# A Linear Parameter Varying Control Approach for DC/DC Converters in All-Electric Boats

Authors: Soroush Azizi; Mohammad Hassan Asemani; Navid Vafamand; Saleh Mobayen; Mohammad Hassan Khooban
Journal: Complexity
 (2021)

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## Abstract

Utilization of renewable energies in association with energy storage is increased in different applications such as electrical vehicles (EVs), electric boats (EBs), and smart grids. A robust controller strategy plays a significant role to optimally utilize the energy resources available in a power system. In this paper, a suitable controller for the energy resources of an EB which consists of a 5 kW solar power plant, 5 kW fuel cell, and 2 kW battery package is designed based on the linear parameter varying (LPV) controller design approach. Initially, all component dynamics are augmented, and by exploiting the sector-nonlinearity approach, the LPV representation is derived. Then, the LPV control method determines the suitable gains of the states’ feedbacks to provide the required pulse commands of the boost converters of the energy resources to regulate the DC-link voltage and supply the power of EB loads. Comparing with the state-of-the-art nonlinear control methods, the developed control approach assures the stability of the overall system, as it considers all component dynamics in the design procedure. The real-time simulation results demonstrate the performance of the designed controller in the creation of a constant DC-link voltage.


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## Body

## 1. Introduction

The popularity of renewable energies results from the reduction of reachable fossil fuels and an increase in their cost. Literature has widely studied the application of renewable energies in the propulsion system of electric vehicles (EVs) [1–3] and electric boats (EBs) [4]. The DC-DC power converters and their control systems have prominent duties in the energy conversion of such vehicles. For instance, low voltage battery packages need step-up converters to provide the required voltage for the DC-link of these vehicles [5]. For this reason, while the conventional boost converters are used as an interface circuit in the structure of the EVs and EBs propulsion system, a suitable robust controller is required to guarantee the correct operation of the converters [6–11].The mathematical and black-box analyses of a power converter are two different approaches to design the controllers of the power converters. In comparison with the black-box analysis, the mathematical analysis of the systems results in a more reliable response in a wide range of operating points. State-space averaging, discrete averaged model, G-parameter modeling, component connection model, describing the function, and linear parameter varying (LPV) are the most used mathematical analysis in the controller designs. In [12], a state-space averaging model has been proposed for a bidirectional converter. For DC-DC converters with pulsating output current, such as boost converters, the use of leading-edge or trailing-edge pulse-width modulation (PWM) brings about distinctly different behavior [13]. This phenomenon cannot be captured by the conventional averaged model but is predicted with a discrete-time model. The results of the proposed approximate discrete-time model in [14] demonstrate the successful performance of the control system of a boost converter. Although the discrete averaged model works well for a single bidirectional converter, the extraction of the averaged model for a system for more than two converters is a challenging procedure. In fact, it is hard to provide an accurate discrete-time averaged model to use two different converters while supplying a load.G-parameter model and component connection model are two outstanding procedures while various power converters work in a system. However, the G-parameter just considers small signals of the model which limit its application for different operating modes. Also, the component connection is a mapping technique that can be used in the stability analysis of a system. Therefore, these techniques cannot work properly in the controller design of converters in EBs or smart grids while different variables can affect their operating modes. Moreover, the described function technique is able to consider single nonlinearity and cannot design a robust controller in the presence of various nonlinearity or uncertainties. It has been proven that the LPV can be employed in the design of large-scale hybrid systems controllers. The LPV technique benefits from its ability in the design of a robust controller for large signal analysis in the presence of a wide range of uncertainties. In [15, 16] and [17], the LPV is used to design a high-performance controller for the pitch angle of wind turbines [18, 19]. Besides, the LPV has been used to design a controller for a single boost converter for a photovoltaic (PV) array which supplies load in [20].In this paper, a comprehensive controller for boost power converters of an electric boat propulsion system is designed based on the polytopic-LPV technique. The considered EB is supplied by a 5 kW solar array, 5 kW fuel cell package, and 2 kW batteries. The gate commands of such a hybrid system are determined by the designed controller to create a stiff DC-link voltage in different solar irradiations and loads. Initially, the topology of the investigated system is described. Then, the polytopic-LPV controller design procedure for the system is designed such that the exponential stability of the overall system is assured. The proposed approach has some advantages over state-of-the-art methods, including the following: (1) it considers the available information of all components in the design procedure compared to [21–23]. Therefore, the overall closed-loop stability is guaranteed theoretically through the Lyapunov stability theory [24–29]. (2) The design procedure of the proposed controller is simple and systematic compared to the other method [30, 31]. Additionally, the controller gains can be computed by using the numerical convex optimization linear matrix inequality (LMI) techniques. (3) The proposed controller is more robust against system parameter variations and uncertainties compared to [32, 33]. Finally, a real-time simulation study is executed on the DC MG power system to examine the performance of the proposed controller.The rest of this paper is as follows: in Section2, the power sources and their dynamics in the overall EB system are discussed. In Section 3, the LPV approach to model the nonlinear dynamics of the propulsion power system and to design the control is presented. In Section 4, simulation results are provided, and Section 5 ends the paper by giving some concluding remarks and future perspectives.


## 2. The Power Propulsion System of the EB

Figure1 shows the power propulsion system of the EB investigated in this paper. It is obvious that the propulsion system, consisting of the photovoltaic array, fuel cell, and battery package, aims to supply the 10 kw load available on their common DC-link. The boost converters which are used as the interface circuit of these power sources must be controlled properly to provide the required DC-link voltage. In this section, the characteristics of each power source and its state-space model are studied.Figure 1
The considered power resources to supply 10 kw load.


### 2.1. Solar Power Plant

A 5 kw solar power plant with the maximum voltage and current of 200 V and 25 A is one of the energy resources of the EB propulsion system. As it is shown in Figure2, the output of the PV array is connected to a DC-DC boost converter which supplies the DC-link capacitor. The dynamics of the PV cell linked to the boost converter is as follows [34]:(1)VPVdt=IPV−ILPVCPV,dILPVdt=VPV−1−DpVDCLPV,where IPV, VPV, VDC, and DP are the PV array current, PV array voltage, DC-link voltage, and duty cycle of the boost converter switch, respectively. Also, LPV and CPV show the inductance and input capacitor of the boost converter available in the interface circuit of the solar power plant. The boost converter switching control plays a significant role in the determination of the output voltage of the solar array to generate the maximum power. Therefore, a suitable controller must be designed to achieve the maximum power from the PV array in different solar irradiations. Also, the boost converter is to provide the required DC-link voltage. Hence, the boost converter state-space model is extracted to use this model during the controller design procedure. The converter state-space equation is written by (2) which is used in the controller design procedure, where U1 is the command signal for the PV array boost converter (Q1).Figure 2
The photovoltaic (PV) array and its interface circuit used in the system.As it is expected, the effect of load shown by (RL) appears in the DC-link voltage equation:(2)dVPVdt=IPVCPV−ILPVCPV=IPVCPVVPVVPV−ILPVCPV,dILPVdt=VPPVVLPV−VDCU1LPV,dVDCdT=IPVU1CDC−VDCCDCRL.


### 2.2. Fuel Cell

If the 5 kw fuel cell package is replaced with the PV array represented in Figure2, the second energy resource of the system can be connected to the DC-link capacitor. The inductance of the fuel cell boost converter is equal to LPV, because of the same power rating and switching frequency of both converters. Also, the DC voltage at the output of the fuel cell remains constant while it supplies the load. Thus, as it is shown in Figure 3, the boost converter of the fuel cell does not require any capacitor in its model. The state-space equation of the fuel cell, working in parallel with the solar array, can be written as follows:(3)dILFdt=VFLF−VDCU2LF=VFLFIFIF−VDCU2LF,dVDCdt=ILFU2CDC−VDCCDCRL.where LF and U2 represent the boost converter inductance and duty cycle of the switching for the boost converter of the fuel cell. Moreover, the fuel cell output current and voltage are shown by IF and VF, respectively.Figure 3
The fuel cell package and boost converter used in the system.


### 2.3. Energy Storage

A 2 kw battery package is investigated to work in parallel with the introduced energy sources to regulate the DC-link voltage of the system when two other sources cannot supply the load. In addition, when the solar power plant generates a higher energy level than the load, the energy storage starts to store the extra generated energy. Therefore, a bidirectional converter is required for the energy storage to balance the energy level on the DC-link. The lead-acid battery can be simply modeled by the following equation:(4)VB=VG−RsIB,where VG is the plate voltage level of the battery package and RS is the equivalent series resistance of the battery. Therefore, the output voltage of the battery shown with VB depends on the output current of the battery. The dynamics of the bidirectional converter presented in Figure 4 is as follows:(5)dILBdt=VB−DBVDCLB,dVBdt=IB−ILBCB,where LB and CB show the bidirectional converter inductance values and DB is the duty cycle of the Q3 shown in Figure 4. The state-space model of the energy storage, written in (6), connected to the same DC-link with two other introduced resources is required to design the controller based on the LPV approach:(6)dVBdt=IBCB−ILBCB=IBCBVBVB−ILBCB,dILBdt=VBLB−VDCU3LB,dVDCdt=ILBU3CDC−VDCCDCRL,where IB and VB are the current and voltage of the battery package, respectively. Also, U3 represents the Q3 command signal, and the bidirectional converter inductor current is shown by IB.Figure 4
The energy storage and its interface circuit used in the system.


### 2.4. Overall State-Space Model

Based on the nonlinear dynamics of the solar power plant, fuel cell, and their boost converters and the bidirectional converter of the energy storage system, the following overall state-space representation is obtained:(7)x˙t=Atxt+Btut,where xt=VPV,ILPV,ILF,VB,ILB,VDC,ZT is the state vector and At and Bt matrices are(8)At=IPVCPvVPV−1CPV000001LPV00000000VFLFIF0000000IBCBVB−1CB000001LB00000000−1RLCDC000000−10,Bt=0−VDCLPV000ILPVCDC000−VDCLF00ILFCDC00000−VDCLBILBCDC0T.As can be seen in (7), the overall dynamics are nonlinear and time-varying due to the fact that the matrices At and Bt contain system states and time-varying parameters of the nonlinear power elements. Additionally, the duty cycle control inputs in ut are amplitude bounded. These issues make the controller design procedure a hard task. To design a controller that theoretically guarantees closed-loop stability, the polytopic-LPV representation is utilized in the following section.


## 2.1. Solar Power Plant

A 5 kw solar power plant with the maximum voltage and current of 200 V and 25 A is one of the energy resources of the EB propulsion system. As it is shown in Figure2, the output of the PV array is connected to a DC-DC boost converter which supplies the DC-link capacitor. The dynamics of the PV cell linked to the boost converter is as follows [34]:(1)VPVdt=IPV−ILPVCPV,dILPVdt=VPV−1−DpVDCLPV,where IPV, VPV, VDC, and DP are the PV array current, PV array voltage, DC-link voltage, and duty cycle of the boost converter switch, respectively. Also, LPV and CPV show the inductance and input capacitor of the boost converter available in the interface circuit of the solar power plant. The boost converter switching control plays a significant role in the determination of the output voltage of the solar array to generate the maximum power. Therefore, a suitable controller must be designed to achieve the maximum power from the PV array in different solar irradiations. Also, the boost converter is to provide the required DC-link voltage. Hence, the boost converter state-space model is extracted to use this model during the controller design procedure. The converter state-space equation is written by (2) which is used in the controller design procedure, where U1 is the command signal for the PV array boost converter (Q1).Figure 2
The photovoltaic (PV) array and its interface circuit used in the system.As it is expected, the effect of load shown by (RL) appears in the DC-link voltage equation:(2)dVPVdt=IPVCPV−ILPVCPV=IPVCPVVPVVPV−ILPVCPV,dILPVdt=VPPVVLPV−VDCU1LPV,dVDCdT=IPVU1CDC−VDCCDCRL.


## 2.2. Fuel Cell

If the 5 kw fuel cell package is replaced with the PV array represented in Figure2, the second energy resource of the system can be connected to the DC-link capacitor. The inductance of the fuel cell boost converter is equal to LPV, because of the same power rating and switching frequency of both converters. Also, the DC voltage at the output of the fuel cell remains constant while it supplies the load. Thus, as it is shown in Figure 3, the boost converter of the fuel cell does not require any capacitor in its model. The state-space equation of the fuel cell, working in parallel with the solar array, can be written as follows:(3)dILFdt=VFLF−VDCU2LF=VFLFIFIF−VDCU2LF,dVDCdt=ILFU2CDC−VDCCDCRL.where LF and U2 represent the boost converter inductance and duty cycle of the switching for the boost converter of the fuel cell. Moreover, the fuel cell output current and voltage are shown by IF and VF, respectively.Figure 3
The fuel cell package and boost converter used in the system.


## 2.3. Energy Storage

A 2 kw battery package is investigated to work in parallel with the introduced energy sources to regulate the DC-link voltage of the system when two other sources cannot supply the load. In addition, when the solar power plant generates a higher energy level than the load, the energy storage starts to store the extra generated energy. Therefore, a bidirectional converter is required for the energy storage to balance the energy level on the DC-link. The lead-acid battery can be simply modeled by the following equation:(4)VB=VG−RsIB,where VG is the plate voltage level of the battery package and RS is the equivalent series resistance of the battery. Therefore, the output voltage of the battery shown with VB depends on the output current of the battery. The dynamics of the bidirectional converter presented in Figure 4 is as follows:(5)dILBdt=VB−DBVDCLB,dVBdt=IB−ILBCB,where LB and CB show the bidirectional converter inductance values and DB is the duty cycle of the Q3 shown in Figure 4. The state-space model of the energy storage, written in (6), connected to the same DC-link with two other introduced resources is required to design the controller based on the LPV approach:(6)dVBdt=IBCB−ILBCB=IBCBVBVB−ILBCB,dILBdt=VBLB−VDCU3LB,dVDCdt=ILBU3CDC−VDCCDCRL,where IB and VB are the current and voltage of the battery package, respectively. Also, U3 represents the Q3 command signal, and the bidirectional converter inductor current is shown by IB.Figure 4
The energy storage and its interface circuit used in the system.


## 2.4. Overall State-Space Model

Based on the nonlinear dynamics of the solar power plant, fuel cell, and their boost converters and the bidirectional converter of the energy storage system, the following overall state-space representation is obtained:(7)x˙t=Atxt+Btut,where xt=VPV,ILPV,ILF,VB,ILB,VDC,ZT is the state vector and At and Bt matrices are(8)At=IPVCPvVPV−1CPV000001LPV00000000VFLFIF0000000IBCBVB−1CB000001LB00000000−1RLCDC000000−10,Bt=0−VDCLPV000ILPVCDC000−VDCLF00ILFCDC00000−VDCLBILBCDC0T.As can be seen in (7), the overall dynamics are nonlinear and time-varying due to the fact that the matrices At and Bt contain system states and time-varying parameters of the nonlinear power elements. Additionally, the duty cycle control inputs in ut are amplitude bounded. These issues make the controller design procedure a hard task. To design a controller that theoretically guarantees closed-loop stability, the polytopic-LPV representation is utilized in the following section.


## 3. Polytopic-LPV Technique

The polytopic-LPV method provides a systematic and straightforward procedure to analyze the stability and design of a stabilizing controller for a nonlinear system. To perform this, initially, the nonlinear system is rewritten by an equivalent polytopic-LPV model. Then, the gains of a polytopic-LPV control law are chosen such that the closed-loop system assures the Lyapunov stability [35].


### 3.1. Polytopic-LPV Model of DC MG

In order to achieve the objectives of the microgrid, the polytopic-LPV control approach is utilized in this paper. In the state-space model, there are some time-varying termsIpv/CpvVpv, VF/LFIF, and IB/CBVB in the matrix A and state variables ILPV, ILB, ILF, and VDC in the matrix B. Let(9)θ=θ1θ2…θ7T=IPVCPVVPVVFLFIFIBCBVBVDCILPVILFILBTto be the vector of varying parameters, where each parameter is bounded in a set, θɛθ¯,θ¯, with θ¯ and θ¯ as the upper and lower limits, respectively. The values of vector θ are contained in a polytopic domain with 27 vertices. Finally, by applying the so-called sector nonlinearity [36, 37], the following polytopic-LPV system is obtained:(10)x˙t=∑i=1128ρiθAixt+Bivt,yt=Cxt,where vt=satut=satu1t,…,satu3tT and(11)satuit=1,uit≥1,uit,for1<uit<1,−1,uit≤−1.Note that, to design the pulse-width modulation (PWM) signal for the converters, it is necessary that the duty cyclesui for i=1,2,3 are bounded by ±1. Thereby, the lower and upper bounds in (10) are −1 and +1, respectively. The goal is to design the control input vector ut such that the VPV tracks the voltage of the VPV−ref and the DC bus voltage converges to VDC∗. The reference value of the DC bus voltage is mainly constant. However, the maximum power point voltage reference must be chosen accordingly.


### 3.2. LPV Controller Design

In order to achieve the objectives of the microgrid, the polytopic-LPV control approach is utilized in this paper:(12)ut=∑i=1128ρiθFixt,where Fi for i=1,…,128 are the controller gains, which should be obtained to assure the closed-loop stability and convergence of the states VPV and VDC to their references. By adding the null term ∑i=1rhiztBi1+ε/2ut−ut with 0<ε<1 to (10), one has(13)x˙t=∑i=1128ρiθAixt+Bi1+ε2ut+Bivt−1+ε2ut=∑i=1128ρiθAixt+Bi1+ε2ut+Birt,where rt=vt−1+ε/2ut. The closed-loop system is obtained as(14)x˙t=∑i=1128∑j=1128ρiθρjθAi+1+ε2BiFjxt+Birt.Lemma 1.
(see [38]). For the saturation constraint defined by (11), as long as ut<1/ε with 0<ε≤1, one has(15)rTtrt≤1−ε22uTtut,which is shown in Figure 5.Figure 5
The functions of Lemma1 (the rt by the solid blue line and the 1−ε/2ut by the dashed red line).Lemma 2.
(S-procedure) (see [39]). Consider that the conditions(16)ζTT0ζ≥0,where ζ≠0 and satisfies the constraints(17)ζTTiζ≥0,fori=1,…,p.Then, condition (16) subject to constraint (17) results in(18)T0−∑i=1pτiTi≥0,where τi for i=1,…,p are arbitrary nonnegative scalars.Lemma 3.
(see [40]). The time-varying parameter-dependent constraint ∑i=1q∑j=1qρiθρjθxTΓijxt<0 is implied by the following time-varying parameter independent conditions:(19)Γii<0,fori=1,…,q,Γij+Γji<0,fori<jandj=1,…,q.Lemma1 is essential to eliminate the time-varying parameters ρiθ and the state vector x in the controller design conditions. The following theorem provides the controller design conditions in terms of LMIs such that the exponential stability of the states to the equilibrium point is assured.Theorem 1.
System (14) is exponentially stabilizable in the local region Ω=x|xTP−1x<ρ, if for the given decay rate σ>0, there exists matrices S=ST and Mi such that(20)S>0,(21)Γ¯ii<0,(22)Γ¯ij+Γ¯ji<0,(23)ulimε2MikMiTkρS>0,where(24)Γ¯ij=SAiT+AiS+1+ε2MjTBiT+1+ε2BiMj+σSBiT1−ε2MjTBi−τI01−ε2Mj0−τ−1I.Proof.
Consider the exponential Lyapunov stability conditions with the decay rateσ, as(25)V˙+σV<0,where the Lyapunov candidate is chosen as(26)V=xTtPxt,where P is a symmetric positive definite matrix. Substituting (14) into (25), the exponential stability condition is continued as(27)V=xTt∑i=1128∑j=1128ρiθρjθAi+1+ε2BiFjTP+PAi+1+ε2BiFj+σPxt+rTt∑i=1128ρiθBiTPxt+xTtP∑i=1128ρiθBirt=∑i=1128∑j=1128ρiθρjθxTtrTtQijxtrt,where(28)Qij=Ai+1+ε2BiFjTP+PAi+1+ε2BiFj+σPBiTPPBi0.
Applying Lemma2 on (27) subject to (15) results in(29)∑i=1128∑j=1128ρiθρjθxTtrTtQ¯ijxtrt,where(30)Q¯ij=Ai+1+ε2BiFjTP+PAi+1+ε2BiFj+τFiTFi1−ε22+σPBiTPPBi−τI.
Applying Lemma1 on (27) results in(31)Q¯ii<0,fori=1,…,128,Q¯ij+Q¯ji<0,fori<jandj=1,…,128.
Considering the Schur complement, (31) leads to(32)Γii<0fori=1,…,128,Γij+Γji<0fori<jandj=1,…,128,where(33)Γij=Ai+1+ε2BiFjTP+PAi+1+ε2BiFj+σPBiTP1−ε2FiTPBi−τI01−ε2Fi0−τ−1I.
Pre- and postmultiplying (32) by diagP−1,I,I and defining the LMI variables S=P−1 and Mi=FiP−1 result in (21) and (22). Moreover, the constraint P=PT>0 leads to the LMI (20). On the other hand, (15) holds in the local region Ω1=x|u<1/ε. Consequently, inspired from the set inverse analysis [41] and considering the same procedure discussed in [42], the condition which guarantees the ellipsoid Ω=x|xTPx<ρ being inside the domain Ω1, i.e., Ω⊂Ω1, is given by(34)FikPρ−1FiTk<1ε2,fori=1,…,r,where Fik indicates the k-th row of Fi. By employing the Schur complement on condition (34), LMI (23) is obtained. The proof is completed.
In most of the existing results on the primary control of power components connected to a microgrid, the control law for each power electronic converters is constructed based on the states of its corresponding component [21–23]. This reduces the complexity of designing the control signal. However, the closed-loop stability overall MG is not guaranteed, as the interconnections and effects of other components are not involved. Here, in this paper, a centralized controller is considered in which all available information of all components is used to design control signals. This consideration assures the closed-loop stability with the expense of increasing the complexity of the design procedure. In other words, as can be seen in (7), the overall system state vector comprises 7 state variables, and the dynamics have 3 nonlinearities and 4 time-varying parameters. Assuring stability for such a system is not a trivial task. Among the nonlinear control techniques, the LPV-based control approach uses linear control and convexity theories to design a stabilizing controller based on a systematic numerical procedure. This feature makes the LPV-based control method suitable for highly nonlinear and high-dimensional systems.
Generally, constraints (20)–(23) are matrix inequalities and contain several unknown variables to be found. Thereby, it is not possible to solve them analytically. This issue persuades using the LMI solver. The way of designing a nonlinear gain-scheduling controller for system (7) is provided in Figure 6. For a given dynamical nonlinear system with physical bounds of nonlinear and time-varying parameters, the equivalent polytopic-LPV representation can be obtained systematically based on the sector-nonlinearity modeling approach [35].
Then, the scheduling parameter and local linear system matrices are achieved. The local linear system matrices are then utilized in Theorem1 to numerically compute the controller gains via the LMI technique. Both unknown controller gains and Lyapunov matrices are calculated so that the constraints in Theorem 1 satisfy. Based on the proof of Theorem 1, the satisfaction of the constraints assures closed-loop stability theoretically. Finally, the LPV controller is constructed based on the scheduling parameters and calculated gains. All steps of controller design are done offline. The proposed controller is designed offline and does not expose an online computational burden, which is the main drawback of the model predictive controller [43, 44]. On the other hand, the linear proportional-integral-derivative (PID) controller [32] and linear quadratic regulator [33, 45] only assure the local stability around the operating point. However, the suggested LPV controller assures the stability of the system on the basis of its physical bounds of the nonlinear and time-varying parameters [46].Figure 6
The LPV controller design procedure.


## 3.1. Polytopic-LPV Model of DC MG

In order to achieve the objectives of the microgrid, the polytopic-LPV control approach is utilized in this paper. In the state-space model, there are some time-varying termsIpv/CpvVpv, VF/LFIF, and IB/CBVB in the matrix A and state variables ILPV, ILB, ILF, and VDC in the matrix B. Let(9)θ=θ1θ2…θ7T=IPVCPVVPVVFLFIFIBCBVBVDCILPVILFILBTto be the vector of varying parameters, where each parameter is bounded in a set, θɛθ¯,θ¯, with θ¯ and θ¯ as the upper and lower limits, respectively. The values of vector θ are contained in a polytopic domain with 27 vertices. Finally, by applying the so-called sector nonlinearity [36, 37], the following polytopic-LPV system is obtained:(10)x˙t=∑i=1128ρiθAixt+Bivt,yt=Cxt,where vt=satut=satu1t,…,satu3tT and(11)satuit=1,uit≥1,uit,for1<uit<1,−1,uit≤−1.Note that, to design the pulse-width modulation (PWM) signal for the converters, it is necessary that the duty cyclesui for i=1,2,3 are bounded by ±1. Thereby, the lower and upper bounds in (10) are −1 and +1, respectively. The goal is to design the control input vector ut such that the VPV tracks the voltage of the VPV−ref and the DC bus voltage converges to VDC∗. The reference value of the DC bus voltage is mainly constant. However, the maximum power point voltage reference must be chosen accordingly.


## 3.2. LPV Controller Design

In order to achieve the objectives of the microgrid, the polytopic-LPV control approach is utilized in this paper:(12)ut=∑i=1128ρiθFixt,where Fi for i=1,…,128 are the controller gains, which should be obtained to assure the closed-loop stability and convergence of the states VPV and VDC to their references. By adding the null term ∑i=1rhiztBi1+ε/2ut−ut with 0<ε<1 to (10), one has(13)x˙t=∑i=1128ρiθAixt+Bi1+ε2ut+Bivt−1+ε2ut=∑i=1128ρiθAixt+Bi1+ε2ut+Birt,where rt=vt−1+ε/2ut. The closed-loop system is obtained as(14)x˙t=∑i=1128∑j=1128ρiθρjθAi+1+ε2BiFjxt+Birt.Lemma 1.
(see [38]). For the saturation constraint defined by (11), as long as ut<1/ε with 0<ε≤1, one has(15)rTtrt≤1−ε22uTtut,which is shown in Figure 5.Figure 5
The functions of Lemma1 (the rt by the solid blue line and the 1−ε/2ut by the dashed red line).Lemma 2.
(S-procedure) (see [39]). Consider that the conditions(16)ζTT0ζ≥0,where ζ≠0 and satisfies the constraints(17)ζTTiζ≥0,fori=1,…,p.Then, condition (16) subject to constraint (17) results in(18)T0−∑i=1pτiTi≥0,where τi for i=1,…,p are arbitrary nonnegative scalars.Lemma 3.
(see [40]). The time-varying parameter-dependent constraint ∑i=1q∑j=1qρiθρjθxTΓijxt<0 is implied by the following time-varying parameter independent conditions:(19)Γii<0,fori=1,…,q,Γij+Γji<0,fori<jandj=1,…,q.Lemma1 is essential to eliminate the time-varying parameters ρiθ and the state vector x in the controller design conditions. The following theorem provides the controller design conditions in terms of LMIs such that the exponential stability of the states to the equilibrium point is assured.Theorem 1.
System (14) is exponentially stabilizable in the local region Ω=x|xTP−1x<ρ, if for the given decay rate σ>0, there exists matrices S=ST and Mi such that(20)S>0,(21)Γ¯ii<0,(22)Γ¯ij+Γ¯ji<0,(23)ulimε2MikMiTkρS>0,where(24)Γ¯ij=SAiT+AiS+1+ε2MjTBiT+1+ε2BiMj+σSBiT1−ε2MjTBi−τI01−ε2Mj0−τ−1I.Proof.
Consider the exponential Lyapunov stability conditions with the decay rateσ, as(25)V˙+σV<0,where the Lyapunov candidate is chosen as(26)V=xTtPxt,where P is a symmetric positive definite matrix. Substituting (14) into (25), the exponential stability condition is continued as(27)V=xTt∑i=1128∑j=1128ρiθρjθAi+1+ε2BiFjTP+PAi+1+ε2BiFj+σPxt+rTt∑i=1128ρiθBiTPxt+xTtP∑i=1128ρiθBirt=∑i=1128∑j=1128ρiθρjθxTtrTtQijxtrt,where(28)Qij=Ai+1+ε2BiFjTP+PAi+1+ε2BiFj+σPBiTPPBi0.
Applying Lemma2 on (27) subject to (15) results in(29)∑i=1128∑j=1128ρiθρjθxTtrTtQ¯ijxtrt,where(30)Q¯ij=Ai+1+ε2BiFjTP+PAi+1+ε2BiFj+τFiTFi1−ε22+σPBiTPPBi−τI.
Applying Lemma1 on (27) results in(31)Q¯ii<0,fori=1,…,128,Q¯ij+Q¯ji<0,fori<jandj=1,…,128.
Considering the Schur complement, (31) leads to(32)Γii<0fori=1,…,128,Γij+Γji<0fori<jandj=1,…,128,where(33)Γij=Ai+1+ε2BiFjTP+PAi+1+ε2BiFj+σPBiTP1−ε2FiTPBi−τI01−ε2Fi0−τ−1I.
Pre- and postmultiplying (32) by diagP−1,I,I and defining the LMI variables S=P−1 and Mi=FiP−1 result in (21) and (22). Moreover, the constraint P=PT>0 leads to the LMI (20). On the other hand, (15) holds in the local region Ω1=x|u<1/ε. Consequently, inspired from the set inverse analysis [41] and considering the same procedure discussed in [42], the condition which guarantees the ellipsoid Ω=x|xTPx<ρ being inside the domain Ω1, i.e., Ω⊂Ω1, is given by(34)FikPρ−1FiTk<1ε2,fori=1,…,r,where Fik indicates the k-th row of Fi. By employing the Schur complement on condition (34), LMI (23) is obtained. The proof is completed.
In most of the existing results on the primary control of power components connected to a microgrid, the control law for each power electronic converters is constructed based on the states of its corresponding component [21–23]. This reduces the complexity of designing the control signal. However, the closed-loop stability overall MG is not guaranteed, as the interconnections and effects of other components are not involved. Here, in this paper, a centralized controller is considered in which all available information of all components is used to design control signals. This consideration assures the closed-loop stability with the expense of increasing the complexity of the design procedure. In other words, as can be seen in (7), the overall system state vector comprises 7 state variables, and the dynamics have 3 nonlinearities and 4 time-varying parameters. Assuring stability for such a system is not a trivial task. Among the nonlinear control techniques, the LPV-based control approach uses linear control and convexity theories to design a stabilizing controller based on a systematic numerical procedure. This feature makes the LPV-based control method suitable for highly nonlinear and high-dimensional systems.
Generally, constraints (20)–(23) are matrix inequalities and contain several unknown variables to be found. Thereby, it is not possible to solve them analytically. This issue persuades using the LMI solver. The way of designing a nonlinear gain-scheduling controller for system (7) is provided in Figure 6. For a given dynamical nonlinear system with physical bounds of nonlinear and time-varying parameters, the equivalent polytopic-LPV representation can be obtained systematically based on the sector-nonlinearity modeling approach [35].
Then, the scheduling parameter and local linear system matrices are achieved. The local linear system matrices are then utilized in Theorem1 to numerically compute the controller gains via the LMI technique. Both unknown controller gains and Lyapunov matrices are calculated so that the constraints in Theorem 1 satisfy. Based on the proof of Theorem 1, the satisfaction of the constraints assures closed-loop stability theoretically. Finally, the LPV controller is constructed based on the scheduling parameters and calculated gains. All steps of controller design are done offline. The proposed controller is designed offline and does not expose an online computational burden, which is the main drawback of the model predictive controller [43, 44]. On the other hand, the linear proportional-integral-derivative (PID) controller [32] and linear quadratic regulator [33, 45] only assure the local stability around the operating point. However, the suggested LPV controller assures the stability of the system on the basis of its physical bounds of the nonlinear and time-varying parameters [46].Figure 6
The LPV controller design procedure.


## 4. Real-Time Simulation Results and Discussion

In this section, the designed controller based on Theorem1 is applied to the propulsion system of the EB dynamics (7). In order to reduce the controller design time burden and controller implementation, in Theorem 1, it is considered that Mi=M. Also, the nominal powers of the solar power plant, fuel cell, and energy storage are set as 5 kW, 5 kW, and 2 kW, respectively. To show the applicability of the suggested approach, a model-in-the-loop (MiL) real-time simulation is carried out as shown in Figure 7, where the dSPACE 1202 board is deployed for the rapid prototyping solution.Figure 7
MiL real-time setup and configuration of the marine DC MG benchmark.In the simulations, it is assumed that the solar irradiation is constant, and so, the solar cell voltage to extract the maximum power is fixed. To show the merits of the suggested control approach (Prop.), it is compared with the sliding mode control (SMC) [47] and adaptive backstepping control (ABP) [48]. Additionally, it is shown in [47] that the SMC approach outperforms the conventional proportional-integral (PI) controller. It should be noted that the DC MG structure considered in [47] is different from this paper. However, the SMC design procedure is modified by [48], which is used for the comparison. Moreover, the control approaches [47, 48] do not consider the issue of input saturation in their design procedures.The closed-loop system is simulated in MATLAB software and the states and the averaged control inputs are shown in Figures8 and 9.Figure 8
The voltages of the DC MG EB system. (a) DC-link, (b) battery, and (c) solar power plant.
(a)(b)(c)Figure 9
The control inputs of the DC MG EB system converters. (a) Solar plant, (b) fuel cell, and (c) battery.
(a)(b)(c)Also, to quantitatively compare the results of the controllers, the norms 2 of the tracking errors of the voltagesVDC, VB, and VPV for these controllers are given in Table 1. As can be seen in Table 1, the proposed approach results in a smaller norm 2 for all voltages. Thereby, the tracking error based on the proposed approach outperforms the other approaches, since the proposed approach develops exponential stability and convergence in comparison with the asymptotic stability in the ABP and SMC.Table 1
Norms 2 of the considered controller approaches.
Prop.ABPSMCVDC6.48911×1048.3269×1041.6737×105VB77.6278150.1197558.4585VPV582.51912.2616×1031.0392×103As can be seen in Figure8, the proposed approach exponentially stabilizes the nonlinear system. Additionally, from Figure 9, one infers that the control laws [47, 48] experience high oscillations for the early stage of the simulation. One of the reasons is that in those approaches, the input saturation is not involved in the design procedure. For the case study of this paper, although the ABP and SMC controllers regulate the closed-loop system with the input saturation, the transient performance is degraded. Further, by changing the initial conditions and changing the operating point, they may not stabilize the overall system.


## 5. Conclusion

In this paper, the problem of the propulsion power system regulating the electric boat was investigated. A robust controller strategy was developed to optimally utilize the energy resources available in the DC power system. A nonlinear power system with a 5 kW solar power plant, 5 kW fuel cell, and 2 kW battery package is considered. By the means of the linear parameter varying (LPV) method, the suitable gains of the states’ feedback controller are determined to provide the required PWM switching signals of the boost converters of the energy resources. The suggested control approach has a systematic offline design procedure based on the linear matrix inequality (LMI) technique. Therefore, it is a suitable approach for highly nonlinear and complex systems to assure closed-loop stability. The simulation results demonstrate the performance of the developed control method to effectively regulate the DC-link voltage fast and without fluctuations. For future work, considering nonlinear loads such as constant power loads (CPLs) and several fuel cells and batteries in the propulsion system to manage the power of the generators is suggested. Comparing the transient and steady-state of the suggested controller with other nonlinear approaches is recommended. Also, evaluating the effect of environmental changes, such as solar irradiation and faults, is a good research topic.



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*Source: 8848904-2021-02-24.xml*