter Rw present in the cost function of MPC, which plays an important role in minimizing the frequency deviation in the system. The closed loop system response obtained by the proposed fuzzy MPC has been found faster and adaptable for different scenarios in the system. The effectiveness has been evaluated with performance index integral time square error (ITSE) value and has been compared with MPC with constant tuning parameter value also with the proportional-integral controller response. Thus, the efficacy of using proposed fuzzy MPC in secondary load frequency control has been validated thereof.

SECTION I. Introduction

A reliable load frequency control is an important function of modern power system which is dispersed geographically over a large scale and interconnected with multiple generations in a complex manner. This load frequency control equally requires greater attention for microgrid operation in the distribution network especially, when the off-grid microgrids are remotely operated in rural areas. Since the microgrid accommodates converter based, low inertial and intermittent renewable Distribution Generations (DGs), it involves greater challenges and demands advanced control strategies to ensure continuous supply to loads and to smoothen frequency fluctuation in the system. Even though there are several works on control and management of microgrid in different perspective over the past two decades [1], [2], the regulation of frequency and voltage within specified nominal values in isolated microgrid operation is really an important aspect for reliable system operation [3] and one that has not received adequate attention.

Hitherto, several control schemes and methods have been reported in the area of LFC in standalone microgrid to study the system dynamics under different system disturbances. These methods range from classical droop controls to various advanced control strategies that contribute to the secondary load frequency control of conventional and distributed generation power system and comprehensibly summarized in [4]. Several such methods are applied to investigate different aspects of secondary load frequency control in microgrid [5]–​[9]. In [5], a hierarchical droop control method is proposed for microgrid load frequency control whereas in [6], a fuzzy based PID controller is employed for coordinating the aqua electrolyser and fuel cell to control the power fluctuation in the microgrid. In [7] and [8], meta heuristic optimisation algorithms such as combined Particle Swarm Optimisation (PSO) with fuzzy logic and PSO based mixed H₂/$\text{H}_{\mathrm {\infty }}$ are proposed to enhance the frequency control in microgrid. Numerical modeling of system dynamics with active Distributed Secondary Loads with DGs sharing is proposed for better frequency regulation in [9]. The uncertainties and nonlinear complexities in the system necessitate robust control techniques to get trade-off between the robust system performance and the stability in the closed loop response against system uncertainties and such robust techniques are discussed in [10]–​[13]. In recent times, Model Predictive Control (MPC) is also widely adopted due to its simple and fast implementation. Even though MPC is proved efficient indigenously in process industry, due to its modelling flexibility that involves straight forward design procedure, acceptable computational time and ease in constraint handling, it is well received in all control application and it is widely adopted in industries such as petrochemical industry, electrochemical, power and water management sectors etc.

Although so many variants of MPC are proposed in the literature for different applications, the classical dynamic matrix control (DMC) of MPC is predominantly applied to the load frequency control problem. Some research which applied MPC to the load frequency control are discussed in [14]–​[19]. In [14], a distributed MPC (DMPC) is proposed for load frequency control of four-area hydro–thermal interconnected power system. In addition to that, the limit position of the governor valve is controlled by a fuzzy model and the local predictive controllers are incorporated into the non-linear control system. LFC design using distributed model predictive control (DMPC) technique for the multi-area interconnected power system is proposed in [15]. The effect of wind turbine generator participation on the load frequency control of two area system is analysed with MPC structure and the overall closed loop system performance has demonstrated the robustness of control in face of uncertainty [16]. Model predictive control is employed for load frequency control of micro grid in the presence of Plugged-in Hybrid Electric Vehicle (PHEVs) [17], [18]. In [17], an aggregator utilizing MPC strategy is proposed to impose control actions on the participating units such as a cluster of PHEVs and controllable thermal appliances and a decentralized Combined Heat Plant (CHP) unit for load frequency control. Whereas in [18], to avoid a large number of PHEVs for better frequency regulation, the smoothing of wind power production by pitch angle control using MPC method is proposed. In [19], an Explicit MPC (EMPC) is proposed to give affine control law which is calculated offline by partitioning the state space into Critical Regions (CRs) by solving Multi-Parametric Quadratic Programming (MP-QP). Optimization is implemented online using the look up table of CRs and controls. EMPC is illustrated on the isolated industrial power system frequency control. Most of the MPC used for load frequency control is transformed from centralised to decentralised/distributed MPC but few have investigated improving the adaptability of MPC. This paper proposes a new adaptive MPC where parameter $\text{R}_{\mathrm {w}}$ in its cost function has been tuned by a fuzzy controller to make MPC a robust control for different operating scenarios.

The main contributions of the paper are:

1. It proposes a new fuzzy adaptive model predictive controller for the load frequency control for a typical microgrid and tested for various cases.

2. The input tuning parameter R_w present in the cost function of MPC has been fuzzified using fuzzy logic controller.

This paper is structured as follows: Section II provides a brief outline of the model predictive control and the MPC implementation for load frequency control of an isolated micro grid is discussed in detail in section III. The proposed rule based fuzzy inference system for parameter ($\text{R}_{\mathrm {w}})$ tuning is explained in section IV. Simulations of various cases of load frequency control and the discussions on the obtained results are presented in section V. Section VI concludes the paper.

SECTION II. Outline of Model Predictive Control (MPC)

An MPC is a model based advanced control strategy which employs an optimization procedure at each sampling time over prediction horizon to calculate the optimal control actions. As there is extensive literature on MPC, this section intends to present only the outline of MPC.

The general discrete state-space model representation is given by:\begin{align} x\left ({ k+1 }\right )=&A_{d}x\left ({ k }\right )+B_{d}u\left ({ k }\right )+E_{d}w(k) \\[5pt] y\left ({ k }\right )=&C_{d}x\left ({ k }\right )+D_{d}u(k) \end{align} Where, $u$ - input variable vector; $y$ - Process output vector; $x(k)$ - State variable vector at $\text{k}^{\mathrm {th}}$ step and $w(k)$ is the disturbance vector at $\text{k}^{\mathrm {th}}$ step.

Since moving horizon control requires current information of the plant for the prediction and control, it is implicitly assumed that u(k) cannot affect output y(k) but u(k-1) at the $\text{k}^{\mathrm {th}}$ instant can [20]. So, on taking difference on both sides of (1) and rearranging \begin{align} \left [{ {\begin{array}{*{20}c} \Delta x(k+1)\\[5pt] y(k+1)\\[5pt] \end{array} } }\right ]=&A\left [{ {\begin{array}{*{20}c} \Delta x(k)\\[5pt] y(k)\\[5pt] \end{array} } }\right ]+Bu\left ({ k }\right )+Ew(k) \\[5pt] y\left ({ k }\right )=&C\left [{ {\begin{array}{*{20}c} \Delta x(k)\\[5pt] y(k)\\[5pt] \end{array} } }\right ] \end{align} Where, \begin{align*} A=&\left [{ {\begin{array}{*{20}c} A_{d} & {0}_{d}^{T}\\[5pt] C_{d}A_{d} & 1\\[5pt] \end{array} } }\right ]; \quad B=\left [{ {\begin{array}{*{20}c} B_{d}\\ C_{d}B_{d}\\[5pt] \end{array} } }\right ]; ~ E=\left [{ {\begin{array}{*{20}c} E_{d}\\ C_{d}E_{d}\\[5pt] \end{array} } }\right ]\\ C=&\left [{ {\begin{array}{*{20}c} {0}_{d}^{T} & 1\\ \end{array} } }\right ] \quad {0}_{d}=\overbrace {[{0~~0~~0~~0}]} \end{align*} Where, m is the number of state variables. A, B and C are augmented state space model used in design of predictive control. This prediction is described within an optimization window of length Np. Since the disturbance in current step is not reflected in the future, the disturbance matrix is omitted in the prediction window [20].

Suppose at sampling instant k, $\text {k}>0$ , the future control trajectory at sampling instant k is given by \begin{equation} \Delta u\left ({ k }\right ), \Delta u\left ({ k+1 }\right ),\cdots , \Delta u\left ({ k+N_{c}-1 }\right ) \end{equation} While the rest of $\Delta \text{u}$ (k) for $\text {k}=\text {Nc}$ , Nc+1,$\ldots \ldots $ Np is assumed to be zero. Where Nc is the control horizon that gives the number of future control inputs to be predicted and it is chosen to be less than Np. With given x(k) and predicted control inputs, future state variables and output are predicted sequentially over the prediction window. This has been shown in Appendix-A. The compact form of output is shown (6).\begin{align} Y=&Fx\left ({ k }\right )+\Phi \Delta U\notag \\[6pt] F=&\left [{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} CA\\ CA^{2}\\ \end{array} }\\ {\begin{array}{*{20}c} CA^{3}\\ {\begin{array}{*{20}c} \vdots \\ CA^{Np}\\ \end{array} }\\ \end{array} }\\ \end{array} } }\right ]\notag \\[6pt] \Phi=&\left [{ {\begin{array}{*{20}c} CB & 0 & 0 & \ldots & 0\\ CAB & CB & 0 & \ldots & 0\\ CA^{2}B & CAB & CB & \ldots & 0\\ \vdots & \vdots & \vdots & \ldots & \vdots \\ CA^{Np-1}B & CA^{Np-2}B & CA^{Np-3}B & \ldots & CA^{Np-Nc}B\\ \end{array} } }\right ]\notag \\ {}\end{align}

For a given reference signal r(k) at sample k, the objective is to predict an output close to the reference signal. And this r(k) remains constant in the optimization window. The control objective is given by:\begin{equation} J=\left ({ Y_{s}-Y }\right )^{T}\left ({ Y_{s}-Y }\right )+{\Delta U}^{T}\bar {R}_{in}\Delta U \end{equation} To find the optimal control vector $\Delta \text{U}$ that minimize J is found by $\frac {\partial J}{\partial \Delta U}=0$ \begin{align} \Delta U=&\left ({ \Phi ^{T}\Phi +\bar {R}_{in} }\right )^{-1}\Phi ^{T}\left ({ Y_{s}-Fx\left ({ k }\right ) }\right ) \\[6pt] \bar {R}_{in}=&R_{w}\ast I_{Nc\ast Nc}; \quad Y_{s}=\left [{ {\begin{array}{*{20}c} 1 ~~ 1 ~~ \ldots ~ 1\\ \end{array} } }\right ]^{T}r\left ({ k }\right ) \end{align}

SECTION III. MPC Implementation for Load Frequency Control of an Isolated Micro Grid

The simplified load frequency model of a typical microgrid is considered in this paper and it is shown in Fig.1. The micro grid consists of a diesel unit, fuel cell, wind, solar and battery storage of ratings given in [6]. The frequency control in the micro grid is achieved by predicting the future outputs and control signals i.e. frequency deviation and control actions to the controllable units respectively. The renewable sources are assumed to be operated at maximum power point. Hence, diesel unit and fuel cell are considered as controllable units in the microgrid. The prediction was accomplished by using a Model Predictive Control (MPC) design where a state space model of the system is used. From the model of the system shown in Fig.1, the dynamics of the system is defined with nine state equations with nine state variables for the micro grid considered. This has been explained in the Appendix-B for case of generator-load transfer function model. The state equations are shown in (10)–(18):\begin{align} \dot {\Delta f}=&\frac {1}{2H}\left ({ {\begin{array}{l} {\Delta P}_{s\_{}filt}+{\Delta P}_{md}+{\Delta P}_{w}-{\Delta P}_{L}+ \\ {\Delta P}_{f\_{}filt}-{\Delta P}_{bat}-D\ast \Delta f \\ \end{array}} }\right )\qquad \\ \dot {\Delta P}_{s\_{}inv}=&\frac {1}{T_{inv}}\left ({ {\Delta P}_{s}-{\Delta P}_{s\_{}inv} }\right ) \\ \dot {\Delta P}_{s\_{}filt}=&\frac {1}{T_{filt}}\left ({ {\Delta P}_{s\_{}inv}-{\Delta P}_{s\_{}filt} }\right ) \\ \dot {\Delta P}_{gd}=&\frac {1}{T_{g}}\left ({ {\Delta P}_{cd}-{\frac {\Delta f}{R}-\Delta P}_{gd} }\right ) \\ \dot {\Delta P}_{md}=&\frac {1}{T_{t}}\left ({ {\Delta P}_{gd}-{\Delta P}_{md} }\right ) \\ \dot {\Delta P}_{fc}=&\frac {1}{T_{fc}}\left ({ {\Delta P}_{cf}-{\frac {\Delta f}{R}-\Delta P}_{fc} }\right ) \\ \dot {\Delta P}_{f_{i}nv}=&\frac {1}{T_{inv}}\left ({ {\Delta P}_{fc}-{\Delta P}_{f\_{}inv} }\right ) \\ \dot {\Delta P}_{f\_{}filt}=&\frac {1}{T_{filt}}\left ({ {\Delta P}_{f_{i}nv}-{\Delta P}_{f\_{}filt} }\right ) \\ \dot {\Delta P}_{bat}=&\frac {1}{T_{b}}\left ({ \Delta f-{\Delta P}_{bat} }\right ) \end{align}

Where,

AbbreviationExpansion

$\Delta f$	- Change in frequency
$D$	- Damping Coefficient; R-Frequency droop
$H$	- Inertia of rotating masses of the micro grid
${\Delta P}_{L}$	- Change in load power; ${\Delta P}_{s}$ - Change in solar power
${\Delta P}_{s\_{}inv}$	- Change in power outputs from inverter circuit of solar unit
${\Delta P}_{s\_{}filt}$	- Change in power outputs of filter circuit of solar unit.
${\Delta P}_{md}$	- Change in turbine mechanical power output from the diesel unit
${\Delta P}_{gd}$	- Change in governor output of diesel unit
${\Delta P}_{f\_{}filt}$	- Change in power from filter circuit of fuel cell
${\Delta P}_{f\_{}inv}$	- Change in power from filter circuit of fuel cell
${\Delta P}_{fc}$	- Change in power output from fuel cell
${\Delta P}_{bat}$	- Change in the battery power
$P_{w}$	- Power generated by wind turbine
${\Delta P}_{Cd}$	- Control inputs given to the diesel
${\Delta P}_{Cf}$	- Control inputs given to the fuel cell
$T_{t}$	-Turbine time constant; $T_{g}$ – Governor time constant
$T_{filt}$	- Time constant of filter circuit
$T_{inv}$	- Time constant of inverter circuit
$T_{fc}$	- Time constant of the fuel cell
$T_{b}$	- Time constant of the battery

FIGURE 1.

Load frequency control model of an isolated micro grid.

The above developed nine state equations of the system dynamics and the output equation are represented in a compact form and are shown in (19) and (20) respectively.\begin{align} \dot {X}=&AX+BU+EW \\ Y=&CX+DU \end{align} Where, the matrices ${X}$ , ${A}$ , ${B}$ , ${U}$ , and ${C}$ are shown at the bottom of next page.

The frequency regulation is achieved only through the controllable generating units in the system, therefore, the diesel unit and the fuel cell inputs are taken as controlled variable in the MPC formulation to minimize cost function to measure the predicted performance.

Mathematically, it is formulated as follows:\begin{align}&\hspace {-1.5pc}J=\left ({ {\Delta f}^{ref}-{\Delta f}^{pred} }\right )^{T}\left ({ {\Delta f}^{ref}-{\Delta f}^{pred} }\right )+{ \left [{ {\begin{array}{*{20}c} {\Delta P}_{cd}\\ {\Delta P}_{cf}\\ \end{array} } }\right ]}^{T}\notag \\&\hspace {11pc}\times \, \bar {R}_{in} \left [{ {\begin{array}{*{20}c} {\Delta P}_{cd}\\ {\Delta P}_{cf}\\ \end{array} } }\right ] \end{align} The first term in the objective function expressions refers to the minimization of error between predicted output and the reference point whereas the second term considers the impact of predicted control input vectors in making $J $ as small as possible. $\text{R}_{\mathrm {in}}$ is a diagonal matrix and is given as \begin{equation} \bar {R}_{in}=R_{w}\ast I_{Nc\ast Nc} \end{equation} The prediction horizon for the output is taken as 10 time steps and the control horizon for the control input is taken as 2 time steps with a sampling interval of 0.01s. These values are considered from the literature of typical load frequency problems using MPC.

Table 1 gives the values of time constant associated with DERs and other parameters used in the frequency control model shown in Fig.1.

TABLE 1 Parameter Values of the Microgrid

SECTION IV. Fuzzy Logic Controller for Parameter ($\text{R}_{\mathrm {w}}$ ) Tuning

MPC is a simple and straightforward procedure with less computational efforts. But like all other control algorithms, it is also parameter driven and it needs to be properly chosen for better performance of MPC. In MPC, we have some such parameters - Prediction horizon Np, Control horizon Nc, Sampling time Ts and Input tuning parameter $\text{R}_{\mathrm {w}}$ . Adaptability of the MPC is achieved only through the extensive analysis of the qualitative and quantitative relationship of these parameters with the behaviour of the control algorithm of MPC.

The impact of these parameters on MPC behaviour is randomly studied by trial and error method for load frequency control. It is found that optimal values of these parameters for ideal behaviour of MPC remains unchanged for different case studies except $\text{R}_{\mathrm {w}}$ . Thus, the idea of fuzzy adaptive MPC is proposed in this paper where ‘$\text{R}_{\mathrm {w}}$ ’ is a scalar and dynamically adjusted by the fuzzy controller over each prediction window of MPC keeping other procedures unchanged.

Three main components of fuzzy logic control are: Fuzzification, fuzzy rules, and defuzzification which are described in the following subsections:

A. Fuzzification

Fuzzification process is mapping the crisp value of inputs to linguistic variables using membership functions. Here inputs to fuzzification block are: Magnitude of Frequency Deviation (FD) and parameter ($\text{R}_{\mathrm {w}})$ and the output is the change in parameter ($\Delta \text{R}_{\mathrm {w}})$ shown in Fig.2. The triangular membership functions are considered for fuzzy mapping and five linguistic variables are considered for each input variable such as VS (very small), S (small), M (medium, B (big), VB (very big) whereas the output variable ($\Delta \text{R}_{\mathrm {w}}$ ) is represented in five linguistic values such as ZE (zero error), PS (positive small), PM (positive medium), PB (positive big), PL (positive large). The membership functions for inputs and outputs are shown in Fig.3. The universe of discourse for the magnitude of frequency deviation is taken as 0-0.25Hz whereas for $\text{R}_{\mathrm {w}}$ and $\Delta \text{R}_{\mathrm {w}}$ it is taken as 1-75 and 1-25 respectively.

FIGURE 2.

Inputs and output of fuzzy logic controller.

FIGURE 3.

Membership Functions a). Magnitude of Frequency Deviation ($\vert \text {FD}\vert $ ) b). Input parameter ($\text{R}_{\mathrm {w}})$ c). Change in $\text{R}_{\mathrm {w}}$ ($\Delta \text{R}_{\mathrm {w}})$ .

B. Fuzzy Inference System: Fuzzy Rules Formulation

Fuzzy rules are formulated using Mamdani-type fuzzy rules which comprise “IF/THEN” conditional statements. In this paper, a total of 25 ($5\times 5=25$ ) rules are formulated using “IF/THEN” statements with the membership functions of two input variables and one output variable which are tabulated in Table 2.

TABLE 2 Fuzzy Rules for Variation of $\Delta \text{R}_{\mathrm {W}}$

In this paper, the impact of parameter $\text{R}_{\mathrm {w}}$ in the cost function of MPC formulation is studied on load frequency control by trial and error method. The system response for different values of $\text{R}_{\mathrm {w}}$ is shown in Fig.4. Thus, based on the relationship of the parameter $\text{R}_{\mathrm {w}}$ with the behaviour of the control algorithm of MPC, the logic for the rule base is established.

FIGURE 4.

System response of the micro grid for different values of $\text{R}_{\mathrm {w}}$ .

Accuracy in solution is achieved by using more tuning rules at the cost of computational complexity. Since frequency deviation is expected to be tracked closely to the minimum value as far as possible, 25 rules are designed to determine the change in input parameter ($\Delta \text{R}_{\mathrm {w}})$ . For example:

• Rule 1:

  IF $\vert \text {FD}\vert $ is S (small) AND $\text{R}_{\mathrm {w}}$ is S (small) THEN the output $\Delta \text{R}_{\mathrm {w}}$ is PS (positive small). As the frequency deviation is small and it can still be reduced to very small values by increasing $\text{R}_{\mathrm {w}}$ to small extent that corresponds to PS (positive small) in output $ \Delta \text{R}_{\mathrm {w}}$ .

• Rule 2:

  IF $\vert \text {FD}\vert $ is B (big) AND $\text{R}_{\mathrm {w}}$ is S (small) THEN the output $\Delta \text{R}_{\mathrm {w}}$ is PL (positive large). The frequency deviation is big and it demands a higher increment in $\text{R}_{\mathrm {w}}$ that corresponds to PS (positive small) in the output $\Delta \text{R}_{\mathrm {w}}$ . All other rules are similarly fired based on the logic established between the inputs and outputs.

C. Defuzzification

Among the 25 designed rules, a maximum of four rules may fire and a minimum of one rule will fire. The output obtained from the fuzzy controller is fuzzy in nature, so defuzzification is required to convert from fuzzy to crisp value. Centroid method is used for defuzzification of output.

SECTION V. Simulation and Results

This section presents the simulation and discussion of various cases of load frequency control in a typical isolated microgrid. The cases include the load and generation variation and parametric variation in the system. The frequency control model shown in Fig.1 is built in MATLAB/Simulink on a personal computer of i5 processor 2.5GHz, 4GB RAM. All the cases have been compared to the system response with constant $\text{R}_{\mathrm {w}}$ value of 15 which is chosen randomly from the Fig.4.

A. Case -1 (Base Case System Response With All DERs)

The load frequency control of a micro grid with all sources such as PV, Wind, fuel cell, diesel and battery storage is simulated in this case. This case presents the frequency deviation response of the system with a step load change of 0.02p.u and solar power change i.e. $\Delta $ Ps taken as 0.2p.u. The mean wind velocity for this case is taken as 7m/s. Appropriate value of $\text{R}_{\mathrm {w}}$ is selected by the fuzzy controller using rule base system. The comparison of system response is shown in Fig.5.(a) The proposed fuzzy MPC gives better and faster response when compared to PI controller. Fig.5.(b) shows the response of cost function evaluated within MPC procedure which has to be minimized to achieve the desired frequency deviation response in the system. Since it is an isolated micro grid, the frequency regulation is supposed to be taken care of by the dispatchable diesel unit and the fuel cell and thus become controlled outputs from the MPC block to the respective units. The response of control inputs i.e. change in diesel and fuel cell units is shown in Fig.5.(c) The simulation period and sampling time is taken as 10s and 0.01s respectively. The prediction horizon and the control horizon are taken as 10 and 2 time steps respectively. The optimal $\text{R}_{\mathrm {w}}$ value from the proposed MPC is 38.5 for this case.

FIGURE 5.

(a). Comparison of system response of the micro grid for case-1 b). Response of Cost functions of MPC over simulation period for case-1 (c). Response of control inputs to diesel and fuel cell for case-1.

B. Case -2 (System Response With Dispatchable DERs)

This case studies the frequency regulation in the system when there are only dispatchable units such as diesel unit, fuel cell and battery in the system. The load change is 0.02p.u. The comparison of system response is shown in Fig.6.(a). Since there are only two dispatchable generation units in this case, the response shows the negative frequency deviation for the load change. The corresponding response of cost function and the change in control inputs are respectively shown in Fig.6.(b) and Fig.6.(c). The tuned $\text{R}_{\mathrm {w}}$ value for this case is 41. There is increase in the control inputs to reduce the negative frequency deviation in the system. The simulation period and sampling time is taken as 10s and 0.01s respectively.

FIGURE 6.

(a). Comparison of system response of the micro grid for case-2 b). Response of Cost functions of MPC over simulation period for case-2 (c). Response of control inputs to diesel and fuel cell for case-2.

C. Case -3 (System Response With Series of Step Load Variation)

This case evaluates the system response of the micro grid with series step changes in the load. The load changes are implemented with increase and decrease in value of $\Delta \text{P}_{\mathrm {L}}$ . The system response for this case is shown in Fig.7.(a). Fig.7.(b) and Fig.7.(c) shows the response of cost function and control inputs respectively for the step load variation in the system. The control inputs are accordingly varied by MPC to meet the load changes for minimum frequency deviation in the system. The optimal value of $\text{R}_{\mathrm {w}}$ is found to be 30 for this case.

FIGURE 7.

(a) Comparison of system response of the micro grid for case-3 b). Response of Cost functions of MPC over simulation period for case-3 (c). Response of control inputs to diesel and fuel cell for case-3.

D. Case-4 (System Response With Wind Perturbation of 2m/s)

In this case, the wind gust component of magnitude 2m/s is introduced for 6s in the wind velocity and the mean velocity of the wind is taken as 6.5m/s. The change in solar power is maintained constant at 0.05p.u. and load change of 0.02p.u. The system response for this case is shown in Fig.8.(a). The performance of the proposed method is better and faster than PI controller. Corresponding cost function and the control inputs are shown in Fig.8.(b). and Fig.8.(c) respectively. The control inputs to diesel and fuel cell are lowered by MPC when there is an increase in wind power generation due to increase in wind velocity in the system. The optimized value of $\text{R}_{\mathrm {w}}$ is found to be 16.5 for this case.

FIGURE 8.

(a) Comparison of system response of the micro grid for case-4 b). Response of Cost functions of MPC over simulation period for case-4 (c). Response of control inputs to diesel and fuel cell for case-4.

E. Case-5 (System Response With Step Changes in Solar Power)

This case considers a series step increase in solar power. In this case, the wind power change ($\Delta \text{P}_{\mathrm {w}})$ is simply taken as 0.05p.u throughout the simulation and the load change of 0.02p.u. The corresponding system response is shown in Fig.9.(a). The cost function of MPC and the control inputs to the controllable units are shown in Fig.9.(b) and Fig.9.(c) respectively. As there is increasing step change in solar power, the control inputs to diesel and fuel cells are lowered accordingly by MPC so as to maintain zero frequency deviation. The obtained value of $\text{R}_{\mathrm {w}}$ for this case is 28.

FIGURE 9.

(a) Comparison of system response of the micro grid for case-5 b). Response of Cost functions of MPC over simulation period for case-5 (c). Response of control inputs to diesel and fuel cell for case-5.

F. Case-6 (All Disturbance Such As $\Delta \text{P}_{\mathrm {L}}$ , $\Delta \text{P}_{\mathrm {w}}$ and $\Delta \text{P}_{\mathrm {s}}$ in the System)

This case presents the system response when all possible disturbances exist in the system. This case applies the disturbance considered in case-3, case-4 and case-5 simultaneously. The frequency deviation response of the micro grid for this case is shown in Fig.10.(a). The cost function of MPC and the control inputs to diesel and fuel cell are shown in Fig.10.(b) and Fig.10.(c) respectively. The optimal value of $\text{R}_{\mathrm {w}}$ for this case is found to be 19.5. The comparison of proposed fuzzy MPC with PI controller is assessed by comparing the performance index ITSE value for all cases of simulation and has been tabulated in Table 3. It is depicted that the optimal value of $\text{R}_{\mathrm {w}}$ obtained in each case of simulation is unique and has to be optimally selected for different operating conditions of the system. Hence, the proposed method is found to be efficient for load frequency control.

TABLE 3 Comparison of Performance Index

FIGURE 10.

(a) Comparison of system response of the micro grid for case-6 b). Response of Cost functions of MPC over simulation period for case-6 (c). Response of control inputs to diesel and fuel cell for case-6.

G. Case-7 (Parametric Variation in the System)

This case introduces the parametric variation in the system model and studies the system response by the proposed fuzzy adaptive Model Predictive Control. The parametric variation is incorporated as follows: R = +30%; D = −40%; H = +50%; $\text{T}_{\mathrm {t}}= -50$ %; $\text{T}_{\mathrm {g}}=+50$ %; Tb = −45%. The change in solar power is kept as 0.05p.u. and the load is 0.02p.u whereas the wind velocity is maintained at 6.5m/s. The response comparison is shown in Fig.11.(a). The respective cost function and the control inputs are shown in Fig. 11.(b) and Fig.11.(c). The optimal $\text{R}_{\mathrm {w}}$ value is found to be 27.5 for this case. The efficiency of the proposed method is tested and found robust for both type of disturbance in the system.

FIGURE 11.

(a) Comparison of system response of the micro grid for case-7 b). Response of Cost functions of MPC over simulation period for case-7 (c). Response of control inputs to diesel and fuel cell for case-7.

SECTION VI. Conclusion

This paper proposed a new fuzzy adaptive MPC for effective and faster load frequency control for an isolated micro grid. The impact of tuning parameter $\text{R}_{\mathrm {w}}$ on the performance of the model predictive control is demonstrated in the paper for load frequency control. The adaptability of MPC is achieved by tuning parameter ‘$\text{R}_{\mathrm {w}}$ ’ using a fuzzy controller. $\text{R}_{\mathrm {w}}$ has been dynamically adjusted with fuzzy “IF/THEN” rule base to make it robust control irrespective of different scenarios of the problem. The proposed fuzzy adaptive MPC is implemented for load frequency control of a typical micro grid. Simulation results and analysis have confirmed the benefits of proposed fuzzy MPC in achieving comparatively better and faster system response with damped oscillations compared to MPC with constant parameter value and also with PI controller response for different cases. Hence, the proposed fuzzy MPC can be used for effective frequency regulation in extended smart grid applications.

FIGURE 12.

The block diagram of generator-load model.

Appendix

SECTION A. Prediction of State Variables and Output at $\text{k}^{th}$ Instant

The future state variables are calculated sequentially using set of future control parameters \begin{align*} xx(k+1\left |{}\right . k)=&Ax(k)+ B\Delta u(k)\\ x(k+2\left |{}\right . k)=&Ax(k+1)+ B\Delta u(k+1) \\=&A^{2}x(k)+AB\Delta u(k)+B\Delta u(k+1) \\\vdots&\\ x(k+N_{p} \left |{}\right . k)=&A^{N_{p} }x(k)\!+\!A^{N_{p} \!-\!1}B\Delta u(k)\!+\!A^{N_{p} \!-\!2}B\Delta u(k\!+\!1) \\&+\cdots +A^{N_{p} -N_{c} }B\Delta u(k+N_{c} -1) \end{align*}

From the predicted state variables, the predicted output variables are calculated by substitution \begin{align*} y(k+1\left |{}\right . k)=&CAx(k)+ CB\Delta u(k)\\ y(k+2\left |{}\right . k)=&CA^{2}x(k)+ CAB\Delta u(k)+CB\Delta u(k+1)\\ y(k+3\left |{}\right . k)=&CA^{3}x(k)+ CA^{2}B\Delta u(k)\\&+\,CAB\Delta u(k+1)+CB\Delta u(k+2) \\\vdots&\\ y(k+N_{p} \left |{}\right . k)=&CA^{N_{p} }x(k)+ CA^{N_{p} -1}B\Delta u(k)\\&+\,CA^{N_{p} -2}B\Delta u(k+1) +\cdots \\&+\,CA^{N_{p} -N_{c} }B\Delta u(k+N_{c} -1) \end{align*}

The predicted variables are formulated in terms of current state x(k) and the future control vectors $\Delta \text{u}$ ($\text{k}+\text{j}$ ), where $\text {j}=0,1,\ldots $ .. Nc-1. Thus the output and input vectors are given as \begin{align*} Y=&\left [{ y(k+1| k)y(k+2| k)y(k+3| k) \cdots y(k+N_{p} | k) }\right ]^{\textrm {T}}\\ \Delta U=&\left [{\Delta \textrm {u(k) }\Delta u(k+1)\Delta u(k+2)\cdots \Delta u(k+N_{c} -1) }\right ]^{\textrm {T}} \end{align*}

SECTION B. Derivations of State Equation With Generator-Load Transfer Function Model

The block diagram of generator-load model from Fig.12 is represented as follows:

If we write power balance equation for the above transfer function model:\begin{align*}&\hspace {-1pc}\Delta P_{s\_{}filt} +\Delta P_{w} -\Delta P_{L} +\Delta P_{f\_{}filt} +\Delta P_{md} -\Delta P_{bat} \\&\hspace {13.3pc}=\left [{ {D+2Hs} }\right ]\Delta f \end{align*} On rearranging the above equation, \begin{align*}&\hspace {-1pc}\Delta \dot {f}=\frac {1}{2H}\Biggl [{ {\Delta P_{s\_{}filt} +\Delta P_{w} -\Delta P_{L} +\Delta P_{f\_{}filt} +\Delta P_{md} }} \\&\hspace {13.2pc}{{-\Delta P_{bat} -D\Delta f} \vphantom {\left [{ {\Delta P_{s\_{}filt} +\Delta P_{w} -\Delta P_{L} +\Delta P_{f\_{}filt} +\Delta P_{md} }}\right .}}\Biggr ] \end{align*} Similarly, all other state equations are derived by evaluating the transfer function model shown in Fig.12.