Control of LPV Modeled AC-Microgrid Based on Mixed H2/H∞ Time-Varying Linear State Feedback and Robust Predictive Algorithm

This paper presents a robust model predictive control (RMPC) method with a new mixed H2/H∞ linear time-varying state feedback design. In addition, we propose a linear parameter-varying model for inverters in a microgrid (MG), in which disturbances and uncertainty are considered, where the inverters connect in parallel to renewable energy sources (RES). The proposed RMPC can use the gain-scheduled control law and satisfy both the H2 and H∞ proficiency requirements under various conditions, such as disturbance and load variation. A multistep control method is proposed to reduce the conservativeness caused by the unique feedback control law, enhance the control proficiency, and strengthen the RMPC feasible area. Furthermore, a practical and efficient RMPC is designed to reduce the online computational burden. The presented controller can implement load sharing among distributed generators (DGs) to stabilize the frequency and voltage of an entire smart island. The proposed strategy is implemented and studied in a MG with two DG types and various load types. Specifically, through converters, one type of DGs is used to control frequency and voltage, and the other type is used to control current. These two types of DGs operate in a parallel mode. Simulation results show that the proposed RMPCs are input-to-state practically stable (ISpS). Compared with other controllers in the literature, the proposed strategy can lead to minor total harmonic distortion (THD), lower steady-state error, and faster response to system disturbance and load variation.


I. INTRODUCTION
In cases where power electricity consumption would be too far from the main power network, for instance, remote villages or isolated islands or communication stations, it will be technically difficult or economically inefficient to deliver power electricity through transmission lines. Under these circumstances, a logically and economically sound approach to the power supply is in the form of MG island mode, which includes RESs like photovoltaic (PV) generators, wind turbines, etc. In these islanding modes of MGs, the DGs in the MG are responsible for controlling the voltage, current, and frequency alone and without any assistance from the main power network [1][2][3]. As a result, it is vital to plan and design a proper and suitable controller that is robust against MG's disturbances and also load variations. In this regard, one of the main aims of this article would be to present a robust nonlinear controller for MGs. To reach this goal, several control approaches were used for MGs. For instance, master-slave control mode, and current control mode, or droop control have been used in references [4][5][6]. These approaches need communication plans, except for the droop controllers, which have attracted a lot of attention for MGs. However, the operating of inverters with droop control can have some problems like voltage-frequency and current control responses, which are excessively and unpleasantly slow or might have a small or uncertain stable operational range [7]. VOLUME XX, 2020

A. BACKGROUND
To tackle the mentioned problems, some articles have proposed different approaches by using modifications of phase feedforward, droop control by first-order dynamics, as well as output impedance control of inverters [8]. However, modification strategies usually do not consider dynamic control of a system with order greater than one or dynamic control of gain scheduled, and also do not utilize progressive methods to control the plan, schedule, and design in a grid. Recent articles have demonstrated the benefits of robust controllers in comparison with common controllers according to traditional control theory to control the applicability of inverters in an MG [9]. In the reference [10], for instance, authors have demonstrated that the operational range of an MG might be risen from 6 percent to 14 percent for changing in all characteristics of a system. For example, in inverter voltages, or line impedances, one utilizes a dynamic control for the grid rather than a fixed gain droop controller. Although this could be remarkable progress, MGs might be subject to several variations and uncertainty; thus, other solutions and methods are required. Subsequent advances in efficiency and stable operational range might be considered via utilizing adaptive gain scheduled controllers. Progressions of 30 percent to 50 percent have been shown by using polytopic adaptive controllers in the paper [11]; nonetheless, such controllers are typically hard to implement and synthesize. The intricacy could be remarkably decreased for a small operating point with bumpless-transfer controllers that have appeared in works as an adaptive ad-hoc control plan [12,13]. References [14,15] have also stated that this method can be easier to synthesize, having less conservative, and also easier to perform in comparison with a polytopic controller and interpolation plan; besides, it would have a better operational range compared with a nominal and common type of controller. In such cases, whenever an inverter with an equal amount of impedance is connected to an MG that is much less than what is predicted by the design engineer, the inverter with a formal controller displays instability in responses and also the inverter with a bumpless controller displays stability in responses [11]. In the references [16,17], for grid-supporting inverter-based systems, a new cascaded control strategy and a generalized droop control have been proposed where compared virtual synchronous generator control and traditional droop control. Other approaches have also been utilized for (uninterruptable power supply (UPS) and DC/AC inverter systems, as those presented in the reference [18], where proportional-integral controllers have been utilized to track current and voltage references. A feedback linearization method was utilized for 3-phase UPS in the paper [19], where the control gains were obtained using a pole placement. Additionally, other alternatives dealing with advanced control methods can be found in the reference [20]; and also for adaptive controllers in reference [21], for sliding mode controllers and for model predictive controllers in the papers [22,23]. It is noted that in the papers [24,25], for power electronics applications, a robust control-based on linear matrix inequalities (LMIs) was addressed. The used strategy is attractive because it can be effectively solved by specialized algorithms, and also it allows for easily including many efficiency specifications. In this way, utilizing of LMIs for stability assessment and control design with practical performance for UPS deserves more investigation.

B. MOTIVATION AND MAIN CONTRIBUTIONS OF THE PAPER
In the islanding mode of MG, the voltage and frequency of the network should be constant. In this regard, in this article, one of the units is in the voltage-frequency control mode. To maintain the voltage and frequency of the network, other units that are not responsible for voltage-frequency control are in the current control mode. If the load current is more than the maximum current supplied by the current control unit, the current will be drawn from the current generation unit. The rest of the current is supplied by the voltage-frequency control unit. Also, if the load current is less than the maximum current of the current control unit, the total load current is supplied by the current control unit. To enhance the control performance and feasibility of the presented method, in this paper, a mixed 2 / ∞ feedback RMPC for systems with both disturbance and uncertainty structured is proposed. The multistep control approach is presented in the proposed mixed H2/H∞ RMPC, in which a sequence of feedback control laws is adopted as a control strategy. Besides this, since the ISpS concept is suitable to assess the stability of the closed-loop system with disturbance, the closed-loop ISpS stability is utilized to validate the proposed RMPC. Furthermore, a novel control strategy is introduced for islanding MG with different DGs according to a mixed 2 / ∞ time-varying linear state feedback robust model predictive control algorithm to control voltage, current, and frequency of the power network system, and also for load sharing among DG units. Another important point in this work is to consider increasing the TDH and root mean square (RMS) value of the voltage profile of MG. One DG is working in the mode of voltage-frequency control; in this case, it will control the voltage and frequency in the power network by forcing the voltage of the grid to follow a reference signal simultaneously trying to lessen the error of output voltage. Similarly, another DG unit is operating in the mode of current control. The reference signal of the current is proportional to the load current that is measured on a real-time basis. The presented method is simulated and validated in different case studies using the MATLAB software.

C. PAPER STRUCTURE
The remainder of this manuscript is organized as follows; in Section II, the dynamic model and concept of the system and the proposed controller will be introduced and explained. In Section III, the results of the simulation of a low voltage MG will be provided and evaluated in different scenarios. Finally, remarks of conclusions will be presented in Section IV.

A. SYSTEM MODEL
In this section, we present the concept of the proposed system. Figure 1 shows an islanded MG with different DG agents that are coupled to the AC bus in parallel mode. In the suggested system, one unit acts in voltage-frequency mode; in this case, it will be in the duty of stabling the MG voltage. The other unit is working in the current control mode and load sharing. Figure 2 depicts the circuit of a 3-phase inverter that is linked to the system. In this case, the output of the LC filter has been employed to decrease the output voltage of harmonic components produced by the pulse width modulation (PWM) inverter. The proposed case is operated in 2 different modes: voltage-frequency or current control mode. Each DG system is composed of LC filters, a DC resource, and a voltage source inverter in common conditions. The power circuit and control schematic diagram of a 3-phase inverter with LC filter is shown in Figure 3.

PV Generation Unit
Wind Generation Unit Battery Energy Storage

Voltage control Mode
FIGURE3. Schematic diagram of a typical voltage source inverter by the inverter, output filter, and loads with mode selection of the controller VOLUME XX, 2020 The model of a 3-phase inverter is presented as: Where: and i i c display the phase currents through filter inductor L. Inverter output voltage vector is displayed in Figure 1 and is defined as bellow: Assuming all the possible combinations of the gating signals, namely a , b and c , eight switching modes and as a result, eight voltage vectors have been retrieved. As v 0 = v 7 , there are just seven different voltage vectors as expressed in Figure  2. Based on KVL: Also, KCL states: State-space representation of each phase is retrieved as: In which ( ) = [ ( ) ( ) ] is the state vector.
Also ( ) defines control command and ( ) denotes disturbance input and: Eq. (6) is discretized to use in the controller design, therefore: where Also, defines the sampling period.

B. PROBLEM FORMULATION
In this part, we present the problem formulation. Hence, a discrete-time polytopic LPV system with disturbance should be considered, where their system matrices should be related to the functions of a parameter vector as follows: With ∈ providing the state vector, ∈ represents the control input vector, ∈ is the control output vector, ∈ gives the disturbance; also: ]. The parameter vector of ∈ depends on the unit simplex where: The matrices of LPV system change inside a corresponding polytope Ω whose vertices include r local system matrices Ω = {( , , , , )} where co represents the convex hull. Lemma 1: It should be supposed that matrices ∈ × and ∈ × are two positive semidefinite matrices ∈ × and ∈ × so that − < 0, − < 0 Then: Lemma 2: It should be supposed that matrices ∈ × , ∈ + and are two positive semidefinite matrices ∈ × and ∈ × have been given. The following matrix inequality will be: When > 0 and ∑ =1 = 1 if: By applying lemma1, it can be seen that equation (10) holds if equation (11) holds.
In the following equations, the problem of system input constraints and measurable modes is considered: In equations (12) and (13): L gives a matrix with as the ℎ line, represents the ℎ element of vector , is the ℎ element of control input , and divides the ℎ element of vector . For the system in equation (9) with constraints equations (12) and (13), the objective is to design a gainscheduled state feedback control law as follows: The disturbance will be bounded as: In the equation (15), ̅ > 0 and q are known. Hence, the following efficiency requirements are satisfied and divided into two following sections: 1. ∞ efficiency requirements: under the zero-initial situation, the controlled output | is satisfied as follows:

2.
2 efficiency requirements: the controlled output | is satisfied as follows:

C. GAIN-SCHEDULED CONTROL LAW DESIGN
By rewriting the closed-loop system equation (9) based on the control law equation (14), we will have equation (18) as follows: With considering the Lyapunov function in the form And by assuming + | = 0, equation (18) will be converted into: By using materials provided in the paper [26], to decrease the conservatism, the state space equation (18) can be presented as follows: Then we have: This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
Thus, if equation (23) holds, ∆ is negative definite, and therefore LPV system of equation (19) will be stable. With considering + | ≠ 0: Lemma 3: By considering the LPV system the equation (9), with disturbance equation (15), the control law that guarantees the efficiency requirements 1 and 2, is given by = if there exist ∈ × and ∈ × satisfy the following situation: [ − + ̅ (11) ̅ (12) ⋇ ̅ (21) − 2 + ̅ (22) Proof: In equation (9), if in the equation (25) becomes negative as the disturbance is satisfied in the equation (15), the disturbance will reduce to zero, and then ≤ 0 implies that ∆ ( + | ) < 0 after some time instant. Thus, ( + | ) goes to zero when l= ∞. As a result, by summing up of both sides of the equation (2)5 from = 0 to → ∞, it yields; consequently, it will be retrieved as: Equation )29) can be transformed into: From the efficiency requirements 1 and 2, when | = 0, it will be under the zero-initial condition, the requirement equation (16) will be equal to: By considering the equation (30), if ≤ 0 holds, the equation (16) is satisfied which in turn the requirement on H∞ efficiency will be satisfied. Under the situation that ≤ 0, the requirement of equation 17 is satisfied if: By multiplying the sides in −1 : According to the above-mentioned assessment, it can be concluded that for the design aim, ≤ 0, is a vital issue, that is:    Consequently, we can define equation (39) as follows: [ − + ̅ (11) ̅ (12) ⋇ ̅ (21) − 2 + ̅ (22) [26] did not take this issue into account, therefore, it cannot guarantee the recursive feasibility of MPC. Herein, we will give the following lemma to overcome this weakness. Lemma 4: For systems as given in the equation (9), if equation (25) is satisfied for | and the following inequalities hold, and then, + | ∈ { | | −1 | ≤ 1}. In this regard, we will have: where + = (1 − )( +1 − ) , = 0, … , − 1, is a matrix variable and b is a parameter taken into account in advance such that < < .

Lemma 5:
With considering the LPV systems equation (9), the closed-loop system under the control law equation (52) satisfies the following conditions:  (12) , ̅ (22) = (22) . Also, u by sing lemma 2 and lemma 4, equation (54) is equal to the following LMI: If equation (57) is solved, the input of control at the current time of is: The process of implementing the proposed controller in MATLAB is shown in the following algorithm for a singlephase system. To control the other phases separately, the controller algorithm is implemented in the same way. To solve the inequality, YALMIP solver is used in MATLAB version 2019a. Codes for implementing the controller are as the following: Define parameters of the system, controller and references Get data from sensors Calculate the error between measured data and reference Controller: Use YALMIP solver to solve and optimize equations 51 and 57 to retrieve control signal gains Calculate the control signal based on the sum of feedback and backward control gains (equations 52 and 58) Send a control signal to the system

A. PARAMETERS AND CASE STUDIES
In this section, the efficiency of the controller is evaluated under different load scenarios, including balanced or unbalanced resistive, inductive, and non-linear loads. Figure 4 displays an islanded MG with 2 DG units. Simulations have been performed by Matlab/Simulink. The specifications of the power network and also control parameters are displayed in Table I. As can be seen from Figure 4, the specifications of the system are measured to be utilized in the input of the controller.
According to Figure 4, the first DG is chosen as the control of voltage and frequency, which is responsible for regulating and adjusting the voltage and frequency of the MG in time of the system encounters every load variation. On the other hand, the second DG works in the mode of output current control, wherein gains have been utilized for sharing loads. The control signal has been utilized as the input of the PWM to control the inverter. A current load factor is also utilized as the reference current signal.

Case A: Symmetrical Resistive Load
In this case, the efficiency of the proposed controller under symmetrical resistive load has been investigated. The values are also repeatedly stated in Table II. In time t = 0.205 s symmetrical load is introduced into the system, respectively. It should be noted that the load value is assumed to be measurable and a load factor is utilized as the reference current signal. Figure 5 depicts the simulation results. Figure 5 (a and d) illustrate the voltage of MG and also the zero steady-state error of phase voltage a. The presented algorithm effectively can track the reference signal with having a minimum error, which approves the robustness of the control strategy against linear load changing. The load current also utilized as the reference current signal, which is shown in Figure 5 Table II. In time t = 0.405 s asymmetric load is introduced into the system. It should be noted that the load value is assumed to be measurable and a load factor is utilized as the reference current signal. Figure 6 depicts the simulation results. Figure 6 (a and d) illustrate the voltage of MG and also the zero steady-state error of phase voltage a. The presented algorithm effectively can track the reference signal with having a minimum error, which approves the robustness of the control strategy against linear load changing. The load current also utilized as the reference current signal which is shown in Figure 6 (b). Besides, Figure  6 (c and e) display the reference signal and output current of the second DG and also the zero steady-state error of phase voltage a. Figure 6 (c and e) show the robustness of the provided current controller. The harmonic spectrum of voltage is analyzed and the amplitude of output voltage is 109.5 and the THD is 0.16 percent.  Table II. At time t = 0.6 s inductive load is introduced into the system. It should be noted that the load value is assumed to be measurable and a load factor is utilized as the reference current signal. Figure 7 depicts the simulation results. Figure 7 (a and d) illustrate the voltage of MG and also the zero steady-state error of phase voltage a. The presented algorithm effectively can track the reference signal with having a minimum error, which approves the robustness of the control strategy against linear load changing. The load current is also utilized as the reference current signal which is shown in Figure 7 (b). Besides, Figure 7 (c and e) display the reference signal and output current of the second DG and also the zero steady-state error of phase voltage a.  Table II. At time t = 0.4 s, a non-linear load is applied to the system. It has to be noted that the load value is assumed to be measurable and a load factor is utilized as the reference current signal. Figure 8 depicts the results of the simulation. Figure 8 (a and d) demonstrate the voltage of MG and the zero steady-state error of phase voltage a. It is evident that the presented algorithm, in this case, is effectively able to track the reference signal with minimum error, which approves the robustness of the control strategy against linear load changing. The load current used as the reference current signal is shown in Figure  8

B. COMPARISON WITH OTHER CONTROLLERS
In this part, the efficiency of the provided controller is compared with the classic sliding mode controller and the classic backstepping controller in Table III. The presented controller can give improved output peak voltage and THD; besides, it can decrease steady-state error in comparison with the classic non-linear controller. To have a better comparison with the suggested controller, details of sliding mode and back stepping controllers are extracted from references [28] and [29] and implemented controllers on the system, which are shown in Figure 4.

IV. CONCLUSION
In this article, we proposed a concept model prediction and H2/ ∞ control plan with input and state constraints with considering disturbances in an MG with two DGs. The model predictive control has used both H2 and ∞ to optimize the concept; and a gain-scheduled MPC, H2/ ∞ algorithm was also studied using LMI techniques. Considering proper upper boundaries as LMI auxiliary variables in terms of stability analysis leads to a reduction in conservatism. The presented algorithm provides a faster response and is added to the controller to avoid obstacles in an algorithmic way like the switching algorithm. Additionally, closed-loop stability and recursive feasibility of the proposed model predictive control have been proven. The main advantages of the proposed approach are the existence of a control law for all conditions concerning the linear and non-linear loads and the ability to avoid obstacles, and decreasing the impacts of disturbances. Not only that, but it was demonstrated that the proposed controller is able to track the reference signal with a minimum steady-state error. Besides, using the presented controller has proven that it can develop the results of the steady-state of the controller, including output voltage peak, RMS, and THD. Obtained simulation results demonstrated that the presented controller is robust against variable load situations. Considering the obtained results, the controller is insensitive to changes in the phase and amplitude of the reference signal. This controller has a decent and fast response to the reference variations and its steady-state error is negligible.