Multi-variable H∞ Control Approach for Voltage Ancillary Service in Autonomous Microgrids: Modelling, Design, and Sensitivity Analysis

This paper proposes a multi-variable robust control scheme for voltage regulation in a diesel-photovoltaic-supercapacitor hybrid power generation system operating in stand-alone mode. First, we study the influence of system parameters on the dynamic behavior of open-loop system measured outputs by means of a stability analysis method based on Monte Carlo simulation. Next, by applying H∞ control theory, an H∞-based voltage controller is proposed to robustly force the voltage magnitude of a point of common coupling such as to satisfy design specifications. A cascaded two-level control architecture, where this controller acts as an upper control level and provides references to classical PI-based current tracking controllers placed on a lower level, is developed. A comprehensive methodology that casts the specific engineering demands of microgrid operation into an H∞ control formalism is detailed. Effectiveness of the proposed voltage robust control strategy is validated via MATLAB®/Simulink® closed-loop time-domain simulations. Finally, we perform a sensitivity analysis of robust performance of the designed H∞ controller in the presence of various load disturbances and model uncertainties through a series of closed-loop time-domain simulations carried out in MATLAB®/Simulink®.


A. MICROGRID CONTROL
M ICROGRIDS (MGs) concept is gaining high momentum as a major, cost-effective solution to integrate distributed energy resources (DERs) into power networks [1]. Moreover, the distinctive autonomous operational capability of MGs has brought in higher reliability measures in supplying power demands when the utility grid is not available. The stability and control issues of autonomous MGs are however among the main challenges due to low inertia, uncertainties, and intermittent nature of DERs [2]. One critical control task in autonomous operation mode is the regulation of the network frequency and voltage magnitude. High-speed storage systems -e.g., lithium-ion batteries, flywheels or supercapacitors -have thus become necessary, leading to new grid configurations, for which more complex robust control structures are needed for dealing with multiple constraints such as unexpected disturbances and model uncertainties.

B. LITERATURE REVIEW
Topics around voltage control in autonomous -otherwise called islanded or stand-alone MGs -are nowadays extensively explored by employing advanced control techniques; among them, robust control design has a major role. In the work of Mohamed et al. [3], a variable-structure voltage controller is integrated with a droopbased power-sharing controller to improve the transient and steady-state response of an MG against voltage disturbances and power angle swings. Based on master/slave control structure, an H ∞ -mixed-sensitivity-based robust voltage controller for the master unit in a single-bus MG is presented in the work of Babazadeh et al. [4]. The proposed decentralized controller provides the good tracking of reference signals and robustly maintains load voltage magnitude despite parameter uncertainties. Similarly, using linear H ∞ mixed sensitivity control method, a new robust two-degree-of-freedom feedback-feedforward control scheme is proposed for the islanded operation of an MG comprising several DG units in the work of Babazadeh et al. [5] to robustly regulate load voltage in the presence of unmodeled dynamics, for example, linear, nonlinear, and highly unbalanced loads. Also, H ∞ -mixed-sensitivityloop-shaping-based robust control design for current sharing improvement in an islanded MG composed of parallelconnected inverter-interfaced DG units is presented in the work of Taher et al. [6]. The designed robust controller is capable of guaranteeing robust stability and performance under line parameter uncertainties and different loads. Hamzeh et al. propose in [7] an H ∞ -based robust controller along with particle swarm optimization algorithm for the autonomous operation of an MG comprising electronically coupled DG units under unbalanced and nonlinear load conditions with unknown dynamics. Voltage control strategy based on feedforward compensation and internal model robust feedback control applied to the autonomous operation of a three/single-phase hybrid multi-MG is detailed in the work of Wang et al. [8]. The designed controller can get better voltage output characteristics and robustness in dealing with micro-source output power fluctuations, loads abrupt change, or nonlinear loads and unbalanced loads. In the above-mentioned methods, the plant transfer function is strictly proper and parametrically uncertain. However, no problem-dependent uncertainty modelling is included in the control design procedure.
Motivated by the aforementioned limitation, Kahrobaeian et al. introduce in [9] a robust system-oriented control approach for voltage performance improvement and suppression of typical interaction dynamics in DG converterbased MGs. The proposed control scheme uses a robust H ∞ voltage controller which is designed under an extended model comprising different converter-MG interactions imposed on the output voltage, e.g., MG impedance variation, local load interactions, uncertainties in AC-side filter parameters. Unlike conventional droop controllers, the proposed strategy generates a two-degree-of-freedom controller, leading to stable and smooth power sharing performance over a wide range of loading conditions. In the work of Davari et al. [10], a robust multi-objective control approach based on D-K iteration µ-synthesiswhose plant nominal model is unstructured uncertainis applied to a voltage-source-converter-based DC-voltage power port in hybrid AC/DC multi-terminal MGs. The developed controller ensures tracking performance, robust disturbance rejection, and robust stability against operating point and parameter variations, as well as has a simple structure, low order, and fixed parameters, which makes it very attractive for industrial applications. Similarly, on the basis of µ-synthesis and H ∞ loop shaping control through genetic algorithm and particle swarm optimization, the reactive power compensation and voltage stability of a PV-wind-diesel hybrid power generation system -where unstructured uncertainty modelling is adopted -can be found in the work of Mohanty et al. [11]. The proposed controller combines robustness and simplicity, and it is quite adaptable in its nature under different wind power input and load variations.
Unlike the augmented unstructured uncertainty modelling approach used with conventional control, structured uncertainty modelling is adopted to enable the realization of a less conservative robust controller. In the work of Karimi et al. [12], a linear time-invariant robust servomechanism voltage controller is proposed for a single DG MG whose plant nominal model is structurally uncertain. Despite the uncertainty of load parameters, the proposed controller guarantees robust stability and prespecified performance criteria, e.g., fast transient response, negligible interaction, and zero steady-state error. Similarly, fundamental concepts of a central power-management system and a linear, time-invariant, multi-variable, robust, decentralized, servomechanism control strategy -whose plant nominal model is structurally uncertain -are detailed for a multiple-DER MG in the work of Etemadi et al., [13] and [14] respectively. Each control agent guarantees fast tracking, zero steady-state error, and robust performance despite the uncertainties of MG parameter, topology, and operating point. On the basis of H ∞ control method, a robust multi-objective centralized control strategy -whose plant nominal model is norm-bounded uncertain with an integral quadratic constant -is presented in the work of Hossain et al. [15] to optimize the active and reactive power sharing performance of a PV-wind MG in the presence of network changes, nonlinear uncertainties, and interaction dynamics. Also, decentralized robust control strategies subject to polytopic uncertainty are designed for plug-and-play voltage stabilization in islanded inverter-interfaced MGs and in islanded DC MGs in the work of Sadabadi et al., [16] and [17] respectively. The performance of the proposed control strategy is verified in terms of voltage tracking, MG topology change, plug-and-play capability features, and load changes. Li et al. propose in [18] a robust control scheme for an MG with a power-factor correction capacitor connected at the point of common coupling (PCC). An integrated control strategy composed of an outer H ∞ voltage controller, an inner filter inductor current feedback controller, and a virtual resistance compensator is presented. By properly selecting weighting functions with parameter uncertainty being transformed to multiplicative output structure, the designed H ∞ controller can effectively produce desired performance and explicitly specify the degree of robustness against effective shunt filter capacitance variations. However, it might be sensitive to system resonance caused by the switching of power-factor correction capacitor. Similarly, Kahrobaeian et al. present in [19] a direct single-loop µ-synthesis voltage control scheme for the suppression of multiple resonances caused by power-factor correction capacitors and residential capacitive loads in DG MGs. Compared to conventional H ∞ multi-loop control, the proposed single-loop µ-synthesis controller, with structured uncertainty modelling, reduced sensor requirement, and no additional passive or active damping mechanism, can ensure the robust stability and performance of the MG subject to parameter uncertainties and uncertain resonant peaks. However, the controller obtained with this method has high order, therefore it leads to harder design and practical implementation. Another interesting approach can be found in the work of Li et al. [20], where a novel cascaded-loop strategy for the control of a grid-forming inverter is designed.
Even if a precise model for the inverter system is not required, the proposed method can tackle uncertainties and LC filter resonance without any passive or active damping techniques. It comprises a sliding-mode-control-based inner current loop and a mixed-H 2 /H ∞ -control-based outer voltage loop where its uncertain matrix are norm bounded, which provides the benefits of constant switching frequency, low total harmonic distortion, robustness against parameters variations, and fast transient response. Moreover, the proposed control strategy ensures better transient and steady performance compared to conventional PI-based nested-loop control. Nevertheless, it may not be sufficiently robust against large system parameter variation and exogenous disturbances as well as well-performing due to the issue of conservatism of H 2 /H ∞ control.

C. PAPER CONTRIBUTION
Unlike the majority of the aforementioned power sharing and voltage control schemes, where MG interaction dynamics are completely ignored or assumed as lumpsum external disturbance, a design approach that is more robust to interaction dynamics can be synthesized when interaction dynamics are qualitatively considered in the design process by using the accurate modelling of overall MG dynamics. MG voltage comprehensive control system development using a complete diesel-PVsupercapacitor/converter dynamic model to effectively reject the wide band of interaction dynamics and uncertainties is conceptually designed in this work.
In addition, this study addresses the voltage stability and regulation issues of islanded MGs with high penetration of renewable energy by making use of energy storage devices. As a matter of fact, voltage magnitude deviation can be significantly reduced by using relatively small storage units, as long as saturation conditions are avoided, by dynamically coordinating storage with other generation sources [8]. Similarly, in the work of Kim et al. [21], a cooperative control strategy has been proposed for sharing the reactive power demand among generation sources and fast-acting energy storage systems (ESSs) during the islanded mode of MGs operation. The PCC voltage magnitude can be regulated at the rated value by an adequate reactivepower-balance matching process, provided that the available capacity of the ESSs does not exceed their limitations. In this paper, an H ∞ -based multi-variable robust controller for PCC voltage magnitude regulation that copes with load reactive power variations in a diesel-PV-supercapacitor hybrid power generation system operating in stand-alone mode is proposed. A two-level control scheme, where this controller acts as an upper control level and generates references to current controllers placed on a lower level, is detailed, leading to the low order H ∞ controller and facilitating the practical implementation. This paper extends results presented in [22], [23], namely by the following points • The influence of system parameters on the dynamic behavior of open-loop system measured outputs is studied by means of a stability analysis method based on Monte Carlo simulation; • Also, in this work, a systematic design of a multivariable robust control structure for voltage regulation in stand-alone MGs -with a comprehensive methodology casting the specific engineering demands of MG operation into an H ∞ control formalism -is developed; • For the considered MG, the performance and robustness of the designed H ∞ controller in the presence of various load disturbances and model uncertainties are analyzed through a series of MATLAB ® /Simulink ® time-domain simulations.
The remainder of this paper is structured as follows. Section II describes system configuration and design specifications. Mathematical modelling for H ∞ control is provided in Section III. Section IV is dedicated to a stability analysis of open-loop system behavior. Section V is devoted to H ∞ control design along with some closed-loop timedomain simulations performed in the rated operating point. A robust performance analysis of the synthesized H ∞ controller, subject to the uncertainties in the steady-state value of the supercapacitor voltage V sc and the length of the transmission line l connecting the diesel engine generator to the PCC, is investigated in Section VI. Some concluding remarks and perspectives are stated in Section VII.

II. SYSTEM CONFIGURATION AND DESIGN SPECIFICATIONS A. MICROGRID STRUCTURE
This paper focuses on an islanded MG constituted by a diesel engine generator, a PV source, and a supercapacitor-based energy storage unit. The studied MG setup is illustrated in Fig. 1, where power sources are connected in parallel to a PCC and feed a common load. The diesel engine generator is connected to the PCC via a three-phase step-up transformer VOLUME

Rn
Step-up transformer Step-down transformer Step-up transformer (a)  and a transmission line. The PV panel bank is connected to the PCC through a power-electronic conversion system and a three-phase step-up transformer allowing only unidirectional power flow; whereas the supercapacitor-based energy storage unit is connected to the PCC by means of a three-phase step-up transformer, which allows charging or discharging. A three-phase step-down transformer connects the aggregated load to the PCC.

B. DESIGN SPECIFICATIONS
In stand-alone MGs with high penetration of PV energy sources, the PCC voltage is risky to be unstable or its magnitude variations can be considerable and unacceptable.

Rt Lt
Step-up transformer

III. MATHEMATICAL MODELLING FOR H∞ CONTROL
The schematic diagram of the MG under consideration for voltage stability analysis is depicted in Fig. 4 [22]. In this work, dynamical equations and equivalent averaged models of power-electronic converters are used to describe the behavior of the studied MG. By linearizing these equations for a given equilibrium state, a linear averaged model can be obtained [23]. It is often more convenient to work with per-unit models of power systems [25], [26]. In particular, when it comes about designing control structures, per-unit models of power systems are generally better conditioned, especially when it is about high-order, possibly multi-scale systems [25], [26]. Moreover, the use of a per-unit system can improve numerical stability of automatic computation methods. Therefore, in this work, the dynamic model of the studied MG is converted from real unit into per unit in order to serve the control design procedure, which is detailed in Section V.
For the considered ESS, the base values for AC-side quantities are first selected and serve to determine the DCside base values. The linear averaged model of the considered MG using the state variables shown in Fig. 4 can be expressed by the following per-unitized set of equations [23] where the state variables are: the supercapacitor voltage variation ∆V sc , the DC-bus voltage variation ∆V dc , the filter output voltage variations in d-axis and q-axis, ∆V rd and ∆V rq respectively, the PCC voltage variations in daxis and q-axis, ∆V P CCd and ∆V P CCq respectively, the ESS step-up transformer output current variations in d-axis and q-axis, ∆I td and ∆I tq respectively, the diesel generator output current variations in d-axis and q-axis, ∆I gd and ∆I gq respectively, the supercapacitor output current variation ∆I s , and the inverter output current variations in d-axis and q-axis, ∆I rd and ∆I rq respectively. α c is the pulsewidth modulation (PWM) duty cycle of the chopper, β d and β q are the average values of the inverter switching functions in d-axis and q-axis respectively, ω grid is the MG pulsation. The external disturbances applied to the system are the direct and quadrature components of the load and PV system output current variations, i.e., ∆I loadd − ∆I P V d and ∆I loadq − ∆I P V q respectively. It should be noted that this averaged model takes into consideration only the primary voltage control of the diesel engine generator. Active and reactive power of the sources and load can be computed by the following equations where P s is storage device active power, P t and Q t are stepup transformer output active and reactive power respectively, P g and Q g are diesel engine generator output active and reactive power respectively supplied at the PCC, P P V and Q P V are PV output active and reactive power respectively, and P load and Q load are load active and reactive power respectively. Let us define here that • For sources: P > 0 if they supply active power, P < 0 if they absorb active power; Q > 0 if they supply reactive power, Q < 0 if they absorb reactive power; • For loads: P > 0 if they absorb active power, P < 0 if they supply active power; Q > 0 if they absorb reactive power, Q < 0 if they supply reactive power. The equations which express power balance at the PCC are given by Subscript e indicates the steady-state equilibrium point. Steady-state real-unit values are given in Table 1. Note that the V sce value chosen for H ∞ control design corresponds here to the midrange between the minimum and maximum supercapacitor voltages. In this case, V sce = 585 V, where 390 V (SoC sc = 25%) is taken as the minimum V sc and 780 V (SoC sc = 100%) is the maximum V sc . The minimum value of the transmission line length l is chosen for H ∞ control design. In this case, l = l min = 1 km, where 10 km has been chosen as the maximum value of l in order to ensure the voltage drop on the transmission line at the high-voltage level of 20 kV being less than or equal to the admissible limit of 8% of the rated voltage.

IV. OPEN-LOOP SYSTEM BEHAVIOR
Based on Monte Carlo simulation, a stability analysis is detailed in this section in order to emphasize the influence of system parameters on the dynamic behavior of open-loop system measured outputs.
With MG parameter values, the zeros of the openloop sensitivity functions ∆V P CCd / (∆I loadd − ∆I P V d ) or ∆V P CCq / (∆I loadq − ∆I P V q ) and ∆V P CCd / (∆I loadq − ∆I P V q ) or ∆V P CCq / (∆I loadd − ∆I P V d ) of the linearized system (1)-(13) are given in Table 2. The transfer function ∆V P CCd / (∆I loadq − ∆I P V q ) or ∆V P CCq / (∆I loadd − ∆I P V d ) has right-half-plane zeros z 2 and z 5,6 (i.e., positive real part). This results in the open-loop step response taking a particular shape, namely, the measured output variation ∆V P CCd or ∆V P CCq first evolves in the opposite sense in relation to the disturbance input variation ∆I loadq − ∆I P V q or ∆I loadd − ∆I P V d respectively, then come back to the positive sense. This is called a non-minimum phase response [27] and leads to an exhibition of non-minimum phase behavior in the PCC line-to-line voltage magnitude variation ∆U P CC (with U P CC = V 2 P CCd + V 2 P CCq ). Moreover, the transfer function ∆V P CCd / (∆I loadd − ∆I P V d ) or ∆V P CCq / (∆I loadq − ∆I P V q ) has left-half-plane complex conjugate zero pairs z 1,2 , z 3,4 , z 5,6 , z 7,8 , and z 10,11 that induces oscillations in ∆V P CCd or ∆V P CCq in response to ∆I loadd − ∆I P V d or ∆I loadq − ∆I P V q respectively. This leads to an oscillation in ∆U P CC , where the oscillation damping is characterized by the ratio |Im (z k,k+1 ) /Re (z k,k+1 )|. Namely, the smaller this ratio is, the more damped the oscillation is. It should be noted here that this ratio, as well as the aforementioned zeros, are completely dependent on MG parameter values and the MG steady-state operating point. Hence, the minimization of this ratio by changing MG parameter values could be a feasible solution in order to minimize the oscillation amplitude in ∆U P CC , as well as to limit the non-minimum phase behavior.
Given admissible variation ranges of MG parameter values, we propose a stability analysis method with the aim of finding out a combination of these MG parameter values that allows to minimize the oscillation amplitude in ∆U P CC . This method is based on Monte Carlo simulation and carried out in the following steps 1) Define a domain of possible values of m input independent random variables: MG parameter value random variables X i (i = 1, m) are here assumed to vary from − 50% to + 50% of their rated values X in , i.e., 0.5X in ≤ X i ≤ 1.5X in ; 2) Sample the random variables X i that are distributed over the continuous ranges [0.5X in , 1.5X in ]. This is randomly done with the MATLAB ® random variable generator "rand"; 3) Compute a steady-state operating point X e , as well as matrices A, B 1 , B 2 , C, D 1 , and D 2 of the linear system, with the sampled random variables X i generated in Step 2; 4) Analyze the structural properties of stability and observability of the linear system based on its matrices A, B 1 , and C to verify if the system is stable and observable or not. If this is not fulfilled, Step 2 then Step 3 are re-conducted; 5) Determine open-loop sensitivity functions of the linear system, then for each sensitivity function ∆y/∆w identify all right-half-plane zeros which characterize non-minimum phase behavior, as well as all lefthalf-plane complex conjugate zero pairs where q/2 is the number of left-half-plane complex conjugate zero pairs of the system) which cause oscillation in the measured output variation ∆y consequent to the disturbance input variation ∆w; 6) For each open-loop sensitivity function ∆y/∆w of the linear system compute the values of output variables Y k,k+1 = |Im (z k,k+1 ) /Re (z k,k+1 )| = 1 − ζ 2 k /ζ k of the left-half-plane complex conjugate zero pairs z k,k+1 , which characterize the oscillation amplitudes in the measured output channel variation ∆y in response to the input channel disturbance ∆w; 7) Suppose that N samples of each random variable X i are generated, then all the samples of random variables constitute N sets of inputs, i.e., X r = (X r−1 , X r−2 , ..., X r−i , ..., X r−m ) (r = 1, N ). Solve the problem from Step 2 to Step 6 N times deterministically yields N sample sets of output variables, i.e., , which is available for statistical analysis and estimation of the characteristics of the output variables Y r . Let us remark here that the accuracy of Monte Carlo simulation depends on the number of simulations N . The higher the number of simulations is, the more accurate the estimate will be; identify the sample Y r−k that has the greater influence on the minimization of the oscillation amplitude in ∆U P CC , then the corresponding sample set of the input variables X r = (X r−1 , X r−2 , ..., X r−i , ..., X r−m ). This input set will be the desired combination of the MG parameter values.
By applying the aforementioned stability analysis method with the input-independent random variables being X = , R lp supposed to vary from − 50% to + 50% of their rated values X in and a sufficiently large number of samples of N = 1000 gives each 1000 samples Y k,k+1 = |Im (z k,k+1 ) /Re (z k,k+1 )| of the left-half-plane complex conjugate zero pairs z k,k+1 of the open-loop sensitivity functions ∆V P CCd / (∆I loadd − ∆I P V d ) or ∆V P CCq / (∆I loadq − ∆I P V q ) and ∆V P CCd / (∆I loadq − ∆I P V q ) or ∆V P CCq / (∆I loadd − ∆I P V d ), as depicted in Fig. 5 and Fig. 6 respectively. Let us remark here that, since V P CCq ≪ V P CCd , deducing that U P CC ≈ V P CCd or ∆U P CC ≈ ∆V P CCd , therefore the variation ∆U P CC is much more influenced by ∆V P CCd than by ∆V P CCq . Moreover, only the quadrature component of the load current variation ∆I loadq is regarded as a disturbance, whereas its direct component is approximately equal to zero, i.e., ∆I loadd ≈ 0. As a result, the dynamic effects exhibited in ∆U P CC are mainly affected by the values Y k,k+1 of the left-halfplane complex conjugate zero pairs z k,k+1 of the sensitivity   Table 3. This desired parameter combination of the studied whole MG allows to minimize the oscillation amplitude in ∆U P CC .

V. H∞ CONTROL DESIGN
This section details a robust control design approach used to satisfy the dynamic specifications in Subsection II-B. Here the basic idea is to consider the current variations ∆I s , ∆I rd , and ∆I rq in (11)-(13) as control inputs for the linearized system (1)- (10), which should result from the disturbance rejection requirement ∆I loadd − ∆I P V d and ∆I loadq − ∆I P V q . A hierarchical two-level control strategy, where the outer control loop deals with output regulation imposing low-frequency dynamics (e.g., ∆V dc , ∆V P CCd , ∆V P CCq ) and the inner loop concerns current reference tracking of high-frequency dynamics (e.g., ∆I s , ∆I rd , ∆I rq ), is adopted for voltage control. The validity and effectiveness of the proposed robust control strategy are demonstrated through a series of closed-loop time-domain simulations carried out in MATLAB ® /Simulink ® .

A. GLOBAL CONTROL CONFIGURATION BLOCK DIAGRAM
As can be seen in most energy management systems and according to control objectives, it is preferred -from the application point of view -to consider power sources acting as current sources [28]. Therefore, a cascaded twolevel control structure -where an H ∞ -based multi-variable controller acts as an upper control level and provides references to classical PI-based current tracking controllers placed on a low control level -is developed to ensure the previous dynamic specifications. The proposed global control structure is illustrated in Fig. 7 [23].
From the control viewpoint, note that V dc appears as a gain in the low-level current control loops, such that V dc being constant is an assumption making that classical PI controllers to be sufficient as this level (i.e., if V dc is not kept within admissible limits, classical PI controllers would not be able to ensure stability and the required tracking performance).

B. CURRENT CONTROL LEVEL
The current variations ∆I s , ∆I rd , and ∆I rq must be controlled and prevented from exceeding admissible limits. Current control loops have fast closed-loop dynamics compared to the H ∞ control loop.
∆I rq in (12) appears as a high-frequency perturbation; the same applies for ∆I rd in (13). Their reciprocal influence is significantly reduced by the d−q decoupling structure. ∆V sc , ∆V dc , ∆V rd , and ∆V rq in (11), (12), and (13) are regarded as low-frequency perturbations that are rejected by the upperlevel control. ω grid is considered a time-invariant parameter in voltage regulation, i.e., ω grid = ω gride or ∆ω grid = 0 (the MG frequency is supposed to be well-regulated at its nominal value of 50 Hz). Hence, first-order transfer functions relating the current variations with the duty ratio variations (inner plants) are computed straightforwardly from (11) The supercapacitor voltage V sc is considered as a time-invariant parameter in the H ∞ controller synthesis, i.e., ∆V sc = 0 (its dynamic equation ∆V sc = f (∆V sc , ∆I s ) in (1) being neglected). The diesel engine generator output voltage is supposed to be well regulated at its setpoint value, i.e., ∆V gd = 0 and ∆V g = 0. Thus, the state-space form of the linear system (2)-(10) can be written as follows (with the terms

Sensitivity function
Sensitivity function or ∆V P CCq / (∆I loadq − ∆I P V q ) and ∆V P CCd / (∆I loadq − ∆I P V q ) or ∆V P CCq / (∆I loadd − ∆I P V d ) of the linear system (25) are given in Table 4. The existence of undesired dynamic effects (e.g., non-minimum phase behavior, oscillation) exhibited in the PCC line-to-line voltage magnitude variation ∆U P CC in response to the disturbance input variation ∆I loadq − ∆I P V q or ∆I loadd − ∆I P V d can be proved in a similar way as the analysis of zeros presented in Section IV. . This results in a non-minimum phase behavior in ∆U P CC . Suppose that the system operates in a closed loop with an H ∞ controller. It will not be possible to compensate exactly this zero in closed loop, because the controller itself will be unstable. In this case, the control requires specific solutions that preserve control quality, meanwhile avoiding controller instabilities. An analysis of left-halfplane complex conjugate zero pairs similar to the one carried VOLUME 4, 2016 out in Section IV allows to conclude that there is an another dynamic effect (e.g., oscillation) exhibited in ∆U P CC in response to the control input variation ∆I ref rd or ∆I ref rq . The design procedure of an H ∞ controller hereafter proposed will take into account influences of these dynamic effects.

3) Control Configuration in the P − K Form
Multi-variable H ∞ control design for the linear system (25) is cast into the formalism in Fig. 8. A linear approach using the small-signal state-space model of the system is here adopted. In this case, focus is on rejecting disturbance due to the direct and quadrature component variations of the load and PV system output currents. Moreover, it is supposed that the measurement noise is relatively small. The tracking problem is not considered and therefore the S/KS mixedsensitivity optimization problem in the standard regulation form must typically be solved [27]. The inclusion of KS as a mechanism for limiting the size and bandwidth of the H ∞ controller, and hence the control energy used, is also important. The key of this design method is the appropriate definition of weighting functions W perf (s) to guarantee the performance specifications and weighting functions W u (s) to translate the constraints imposed to the control inputs.
The generalized plant P has five inputs, namely, the direct and quadrature components of the load and PV system output current variations, ∆I loadd − ∆I P V d and ∆I loadq − ∆I P V q respectively, acting as disturbance inputs ∆w and the current reference variations ∆I ref s , ∆I ref rd , and ∆I ref rq , which are the control inputs ∆u. The measured output vector ∆y is composed of the DC-bus voltage variation ∆V dc and the direct and quadrature components of the PCC voltage variation, ∆V P CCd and ∆V P CCq respectively. The desired performances are expressed in the form of weighting functions on the chosen performance outputs. ∆V dc , ∆V P CCd , ∆V P CCq , as well as ∆I ref

Plant together with weighting functions
Multi-variable fullorder controller are chosen as performance outputs. Their vector is noted ∆z in Fig. 8. The S/KS mixed-sensitivity optimization of the H ∞ control framework consists in finding a stabilizing controller which minimizes the norm W perf S W u KS ∞ .

4) Weighting Functions Selection
Performance specifications on the direct and quadrature components of the PCC voltage variation, ∆V P CCd and ∆V P CCq respectively, in response to a load reactive power step disturbance, illustrated in Fig. 9 and Fig. 10 respectively, are deduced from the grid code requirements on the PCC line-to-line voltage magnitude variation ∆U P CC , where U P CC = V 2 P CCd + V 2 P CCq . It should be noted here that since V P CCq ≪ V P CCd , meaning that U P CC ≈ V P CCd or ∆U P CC ≈ ∆V P CCd , therefore the performance template for ∆V P CCd is chosen to be quasi-identical to that for ∆U P CC .
The choice of linear time-invariant weighting functions is the key to deal with the performance requirements. The DCbus voltage variation ∆V dc and the direct and quadrature components of the PCC voltage variation, ∆V P CCd and ∆V P CCq respectively, are bounded by first-order weighting functions W perf (s) of the following form [27] 1 The function 1/W perf (s) can be representative of timedomain response specifications, where the high-frequency gain M s has an impact on the system overshoot, whereas the desired response time is tuned by the cut-off frequency ω b and the steady-state error can be limited by appropriate choice of the low-frequency gain A ε [29]. In order to satisfy the tight control requirement at low frequencies in the presence of the right-half-plane zeros of the plant, the selection of the value ω b of the function 1/W perf (s) is of great importance. Let G (s) be a multiple input multiple output (MIMO) plant with one right-half-plane zero at s = z and W perf (s) be a scalar weight, then closed-loop stability is ensured only if ∥W perf (s) S (s)∥ ∞ ≥ |W perf (z)| [27].
For the design problem, if the H ∞ controller meets the requirements, then ∥W perf (s) S (s)∥ ∞ ≤ 1. Therefore, with W perf (s) being given in (26), a necessary condition for closed-loop stability is |W perf (z)| ≤ 1, which corresponds to [27] z Consider the case when z is real and positive, then (27) is equivalent to The parameters of the weighting functions W perf (s) and W u (s) are given in Table 5 and Table 6 respectively [23], where 1/T 0s and 1/T 0 rdq are ∆I s and ∆I rdq inner imposed closed-loop bandwidths, respectively. Let us note here that the selection of the values ω b2 and ω b3 of the functions 1/W perf2 (s) and 1/W perf3 (s) respectively satisfies the condition (28), which means that closed-loop stability is implicitly ensured in the presence of the right-half-plane zero z 4 of the open-loop transfer function ∆V P CCd /∆I ref rq

5) H∞ Controller Synthesis
The robust control toolbox in the MATLAB ® software environment is used to design a full-order H ∞ controller based on the system modelling and the selected weighting functions. The obtained result corresponds to the minimization of the norm Design procedure may yield unstable controllers. Therefore, the value γ min must be slightly increased in this case to relax the constraint on the control and lead to stable controllers.  to be approximately equal to zero, i.e., ∆I loadd ≈ 0, which corresponds to a load step change of + 5% of the load rated reactive power (i.e., + 45 kVAr). The supercapacitor voltage V sc , initially regarded as a time-invariant parameter in the H ∞ controller synthesis, is considered now as a time-variant one, i.e., ∆V sc ̸ = 0 (its dynamic equationV sc = f (V sc , I s ) being taken into account in the simulation models). Fig. 11, Fig. 12, Fig. 13, and Fig. 14 present a comparison of topological and averaged modelling for the time-domain responses of the PCC line-to-line voltage magnitude U P CC , the direct and quadrature components of the PCC voltage V P CCd and V P CCq , and the DC-bus voltage V dc respectively. The results show a good concordance with both of these models despite high-frequency oscillations seen in the topological model. Therefore, the averaged models themselves are validated when compared to the topological model results. It can be observed from Fig. 12 and Fig. 13 that  the desired time-domain performances of V P CCd and V P CCq are successfully achieved in terms of overshoot, response time, and steady-state error with respect to the choice of the weighting functions W perf2 (s) and W perf3 (s) respectively. Fig. 11 shows that the dynamic performance specification imposed on U P CC in Fig. 2 is well respected. The system is also always stable. It should be noted here that, since focus is on primary voltage control only, the PCC voltage error cancellation cannot be obtained, which means that the PCC voltage cannot be brought back to its initial steady-state value after the disturbance. It can be seen from Fig. 14 that the desired time-domain performances of V dc corresponding to the tuning of the weighting function W perf1 (s) are satisfied. Hence, the synthesized H ∞ controller guarantees the desired performance specifications. Fig. 15, Fig. 16, and Fig. 17 compare the time-domain responses of the supercapacitor output current I s and the direct and quadrature components of the inverter output current I rd and I rq , respectively, obtained with both topological and averaged models. The results show a good agreement between these models, except for high-frequency oscillations observed in the topological model. Therefore, the averaged models precisely represent the dynamics of the ESS in that frequency bandwidth that is interesting for control purposes. Moreover, it can be seen that the admissible limit is guaranteed for these currents, which means that their imposed dynamic performance specifications are satisfied. For the sake of simplicity and without loss of generality, the nonlinear averaged model will from now on be used for time-domain simulation.
The time-domain responses of the active power variations of the sources and load are illustrated in Fig. 18. It can be observed that, due to large variations in U P CC , V P CCd , and V P CCq within a 10-ms interval consequent to the disturbance, the active power of the diesel engine generator system output P g and the load P load fluctuate considerably, however they return nearly to their initial values after 10 ms of disturbance; whereas the output active power of the ESS step-up transformer P t and the PV system P P V present slight fluctuations around their initial values within this interval. Fig. 19 shows the time-domain responses of the reactive power variations of the sources and load. In a similar way, one can see that the reactive power of the diesel engine generator system output Q g and the load Q load present dramatic fluctuations resulting from large variations in U P CC , V P CCd , and V P CCq within a 10-ms interval subsequent to the disturbance; whereas the output reactive power of the ESS step-up transformer Q t and the PV system Q P V fluctuate slightly around their initial values within this interval. Moreover, the ESS participation in primary voltage control has reduced the reactive power variation of the diesel generator around the steady-state point after the disturbance, as shown in Fig. 19.  Reactive power (kVAr) Time-domain response of the reactive power variations ESS step-up transformer output : ∆Q t Diesel engine generator system output : ∆Q g + U 2 P CC − U 2 P CCe C l ω grid PV system output : ∆Q P V Aggregated static load : ∆Q load + 5% load step (+ 45 kVAr)

VI. ROBUST PERFORMANCE ANALYSIS
This section details a robust performance analysis of the H ∞ controller, which was designed in the previous section while the system parameters are fixed at their rated values. First, the uncertainties in the steady-state value of the supercapacitor state of charge SoC sce (or supercapacitor voltage V sce ) and the length of the transmission line l connecting the diesel engine generator to the PCC are considered. Then, a sensitivity analysis is performed so as to determine the maximum variation ranges of the values SoC sce (or V sce ) and l for which the imposed closed-loop performances are guaranteed. Next, a series of MATLAB ® /Simulink ® closedloop time-domain simulations are presented to validate the controller robustness and performance in the presence of various load disturbances and the uncertainties in SoC sce and l.

A. PARAMETRIC UNCERTAINTIES
In the case of the studied hybrid system, uncertainties could be associated first with the parameters of the ESS but could also be representative of the variation in the steadystate value of the supercapacitor state of charge SoC sce (or supercapacitor voltage V sce ). They could also represent changes in the parameters of the diesel engine generator system (e.g., inductor L tg and resistor R tg of the stepup transformer; inductor L l , resistor R l , capacitor C l , and parallel resistor R lp of the transmission line). This results in changes in the steady-state operating point of the system. Let us note that only the uncertainties in SoC sce (or V sce ) and L l , R l , C l , and R lp (i.e., due to the change in the length of the transmission line l) are taken into account in the below sensitivity analysis. In order to ensure reliable, efficient, and safe operation of the supercapacitor and prolong its lifespan, an admissible range of [25,100] % should practically be required for SoC sc . The limited variation range, corroborated  with the well-known equation for SoC estimation, points out that the V sc value must be controlled between 390 V and 780 V. The midrange between these two values, i.e., V sc = 585 V, is a suitable choice for designing the nominal H ∞ controller whose robustness is further assessed. Moreover, the minimum value of the transmission line length l = l min = 1 km is chosen for H ∞ control design, where 10 km is the maximum value of l. This maximum value is limited at 10 km so as to ensure the voltage drop on the transmission line at the high-voltage level of 20 kV being less than or equal to the admissible limit of 8% of the rated voltage.

B. SENSITIVITY ANALYSIS
This subsection is devoted to the sensitivity analysis of robust performance of the synthesized H ∞ controller to answer the question whether the closed-loop system remains robust or not (from a performance point of view) to a given  parametric uncertainty level in the steady-state value of the supercapacitor voltage V sce around its design value, 585 V, or to the variation in the transmission line length l from 1 km to 10 km. Without loss of generality, MATLAB ® /Simulink ® closedloop time-domain simulations using the nonlinear averaged model are performed with a load step change of + 5% of the quadrature component of the load rated current I loadqe (i.e., −2.25 A) at t = 1 s, whereas the direct component of the load current variation is regarded to be approximately equal to zero, i.e., ∆I loadd ≈ 0, which corresponds to a load step change of + 5% of the load rated reactive power (i.e., + 45 kVAr). The supercapacitor voltage V sc appears as a timevariant parameter (i.e., ∆V sc ̸ = 0) in the simulation model. The initial value of SoC sc is varied between 25% and 100% in simulation. The length of the transmission line l is changed between 1 km and 10 km. The time-domain responses of the supercapacitor output current I s and the direct and quadrature components of the inverter output current I rd and I rq taking into account the uncertainty in V sce (or SoC sce ) are given in Fig. 24, Fig. 25, and Fig. 26, respectively. Similarly, as depicted in the aforementioned figures, the admissible limit is ensured for these currents with respect to the choice of the weighting functions W u (s), which means that their imposed dynamic performances are fulfilled.

2) Uncertainty in the Transmission Line Length l
The time-domain responses of the PCC line-to-line voltage magnitude U P CC , the direct and quadrature components of the PCC voltage V P CCd and V P CCq , and the DC-bus voltage V dc , taking into account the uncertainty in l, where the steady-state value of the supercapacitor voltage is chosen at V sce = 585 V (or supercapacitor state of charge SoC sce = 0.56), are illustrated in Fig. 27, Fig. 28, Fig. 29, and Fig. 30  of U P CC , V P CCd , V P CCq , and V dc , corresponding to the tuning of the weighting functions W perf (s), are maintained irrespective of the value of l. Therefore, the designed H ∞ controller is demonstrated to be robust in performance to l ∈ [1, 10] km. Let us remark here that the longer the length of the transmission line l is, the more damped the oscillations of U P CC , V P CCd , and V P CCq are, as shown in Fig. 27, Fig. 28, and Fig. 29, respectively. This can be explained due to the fact that if the length of the transmission line is increased, its resistance value is therefore higher, which induces a rise in the damping coefficient of the system. Fig. 31, Fig. 32, and Fig. 33 show the time-domain responses of the supercapacitor output current I s and the direct and quadrature components of the inverter output current I rd and I rq , respectively, taking into account the uncertainty in l. Similarly, as shown in the aforementioned figures, the admissible limit is guaranteed for these currents corresponding to the tuning of the weighting  functions W u (s), which means that their imposed dynamic performances are satisfied. The time-domain responses of the active power variations of the ESS step-up transformer output ∆P t and the diesel engine generator system output ∆P g − U 2 P CC − U 2 P CCe /R lp are depicted in Fig. 34 and Fig. 35, respectively. It can be seen that, due to large oscillations in U P CC , V P CCd , and V P CCq within a 10-ms interval in response to the disturbance, as shown in Fig. 27, Fig. 28, and Fig. 29, respectively, there is a dramatic fluctuation in the active power of the diesel engine generator system output P g , nevertheless it returns nearly to its initial value after 10 ms of disturbance; whereas the active power of the ESS stepup transformer output P t presents a slight fluctuation around its initial value within this interval. Fig. 36 and Fig. 37 depict the time-domain responses of the reactive power variations of the ESS step-up transformer output ∆Q t and the diesel engine generator system output  ∆Q g + U 2 P CC − U 2 P CCe C l ω grid . In a similar way, one can see that the reactive power of the diesel engine generator system output Q g presents a considerable fluctuation which is caused by large oscillations in U P CC , V P CCd , and V P CCq within a 10-ms interval subsequent to the disturbance, as shown in Fig. 27, Fig. 28, and Fig. 29, respectively; whereas the reactive power of the ESS step-up transformer output Q t fluctuates slightly around its initial value within this interval. In addition, the ESS participation in primary voltage control has decreased the reactive power variation of the diesel generator around the steady-state point after the disturbance, as seen in Fig. 36 and Fig. 37. It should be noted here that the longer the length of the transmission line l is, the higher the reactive power variation of the ESS stepup transformer output ∆Q t is and the lower the reactive power variation of the diesel engine generator system output ∆Q g + U 2 P CC − U 2 P CCe C l ω grid is, around their initial values after the disturbance.

VII. CONCLUSION AND PERSPECTIVES
In this paper, we have presented voltage-control-oriented modelling for the diesel-PV-storage hybrid power generation system operating in stand-alone mode. In particular, topological, as well as nonlinear and linear averaged models of each subsystem have first been provided, then a voltage-control-oriented model of the whole system has been obtained by aggregating these sub-models. We have also studied the influence of system parameters on the dynamic behavior of open-loop system measured outputs by means of a stability analysis method based on Monte Carlo simulation. Next, a voltage robust control design approach based on a cascaded two-level control structure -where classical PI-based current tracking controllers are placed on the low control level and receive references from an H ∞ -control-based upper level -has been developed in order to satisfy the required dynamic specifications. Effectiveness of the proposed voltage robust control strategy has been validated via MATLAB ® /Simulink ® closed-loop time-domain simulations. Finally, we have performed a sensitivity analysis of robust performance of the synthesized H ∞ controller to a given parametric uncertainty level in the steady-state value of the supercapacitor state of charge SoC sce (or supercapacitor voltage V sce ), as well as to a given variation in the transmission line length l connecting the diesel engine generator to the PCC. Through a series of MATLAB ® /Simulink ® closed-loop time-domain simulations in the presence of various load disturbances, it has been pointed out that the synthesized H ∞ controller remains robust in performance to a large uncertainty in the initial value of SoC sc (in particular SoC sce ∈ [25, 100] %), as well as to a large uncertainty in l (in particular l ∈ [1, 10] km).
The main prospect of this paper is to design a coordinated robust control strategy of the ESS, the diesel engine generator, the PV system, and the demand-side management   DELPHINE RIU received the Eng. and Ph.D. degrees in electrical engineering from the Grenoble Institute of Technology, Grenoble, France, in 1999 and 2002, respectively. She is currently a Professor with the Grenoble Institute of Technology. Her current research interests include design and control of embedded electrical systems or microgrids, including production with storage devices. VOLUME 4, 2016