Martin Legry
email: martin.legry@ensam.eu

Frédéric Colas

Christophe Saudemont

Jean-Yves Dieulot

Non-Linear Primary Control Mapping for Droop-Like Behavior of Microgrid Systems Martin Legry , Member, IEEE, Jean-Yves Dieulot , Frédéric Colas , Member, IEEE, Christophe Saudemont , and Olivier Ducarme Abstract-Interconnecting microgrids in LV power system presents appealing features such as self-healing or power quality. When networked microgrids are not connected to a strong utility grid, their Point of Common Coupling (PCC) voltage and their power reserves vary with the operating point. An external droop control architecture is proposed that allows active and reactive power sharing among the different microgrids, thereby stabilizing the system frequency and PCC voltage, and the maximum achievable droop gains are supplied. Next, the design of appropriate primary controllers for the Distributed Energy Resources inside each microgrid allows to achieve a specified aggregated external droop controller at the connection point with little communication requirements. This methodology is applied to a modified CIGRE benchmark and shows good results while keeping a standard decentralized control architecture.

Index Terms-Networked microgrids, DER primary control, nonlinear primary control, PCC doop control. Weighting factors.

NOMENCLATURE

I. INTRODUCTION

M ICROGRIDS are considered as a way to increase the penetration of renewable Distributed Energy Resources (DER) [START_REF] Lasseter | MicroGrids[END_REF]. They are able to work in islanded mode or supporting a distribution grid. The traditional hierarchy of controllers in a standard microgrid has three layers [START_REF] Bidram | Hierarchical structure of microgrids control system[END_REF]. The primary control, generally droop-based, is designed to stabilize the grid frequency and voltage by using only local measurements and local DER controllers. The secondary control of the microgrid is an Energy Management System which yields the references of the primary control and is commonly achieved by optimization-based algorithms. The purpose of tertiary control is to manage the power flow between the microgrid and the utility grid and coordinate, whenever appropriate, the operation of grid-connected microgrids. The interconnection of several microgrids to form a networked microgrid system presents appealing features such as self-healing capabilities, load control or power quality to the distribution grids [START_REF] Lasseter | Smart distribution: Coupled microgrids[END_REF]. However, the interactions and cooperation between microgrids or more generally clusters with DER and local intelligence in such a system make think microgrid control anew [START_REF] Alam | Networked microgrids: State-of-the-art and future perspectives[END_REF]. Cooperation and tertiary control may enable market bidding and energy-based transactions. It has been largely covered by the literature as a reformulation of the classical unit commitment (UC) or economic dispatch (ED) optimization problem, see for example [START_REF] Tsikalakis | Centralized control for optimizing microgrids operation[END_REF], [START_REF] Nguyen | Optimal bidding strategy for microgrids considering renewable energy and building thermal dynamics[END_REF], [START_REF] Hawkes | Modelling high level system design and unit commitment for a microgrid[END_REF] and the references therein. Unlike the case of the connection to a strong grid which is wellcovered in the literature, the problem is to find how power can be shared among networked weak microgrids. Primary control and automatic control of the power flow through tie lines are of major concerns. This task can be completed by back-toback converters [START_REF] Zhou | A microgrid cluster structure and its autonomous coordination control strategy[END_REF] or smart transformers [START_REF] De Carne | Real-time primary frequency regulation using load power control by smart transformers[END_REF], [START_REF] Hossain | Design of robust distributed control for interconnected microgrids[END_REF], [START_REF] Majumder | Power management and power flow control with back-to-back converters in a utility connected microgrid[END_REF] at the expense of microgrids cost and converters losses. However, for networked microgrids, one can use static switches and synchronization routines at the price of weak flexibility to manage the interface which is the alternative this paper focuses on. Nonetheless, creating a networked microgrid system may lead to changes in power flows and increase losses and the cost of energy. References [START_REF] Pashajavid | A multimode supervisory control scheme for coupling remote droop-regulated microgrids[END_REF] and [START_REF] Shahnia | Primary control level of parallel distributed energy resources converters in system of multiple interconnected autonomous microgrids within self-healing networks[END_REF] proposed a routine to adapt the power sharing among supporting microgrids. The algorithm computes new droop coefficients, according to the mismatch between the total and desired flows of the tie-line, while ensuring the stability and the consistency of the decentralized local droops. Reference [START_REF] Zamora | Multi-layer architecture for voltage and frequency control in networked microgrids[END_REF] introduced an adaptive modified droop for controlling frequency, when connected through a tie-line converter or for networked microgrids. Still, no specific attention was given to the PCC behavior. Reference [START_REF] Zhang | Hierarchical MPC-based energy management and frequency regulation participation of a virtual power plant[END_REF] proposed a Model Predictive hierarchical controller which embeds a Virtual Power Plant (VPP) into the frequency control scheme. The primary control allocates the power demand between the different levers using an optimization routine, but the internal power system is neglected in the model. Reference [START_REF] Fooladivanda | Control of networked distributed energy resources in grid-connected AC microgrids[END_REF] designed a joint economic dispatch and power flow optimization which minimizes a reference tracking cost and providing a synthesis of the local control laws for each DER. However, the algorithm requires to receive an external signal from an independent operator and no implicit behavior is enforced at the PCC. Reference [START_REF] Pudjianto | Virtual power plant and system integration of distributed energy resources[END_REF] introduced the Technical VPP (TVPP) as an extension of the VPP concept that aggregates and models the characteristics of a system with DERs. Still, to the best of the authors' knowledge, no further publications deal with an automatic control of the primary control for TVPP. In the literature, nothing is proposed to enforce any specific power sharing policy at the microgrid PCC. One of the most promising methods aims to define local controls of Heating, Ventilation and Air-Conditioning (HVAC) units [START_REF] Wu | Hierarchical control of residential HVAC units for primary frequency regulation[END_REF]. The objective is to determine the ON/OFF triggers of HVAC according to frequency references. Under comfort and over actuation constraints, the algorithm results into a virtual P-f droop curve of HVAC aggregation. As for every direct load control modulation of active power, it is based on ON/OFF triggers that do not consider DER control. In order to obtain a specific behavior at the PCC and to ensure a power sharing policy, a first solution would be to consider a centralized supervision strategy based on an external signal. Instead, this paper proposes a hierarchical framework using load and generation flexibilities to optimally tune non linear local controllers in order to provide an automatic primary response at the PCC. The general principle is depicted in Fig. 1. The general capability of the microgrid is estimated (I), and based on global coordination and capabilities, the maximum achievable droop gains at the PCC are obtained (II). Finally, the primary control laws are determined to enforce the specific droop-like behavior at the PCC (III).

The resulting control structure exhibits the following advantages and novelties:

• No external signal is required at the PCC to enforce a microgrid primary controller; • The primary response at the PCC is optimally distributed among the flexibilities (DERs) considering their heterogeneities, constraints and limitations. The nonlinear primary control of the DERs requires only the PCC voltage and frequency signals, and is quite different from the conventional local droop architecture; • The external operator (DSO or coordinator) only requires to know the capabilities and the behavior enforced at the PCC, preserving data and operating privacy. The remainder of this paper is organized as follows. First, based on an extension of the literature on the flexibility aggregation, a routine to determine the global flexibility for different voltage is proposed in Section II which yields the microgrid capabilities and droop gains. Section III addresses the synthesis of the nonlinear mapping of the active and reactive power of each DER as an optimal problem. Finally, the proposed framework is validated by simulation in Section IV on a modified European LV distribution benchmark network.

The proposed methodology aims to support active and reactive powers at the PCC within the framework of primary control mechanism. Usually, this action takes place within few seconds, for at least few minutes. A recent update of the German grid codes requires a change in the reactive power set point to be reached in 6 to 60 seconds [START_REF]Technical Requirements for the Connection to and Parallel Operation With Low-Voltage Distribution Networks[END_REF]. The following developments consider a three-phase balanced network and assume the voltage to be a true sine wave, and therefore, the PCC distortions and power electronics non-linearities are not considered.

II. PCC BEHAVIOR OF NETWORKED MICROGRIDS

A. Flexibility Aggregation and Equivalent PQ Diagram

Microgrid operators must be able to quantify the reserves of the system in a similar manner that for a single inverter through a capability diagram. Therefore, based on the unitary capability diagrams, the routine presented in Fig. 2 aims to determine the aggregated capability at the PCC for each microgrid. Fortunately, a number of recent papers have addressed this point when a sub-part of a grid is connected to a distribution grid with a nonlinear formulation of the power flow problem [START_REF] Silva | Estimating the active and reactive power flexibility area at the TSO-DSO interface[END_REF] or based on a linearization of the Optimal Power Flow (OPF) equations [START_REF] Contreras | Improved assessment of the flexibility range of distribution grids using linear optimization[END_REF]. However, in Low Voltage and weak power systems, PCC voltages are highly fluctuating, and, in turn, any change in the power injections from the microgrid to the external network will induce in a change in the PCC voltage, due to the power sharing between the networked microgrids. This behavior is highlighted in Fig. 3 which presents a diagram for a simple 2-flexibility microgrid. Therefore, in networked microgrid systems, the aggregation of flexibilities should consider a wide range of PCC voltages and should be estimated in real time. Hence, the aggregation routines proposed in the aforementioned papers has been updated to the case where the PCC voltage varies. Specifically, the method proposed in [START_REF] Silva | Estimating the active and reactive power flexibility area at the TSO-DSO interface[END_REF] can be easily extended. First, an OPF estimates the maximum and minimum reachable active and reactive powers at the PCC. This results in four couples (P i , Q i ) with i ∈ {1, 2, 3, 4}. In a second step, an OPF minimizes αP i+1 pcc , and i ≥ 4, α ∈ {1, -1}.

The reactive power is constrained to the mean value of the reactive power between two successive points

Q i+1 pcc = 0.5 × (Q i pcc -Q i-1 pcc ).
Then, the minimum or maximum active power can be found respectively for α = 1 and α = -1. The algorithm iterates for each new pair of points until the distance between two successive points is less than a prescribed tolerance, and the routine is repeated for the full range of PCC voltages. The resulting figure is a diagram that maps the active and reactive power flexibilities according to the voltage at the PCC. The flexibilities are assumed to be ideal and characterized by a unitary PQ diagram (see Fig. 4) and their dynamics are discarded. These diagrams may present slight differences, such as for the wind turbine generators, or synchronous generators in which the excitation systems may reduce the reactive power capabilities [START_REF] Lund | Reactive power capability of a wind turbine with doubly fed induction generator[END_REF]. However, the diagrams are not significantly modified, and the proposed methodology remains valid as long as the diagrams are convex, which is generally the case. Type 1 represents an ideal energy storage system (ESS), constrained by its output current, type 2 represents a renewable system, able to de-rate its output power, type 3 models a conventional generator (see [START_REF]Modalités Du Contrôle De Performances Des Installations De Production Raccordées En haute Tension ( HTA ) Au réseau Public De Distribution Géré par ERDF[END_REF]Figs. 1 and 2]), constrained by its output current and by a minimal output active power, and type 4 models a flexible load. Finally, the modified algorithm is depicted in Fig. 2.

B. Architecture and Coordination of Networked Microgrids System

To discuss the architecture and the coordination technique between the microgrids is beyond the scope of this paper. Still, Fig. 5 exhibits the architecture considered in the following with a coordination agent that assesses the stability of the networked microgrid system to determine the necessary behavior at the different PCCs according to classical criteria (such as the N-1 situation). These gains can also be obtained through a different architecture, such as consensus-based [START_REF] Lu | Consensus-based droop control synthesis for multiple DICs in isolated micro-grids[END_REF]. In both cases, the specified behavior at the PCCs is assumed to be sufficient to ensure the stability of the whole networked microgrids system. The choice between conventional or opposite droop control at the PCC indeed depends on the R ext versus X ext ratio of the transmission line between the microgrid PCC and the external network. Numerous low voltage cables or lines favour the use of opposite droop control that relates the active power to the voltage deviation and the reactive power to the frequency deviation. However, [START_REF] Engler | Droop control in LV-grids[END_REF] and [START_REF] Hou | Conventional p -ω/qv droop control in highly resistive line of low-voltage converter-based AC microgrid[END_REF] pointed out that conventional droop relations are suitable for low voltage network due to the indirect coupling between the active power and the frequency and between the reactive power and the voltage magnitude. Finally, it is important to remark that the choice of the droop relations we want to enforce at the PCC only changes the relations and the virtual droop coefficients in the objective function. In the remainder it is assumed that the stability is ensured with a conventional droop law at the PCCs:

P pcc -P ref pcc = -k p pcc . ω -ω n , Q pcc -Q ref pcc = -k q pcc . V pcc -V n pcc .
(1)

C. Variable PCC Droop Control for Weak Networks

In this section a method to determine the maximum achievable droop gains k p pcc and k q pcc is proposed. These gains can be found by considering the active and reactive power margins at the PCC and the frequency and the voltage deviations. Voltage and frequency deviations must remain within a tolerance band of five percent (see [START_REF]Network Code on Requirements for Grid Connection of Generators[END_REF] type B). To support the droop gains computation, the following values are defined:

P + flex = max P pcc |Q pcc = Q 0 pcc -P 0 pcc , P - flex = min P pcc |Q pcc = Q 0 pcc -P 0 pcc , Q + flex = max Q pcc |P pcc = P 0 pcc -Q 0 pcc , Q - flex = min Q pcc |P pcc = P 0 pcc -Q 0 pcc , (2)
where P + flex and P - flex are the maximum active power that the microgrid is able to modulate, and (P pcc |Q pcc = Q 0 pcc ) denotes the active power P pcc when the reactive power at the PCC is equal to its initial value Q 0 pcc . The same holds for the reactive power. The arrows on Fig. 3 exhibit the power margins from an initial situation without any power exchange between the microgrid and the external network.

Therefore, the droop gains at the PCC are as follows:

k p+ pcc = P + flex ω -ω n , k p- pcc = P - flex ω n -ω . ( 3
)
In the remainder, without loss of generality, the droop is assumed to be symmetric, that is to say

k p pcc = min(k p + pcc , k p - pcc ), and k q pcc = min(k q + pcc , k q - pcc ).
In addition, conventional conditions for stability hold. The power sharing strategy that generates the networked microgrids droop gains is out of the scope of this paper; our method aims to calculate the maximum droop gain that can be actually achieved for a microgrid. We refer the interested readers to [START_REF] Simpson-Porco | Stability, power sharing, & distributed secondary control in droopcontrolled microgrids[END_REF] and [START_REF] Schiffer | Conditions for stability of droop-controlled inverter-based microgrids[END_REF] and the references therein for more in-depth details on droop stability.

Finally, the operating area defined by the droop control (1) is a rectangle as can be seen in Fig. 6. However, some small regions of the rectangle are outside of the capability diagram (hatched areas in Fig. 6) which means that the corresponding operating point cannot be reached. A solution would have been to reduce drastically the active and reactive power margins. This would generate a smaller rectangle that would fit into the capability diagram, thereby yielding lower maximum achievable droop coefficients. However, the methodology detailed in the next section proposes to handle this situation through the formulation of an objective function. Resulting operating region of a 2-flexibility microgrid with V pcc = 0.97 pu.

III. EXTERNAL DROOP CONTROL ACHIEVEMENT VIA DER LOCAL CONTROLLER SYNTHESIS

A. Structure of Local Control Laws

The local controllers of the flexibilities of a microgrid are the only degrees of freedom which enable to enforce a droop-like behavior at the PCC (3), which considers only the frequency and PCC voltage. Hence, a new methodology is required to determine the primary control laws of the DERs f i and g i , that will be based on these two variables, contrary to the traditional local droop controllers that use the local node voltages. The resulting mapping will be nonlinear because of the power flow equations:

P g i = P ref i + f i ω, V pcc , Q g i = Q ref i + g i ω, V pcc . ( 7
)
By combining the conventional power flow equations to the desired behavior at the PCC (4) and the control laws at the inverter nodes, the microgrid system can be represented by ( 4)- [START_REF] Nguyen | Optimal bidding strategy for microgrids considering renewable energy and building thermal dynamics[END_REF], as shown at the bottom of the page.

B. Optimal Tuning of DER Non-Linear Primary Control

The microgrid system presents numerous degrees of freedom thus, in order to determine suitable DER control laws, the system of ( 4)-( 6) is solved using Non Linear Programming (NLP) methods. We recall that the DERs nonlinear primary controllers should consider the frequency and the PCC voltage to enforce a droop-like controller at the PCC. As the frequency ω is a global variable shared by all nodes, the only variable that needs to be communicated to the controllers is the PCC voltage.

The following section presents the objective function and constraints of the NLP.

Objective function: The function to be minimized should consider two main objectives. The first one consists to supply the expected active and reactive power injections at the PCC according to [START_REF] Lasseter | MicroGrids[END_REF]. It is important to note that the method proposed to define the droop coefficient may lead to unreachable states as previously detailed. Thus, the objective function embeds a least square problem with different weighting factors λ p and λ q for the active and reactive power deviation respectively, which also help to prioritize the provision of voltage or frequency support. The second objective aims to share the effort proportionally among the levers. This strategy avoids to request power only from the levers which exhibit a strong coupling with the PCC. Moreover, the EMS yields an economic optimum for the steady state, for which the criterion embeds a weighted sum of the powers quadratic deviations. Hence, in general, proportional power sharing avoids large deviations from a single lever and keeps the operating point close to the economic optimum.

This results in the following objective function

J ω, V pcc = λ p P 0 pcc -k p pcc ω 2 + λ q Q 0 pcc -k q pcc V pcc 2 + i∈R d P i P n i S base -P pcc 2 + i∈R d Q i Q n i S base -Q pcc 2 , (8)
with P i and Q i the active and reactive power changes of the piloted node i ∈ R d .

Constraints: The constraints of the NLP are the conventional constraints of OPF, plus the flexibility diagrams as detailed in the remainder of this section.

P ref pcc -k p pcc • ω -ω n = V pcc j∈N V j G pccj cos θ pcc -θ j + B pccj sin θ pcc -θ j Q ref pcc -k q pcc • V pcc -V n pcc = V pcc j∈N V j G pccj sin θ pcc -θ j -B pccj cos θ pcc -θ j (4) ∀i ∈ R d ⎧ ⎪ ⎨ ⎪ ⎩ P ref i + f i ω, V pcc = V i j∈N V j G ij cos θ i -θ j + B ij sin θ i -θ j Q ref i + g i ω, V pcc = V i j∈N V j G ij sin θ i -θ j -B ij cos θ i -θ j (5) ∀i ∈ {R -R d } ⎧ ⎪ ⎨ ⎪ ⎩ P k = V k j∈N V j G kj cos θ k -θ j + B kj sin θ k -θ j Q k = V k j∈N V j G kj sin θ k -θ j -B kj cos θ k -θ j ( 6
)
• Power flow model: The optimization problem considers the power flow equations ( 4)-( 6) as a set of constraints. The precise model of the network with the admittance matrix Y considers the complex nature of the internal lines and their R versus X ratio are therefore intrinsically considered in the resulting local control laws. • Nodal power balance:

P i = P g i -P l i , ∀i ∈ R, Q i = Q g i -Q l i , ∀i ∈ R. (9)
• Voltage constraints:

V i ≤ V i ≤ V i . ( 10
)
• Unitary capability constraints: Four types of flexibility diagrams are considered as shown in Fig. 3. Type 1:

P i ≤ P g i ≤ P i , P g i 2 + Q g i 2 ≤ S i . ( 11
)
Type 2:

0 ≤ P g i ≤ P g0 i , -0.36 ≤ Q g i ≤ 0.36, -0.36 × P tr i -P g i P tr i ≤ Q g i ≤ 0.36 × P tr i -P g i P tr i , (12)
with P tr i , the minimal active power to provide full reactive power support. Type 3:

δ i P i ≤ P g i ≤ δ i P i , P g i 2 + Q g i 2 ≤ δ i P i , ( 13
)
with P i and P i , the maximal and minimal output active power of the generator when started, and δ i a Boolean which states whether the diesel generator is running or not. However, the start up or shut down of a type 3 flexibility is not considered. Type 4:

Q i = P i cos(φ i ). ( 14
)
Last, there may be generating units for which the local control laws cannot be updated such as droop-controlled inverters. For these units, an additional set of constraints accounts for their behavior, as

P i = P ref i + k p i ω, Q i = Q ref i + k q i V i . ( 15
)
Finally, the optimization problem can be summarized as:

min P i , Q i ,i∈R d J ω, V pcc s. t. (4) -(15) (16)
The overall flowchart is depicted in Fig. 7.

IV. SIMULATION RESULTS AND DISCUSSION

The simulations use a modified version of the CIGRE European LV residential distribution network benchmark [START_REF] Strunz | Benchmark systems for network integration of renewable energy resources[END_REF], which parameters are detailed in Table II-V. The network described in Fig. 9 includes two storage systems (type 1, see Fig. 4), two photo-voltaic (PV) plants (type 2) and a conventional generator (type 3). From now on, a positive power indicates a generation, while a load is labeled as negative. The external network is considered as a load at the PCC (see Fig. 9).

The initial operating point of the microgrid is given in Table V. Following the methodology summarized in Fig. 7, Fig. 9. Microgrid topology -Flexibilities and resulting primary mapping P -{ω, V pcc } and Q -{ω, V pcc }. the capability diagram at the PCC (Fig. 8) can be inferred from the flexibility diagrams of the DERs.

The shape of the capability diagram depends mainly on the grid topology and the DERs flexibilities, but is quite robust with respect to the initial operating point. Equation (2) gives the power margins P = 1.75 pu and Q = 3.43 pu corresponding to the rectangle centered on P 0 pcc = 0.45 pu, Q 0 pcc = 0 pu. The maximum achievable droop coefficients k p pcc = 34.6 and k q pcc = 68.6 are calculated from [START_REF] Lasseter | Smart distribution: Coupled microgrids[END_REF]. The results presented in this section consider the maximum droop coefficients. the corresponding active power injection deviates significantly from the expected linear PCC droop control law, while the reactive power injection is barely affected. This situation occurs for under frequency and extreme PCC voltages (left part of Fig. 8).

A. Complete Control of DERs

Fig. 9 also displays the nonlinear primary control mappings of the DERs, depending on V pcc and ω. The diagrams display the active and reactive power injections scaled with a per unit PCC power. As a general rule, we can see that the control effort is shared proportionally among the levers. Each nonlinear control mapping can be interpreted consistently with the characteristics of the LV network and the operating constraints. A first remark is that the network is mainly resistive ( X R < 1). Hence, only a limited number of DERs exhibit a strong coupling between their active powers and the grid frequency. Specifically, an important share of the required active power at the PCC is allocated to node 2, and the corresponding diagrams are rather similar. This induces a strong constraint on the reactive power at node 2, which in turn drives to distribute the reactive power at the PCC among the other levers. As a result, the reactive power diagrams at nodes 6, 8, 11 and 15 exhibit a shape which is rather similar to that of the PCC reactive power diagram.

A number of saturation constraints explain the behavior at the extreme areas of some diagrams. At node 2, the reactive power is limited by the type 1 diagram constraints, which enlightens the behavior for under frequency. Conversely, the type 2 diagram at nodes 8 and 11 constrains the reactive power generation. The generator at node 15 cannot be shut down and the active power is nonzero even for frequencies over 1.04 pu. Finally, the microgrid voltage constraints limits the reactive power generation for the solar panels when the PCC voltage is close to 1 pu. and extreme frequencies (note that the PV active power is not curtailed). When the operating point is outside of the PCC capability diagram, it is interesting to focus on the impact of λ p and λ q on the active and reactive power provision. λ p and λ q weigh respectively the priority to frequency or voltage support in the criterion defined in [START_REF] Zhou | A microgrid cluster structure and its autonomous coordination control strategy[END_REF]. Table I presents the active and reactive power injections at the PCC for different ratio λ q λ p and a specified operating point P pcc = 2.189 pu, Q pcc = 2.678 pu which lies outside the diagram. It can be seen that a high ratio (higher than 10) generates a small reactive and a high active power error. The converse holds for a small ratio. A ratio of 1 minimizes the error on the apparent power.

B. Inclusion of Non-Controlled DER

In the first test case, each of the flexibilities were entirely controlled both in active and in reactive power injections. In this second test case, the reactive power support at node 8 is assumed to be controlled by a linear local control law, namely a reactive power droop. The local droop coefficient can be inferred from the reactive power capacity of the PV inverters (±0.36 pu, see Fig. 4) and results in a linear gain of 7.2.

Fig. 12 presents the resulting behavior of the PCC. On overall, the behavior remains linear, as expected for a droop-like PCC. The changes appear on the reactive power injection of the PV at node 6. Fig. 13 presents the voltage node 8 (left) and the resulting reactive power injection (right), according to the specified local droop gain. It is interesting to point out that the voltage at node 6 resulting from the optimization routine is mostly dependent of the system frequency and of the active power injections within the micro-grid. Last, the corresponding reactive power is modulated based on the local voltage which does not represent the PCC voltage. This effect can be explained by the low X R ratio and the inverse coupling, compared to conventional power system, for which the droop mechanism has been established. This highlights the inability of usual droop relations to enforce accurately the specified behavior at the PCC.

V. CONCLUSION

A new external droop algorithm has been proposed to achieve the primary control of networked multi-microgrids with DERs. This hierarchical framework only the frequency and PCC voltage signals. The PCC capability diagrams, which vary with the PCC voltage can be found from the grid model and DERs capability diagrams, and the maximum achievable droop gains are computed using power margins. Next, an OPF generates new nonlinear primary control laws of the DERs which achieve the desired behavior at the PCC, while sharing fairly the control effort among the levers. The results obtained with a modified CIGRE benchmark are consistent with intuitive rules.

Further works will focus on the case of a microgrid connected through more than one PCC to one or more external power systems. The estimation of the microgrid capabilities at the PCC can be extended to the case of a microgrid with two or more PCCs. The objective function can be upgraded and embeds weighting factors which will correspond to an allocation of the flexibilities between the different PCCs. Once the PCC PQ diagrams are obtained, the last step in which the local control laws are computed remains the same as in this paper.

The proposed algorithm can consider the uncertainties in both steps of the methodology that is the estimation of the microgrid capabilities at the PCC and the tuning of the local control laws. First, regarding the estimation of the microgrid capabilities at the PCC, the uncertainties can directly be handled by embedding the probability density functions in the optimization routine. This results in an additional dimension of the capability diagram (that is, a three dimensional diagram) which captures the probability of the microgrid to be able to modulate its active and reactive power flows. Second, addressing the uncertainties in the computation of the local control laws requires to change the optimization problem. A linear microgrid network should be considered in order to ensure the global convergence of a confidence level optimization problem as proposed in [START_REF] Buire | Confidence level optimization of DG piecewise affine controllers in distribution grids[END_REF]. The consideration of the uncertainties is a major concern for power system operations and represents the major perspectives of this work.

Fig. 1 .

 1 Fig. 1. General synoptic of droop-like behaviour of the PCC by optimally tuning local non linear primary controllers -(1) flexibility aggregation, (2) virtual PCC droop gains, (3) local control laws synthesis.

Fig. 2 .

 2 Fig. 2. Flexibility aggregation routine for variable PCC voltage.

Fig. 3 .

 3 Fig. 3. Capability diagram of a 2-flexibility microgrid.

Fig. 4 .

 4 Fig. 4. Unitary flexibility diagrams.

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Fig. 5 .

 5 Fig. 5. Global architecture and coordination of networked microgrids.

Fig. 6 .

 6 Fig.6.Resulting operating region of a 2-flexibility microgrid with V pcc = 0.97 pu.

Fig. 7 .

 7 Fig. 7. DER primary control design procedure.

Fig. 8 .

 8 Fig. 8. Flexibility diagram of the 18-node modified European LV benchmark network.

Fig. 10 .

 10 Fig. 10. Resulting active power behavior at the point of common coupling-P pcc -{ω, V pcc }.

Fig. 10

 10 Fig. 10 and 11 display respectively the P -{ω, V pcc } and Q -{ω, V pcc } mappings obtained by the optimal tuning method of Section III-B, which are almost linear. When the operating point is outside of the PCC capability diagram,

Fig. 11 .

 11 Fig. 11. Resulting reactive power behavior at the point of common coupling -Q pcc -{ω, V pcc }.

Fig. 13 .

 13 Fig. 13. Voltage node (left) and reactive power injection (right) at node 6 as function of the system frequency and PCC voltage3.

TABLE I INFLUENCE

 I OF λ p AND λ q ON THE PCC POWER INJECTIONS Fig.12. Resulting active and reactive power behavior at the point of common coupling -P pcc -{ω, V pcc } and Q pcc -{ω, V pcc }.

TABLE III DER PARAMETERS

 IIIPARAMETERS

Authorized licensed use limited to: UNIVERSITE DE LILLE. Downloaded on March 08,2022 at 09:24:44 UTC from IEEE Xplore. Restrictions apply.


Martin Legry
email: martin.legry@ensam.eu

Frédéric Colas

Christophe Saudemont

Jean-Yves Y Dieulot

Olivier Ducarme

Olivier Ducarme Mixed

Mixed integer quadratic programming receding horizon microgrid supervisor

I. INTRODUCTION

Microgrids can be defined as a set of distributed generators, renewables or conventional ones, storage systems and loads that can operate in a coordinated manner and possibly in islanded mode. In addition, their aim is to produce locally what is needed locally, by means of local controllers. Based on conventional power system representation, the microgrid community has developed three-level structure control supervisor (see Figure 1).

Tertiary Control Economical operation

Secondary Control

Nominal operation

Primary Control Power sharing

Fig. 1: Micro-grid control architecture

The primary control ensures at the faster layer the power sharing between all the active generators through well-known techniques such as classical droop control and its variants [START_REF] Han | Review of Active and Reactive Power Sharing Strategies in Hierarchical Controlled Microgrids[END_REF]. The secondary control aims to restore nominal operations of the microgrid following any fluctuation of the load or renewable sources while minimizing losses or the operating cost. Finally, the tertiary control mainly focuses on the economical optimization of the microgrid on a long-term basis. Secondary and tertiary control are also known as Power and Energy Management System (PMS and EMS) respectively. In the latter we refer to microgrid supervisor as the combination of secondary and tertiary control layers.

There are different tools and methods to achieve the supervisor, depending on whether an optimal solution or an approached solution is desired. Among the optimal control techniques, Model Predictive Control (MPC) is one the most promising. Using two-layer MPC supervisor fits the conventional hierarchical control of microgrids previously described. On the longer timescale, an economic optimization is achieved for at least one day ahead. Model and constraints of this layer only include the slow dynamics of the equipment and economic performances of each one. On the faster timescale the cost function embeds a trajectory tracking performance, and some objectives related to the microgrid operation, such as the losses minimization or the voltage and frequency deviations. Each layer embeds a detailed model of the dynamics of interest to optimize specific criteria. However, the convergence of each objective may not be ensured. Coordinating economic constraints with real time operations is crucial for microgrid control. The problem of the cooperation between multi-layer supervisors has been discussed in [START_REF] Zanoli | The importance of cooperation and consistency in two-layer Model Predictive Control[END_REF]. It has been handled by a two-layer MPC with an intermediate module that computes reachable states, constraints and target for the lower layer based on an economical criterion. The authors of [START_REF] Picasso | An MPC approach to the design of two-layer hierarchical control systems[END_REF] proposed a formulation of a two layer MPC and addressed the problem by determining a subset of control action feasible at the higher level. Then, the lower layer solves each of the tracking problems for each feasible control action and regulate the actuators. For microgrid application, the main focus is mainly motivated to provide a solution to stochastic and uncertain forecasts, as explained in [START_REF] Cominesi | Two-layer predictive control of a micro-grid including stochastic energy sources[END_REF] and [START_REF] Velarde | On the comparison of stochastic model predictive control strategies applied to a hydrogen-based microgrid[END_REF]. The first one is a survey on three methods to deal with the uncertainties of the renewable energy by using stochastic MPC. In [START_REF] Velarde | On the comparison of stochastic model predictive control strategies applied to a hydrogen-based microgrid[END_REF], it is proposed to consider such stochastic optimization within the lower level and a deterministic layer at the higher. However, the models and objective functions remain the same in each layer that is to say the power balance of the microgrid and a minimization of the operating costs. In [START_REF] Tatjewski | Advanced control and on-line process optimization in multilayer structures[END_REF], the author suggested to use a multi-layer structure so that each layer includes different model dynamics and objectives. Among the three proposed architectures, the most promising consists in a steady state optimization followed by a target correction at the lower layer side that can handle an additional task. The novelty in this paper is twofold. First, the Model Predictive Control (MPC) technique allows the supervisor to include a trajectory tracking problem into the objective function and thus to consider economic references. Second, the embedded model is able to predict the power injections at each node of the microgrid and, by upgrading the objective function, losses can be minimized.

The rest of this paper is organized as follows. In section II, the supervisor and more specifically the secondary control, its objectives and models will be detailed. To compare the conventional and novel supervisor, two simulations are presented in section III:

• Case 1: No losses minimization, the supervisor only follows the economical references, • Case 2: The supervisor is upgraded to minimize the losses. Finally, we will conclude on the possible improvements of this new supervisor.

II. MPC-BASED SUPERVISOR AND MICROGRID MODEL

A. Global supervisor architecture

The computation of optimal references and economic optimization of the microgrid are out of the scope of this paper, and we refer the interested reader to optimal control reviews such as [START_REF] Fathima | Optimization in microgrids with hybrid energy systems -A review[END_REF] or unit commitment problem under uncertainties reviews [START_REF] Van Ackooij | Large-scale unit commitment under uncertainty: an updated literature survey[END_REF], and finally the global architecture of the supervisor is depicted in Figure 2.

Energy Management System Provides set points for optimal economic dispatch

Power Management System

Provides set points to restore nominal operation

PCC

Grid connection management

Forecast

Provides forecast data and reserve estimations

B. Two-layer MPC-based supervisor

To develop the proposed supervisor, we used the wellknown Model Predictive Control technique which allows to capture the system future behaviour thanks to a prediction model and is able to minimize a multi-objective function with [START_REF] Tatjewski | Advanced control and on-line process optimization in multilayer structures[END_REF]). The model embedded in the supervisor is based on the assumptions that over the considered time scale the transient and high dynamics are not of interest, and that the system remains close to its equilibrium point. Hence, the microgrid system can be linearized and the resulting model formulated as a discrete incremental state space as follows:

x(k + 1) u(k) = A B 0 I x(k) u(k -1) + B I Δu(k) + D 0 Δd(k) y(k) = C 0 x(k) u(k -1)
in which x(k) is the state vector at time step k, u(k) the input vector at timestep k, Δu(k) the change of input references between (k -1) and (k), and Δd(k) the disturbance vector. Matrices A, B, D and C are obtained by the linearization of the equations of the microgrid system as detailed for the main equipment in the next section. As detailed in the introductory paragraph, the objective of the supervisor is to reach as most as possible the economic performances computed by the tertiary control. Several performance indexes can be chosen such as states of charge, produced energy or average power. In the rest of this paper we assume the upper layer uses state of charges references, and finally, the global objective function can be expressed as:

min Δu Nc k=1 α(ỹ -y * (k)) 2 + β(Δu(k)) 2 + λ(P los ) 2 (1)
in which α, β, λ are the weighting factors of the trajectory deviation, the uses of levers and the line losses resp., y (k) and y * (k) are the predicted and reference states for time k. It can be noticed that an additional objective is added in order to prevent saturation of the levers. Due to the difference of timescale between the secondary and tertiary control, a linear interpolation layer is introduced to define the references for the lower layer objective function. At last, the different levers which help to minimize the objective function will be detailed in the next sections with the modelling of the microgrid components.

C. Microgrid network model

The proposed supervisor embeds a modelling of the microgrid network in order to supervise the voltage nodes and the microgrid frequency. The traditional technique for quasi steady-state modelling of the network is based on the linearization of the power flow equations:

P i = V i j∈N V j |Y ij | cos(δ i -δ j -θ ij ) Q i = V i j∈N V j |Y ij | sin(δ i -δ j -θ ij ) (2)
in which, |V j |, |V i |, δ i , δ j are the voltage magnitudes and angles resp., |Y ij | and θ ij are the admittance matrix magnitude and angle resp. We recall that based on (3), we can derive and include the frequency in the linear model [START_REF] Cominesi | Two-layer predictive control of a micro-grid including stochastic energy sources[END_REF].

|Y ij | = 1 R 2 ij + ω 2 L 2 ij θ ij = tan -1 ωL ij R ij (3)
Thus, after linearization and calculation of the Jacobian matrix ((5), the power flow equations are modelled as follows:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ΔV 1 (k + 1) . . . ΔV i (k + 1) Δδ 2 (k + 1)
. . .

Δδ i (k + 1) Δω(k + 1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = J -1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ΔP * 1 (k) . . . ΔP i * (k) ΔQ 1 * (k) . . . ΔQ i * (k) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4)
with

J -1 = ⎡ ⎣ ∂Pi dV x(k) ∂Pi dδ x(k) ∂Pi dω x(k) ∂Qi dV x(k) ∂Qi dδ x(k) ∂Qi dω x(k) ⎤ ⎦ -1 (5) and thus, ⎧ ⎨ ⎩ V i (k + 1) = V i (k) + ΔV i (k) ω i (k + 1) = ω i (k) + Δω i (k) δ i (k + 1) = δ i (k) + Δδ i (k) (6)
The network characteristics are constrained to be within specified boundaries:

ω < ω(k + 1) < ω V < V i (k + 1) < V (7)
where X and X, are the maximum and minimum bounds of the variable X.

D. Power electronics interfaced generation

Based on the considered time scale, it is only relevant to capture the dynamics of the droop control. The conventional double-loop control (see [START_REF] Marwali | Control of Distributed Generation Systems Part I: Voltages and Currents Control[END_REF] for a detailed description) are then neglected, and we refer to [START_REF] Gutiérrez | Droop controller comparison for AC microgrids[END_REF] for a complete review of the droop techniques. The main dynamics of the droop technique are recalled in [START_REF] Van Ackooij | Large-scale unit commitment under uncertainty: an updated literature survey[END_REF]. We refer the set of droop-controlled nodes by N d , and for i ∈ N d , 7) directly introduces the levers of the droopcontrolled power electronic devices that are the active and reactive power references denoted by P genref i and Q genref i , respectively. Finally, with

P gen i (k + 1) -P gen ref i (k + 1) = -k pi .(ω(k + 1) -ω * ) Q gen i (k + 1) -Q gen ref i (k + 1) = -k qi .(V i (k + 1) -V i * ) (8) Equation (
P gen ref i (k + 1) = P gen ref i (k) + ΔP gen ref i (k) Q gen ref i (k + 1) = Q gen ref i (k) + ΔQ gen ref i (k) (9)
we obtain the following model of the droop-controlled inverters:

P gen i (k + 1) = P gen ref i (k) + ΔP gen ref i (k) -k pi . (ω(k + 1) -ω * ) Q gen i (k + 1) = Q gen ref i (k) + ΔQ gen ref i (k) -k qi . (V i (k + 1) -V i * ) (10)
Although we could also use the voltage magnitude and the system frequency as lever, in the remainder, we discarded this possibility. Equation ( 8) contains an algebraic relation between two states P gen i (k + 1) and ω(k + 1) that we can solve by reinjecting ( 6) into [START_REF] Gutiérrez | Droop controller comparison for AC microgrids[END_REF] the same hold for the reactive power and the voltage node. The last dynamics to include in the model is related to the storage capability and the state of charges. The storage devices are considered as ideal and can be represented as:

SoC i (k + 1) = SoC i (k) -η ch P i gen (k + 1).T s 60 (11a) SoC i (k + 1) = SoC i (k) -η disch P i gen (k + 1).T s 60 (11b)
depending on if P i (k + 1) < 0 (11a) or P i (k + 1) > 0 (11b) with the following constraints:

SoC < SoC(k + 1) < SoC P i gen < P b (k + 1) < P i gen E.

Conventional generation

Conventional generators, such as diesel or gas generators are modelled as controlled power sources. The output active power and the apparent power are bounded and can be modelled as:

P gen (k + 1) = P gen (k) + ΔP gen (k) (12)
P gen < P gen (k + 1) < P gen

P gen 2 + Q gen 2 < S max gen F.

Loads and renewable generations

For sake of simplicity, the renewables sources are modelled as negative loads. The supervisor will deal with the loads and renewables sources as known disturbances profiles from a forecast module. Finally, the optimization problem solved by the supervisor is formulated as a MIQP (Mixed Integer Quadratic Program):

min Δu Nc k=1 α( y(k) -y * (k)) 2 + β(Δu(k)) 2 + λ(P los ) 2 (13)
subject to the microgrid dynamics:

X(k + 1) = A.X(k) + B.ΔU (k) + C.ΔD(k) Y (k) = C.X(k) ( 14
)
where the matrices A, B, D and C are obtained based on the linear model of each equipment, and the vector X(k), ΔU (k) are defined as follows, and we denote the set of nodes, the set of droop-controlled nodes, and the set of controlled nodes by

N , N d , N p X(k) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ [SoC l∈N d ] T ω [V i∈N ] T [δ i∈N ] T [P i∈N ] T [Q i∈N ] T P gen ref j∈Np T Q gen ref j∈Np T ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ΔU (k) = ΔP gen ref j∈Np (k) ΔQ gen ref j∈Np (k)

III. SIMULATION RESULTS

The proposed supervisor has been implemented in MAT-LAB with the toolbox YALMIP [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF]. The solver used for the optimization problem is Gurobi. The model of the microgrid is depicted on Figure 10, and the specifications of the storage devices and PV systems are summarized in Table 2. The renewable and load forecasts for both simulation are also presented in Appendix. The reference case presents the microgrid without any physical representation of the microgrid lines. It represents the raw economic optimization and the state of Charge at the end of the day will serve as a reference to compare both supervisors. The results are presented in Figure 4. The final states of charge are 0.22 and 0.59.5 p.u for battery 1 and 2 respectively.

A. Case 1: Classical supervisor

The first simulation case only includes the tracking of the economical references. This simulation presents the conventional proposition for the microgrid secondary control. In this case, for every new economical reference, a power flow routine is computed to define the initial operating point and the matrices to generate the linear model embedded within the supervisor. The voltage set point is set to 1 p.u at the node 1. There is no change in references and the microgrid is only led by the initial set point from the economical optimization, Fig. 4: Economical optimal SoC trajectory unless voltage or frequency deviations exceed the limitations. This supervisor exhibits around 0.007 p.u of mean losses as shown in Figure 5. For the sake of clarity, we only present the voltage of the generators (batteries) and of the extremal nodes (nodes 15, 17, 18, cf. Figure 10). As expected, the voltage at the node 1 is close to 1 p.u, and the voltage and the extrema of the microgrid change up to 1.01 p.u when the renewable produces, and down to 0.97 p.u during the evening peak load. Figure 6 presents the daily power profile. At first, the battery Figure 6 presents the daily power profile. At first, the battery Fig. 5: Losses and voltage profile -case 1 1 is charged with the diesel generator (before 60 min) in order to store enough reserves in the storage devices for this day. Before 200 min, the power predictions are accurate enough so that the profile is very similar to the economical optimization expected values and only the battery at the node 1 supply the load. Fast fluctuations of the renewable 1 lead to a reaction of the droops of battery 1 and 2 as can be seen between 400 and 600 min. Finally, after 800 min, the state of charge of the battery 1 approaches its minimum and battery 2 is thus more called for. It can be noticed that the voltage is maintained within the specified limits (±0.05 p.u) and the frequency between 0.01 p.u. Finally, with this supervisor, the final states of charge are 0.218 and 0.563 p.u. The second simulation presents the results of the tracking of the economical references while minimizing the line losses. With this supervisor, the losses are reduced to 0.006 p.u. It is worth noting that the weight of the line losses is relatively low in this case. As expected, the supervisor tends to increase the voltage to its upper bound and it can be observed some rapid changes of the voltage of the nodes 16, 17 and 18 due to the misleading references from the economical optimization, and the correction applied by the supervisor. It can be noticed that the supervisor is able to react to fast fluctuations of the renewable in order to maintain the voltage within the specified boundaries. Furthermore, in the microgrid, the ratio X/R is relatively low, and thus, to reduce the losses, the supervisor tends to use the active power references, as it can be seen in Figure 10 with the fluctuations of the output powers of both batteries. This lead to an opposite effect to the objective which is desired with the economical references tracking. Finally, with this supervisor, the final states of charge are 0.201 and 0.610 p.u.

C. Comparison

Table 1 presents the main results of both supervisors compared to the predicted optimal references. It can be noticed that in term of total energy the proposed supervisor is closer to the EMS. This represents a difference of about 9 kWh between both supervisors, and no differences compared to the purely economic references. In the first simulation, the position within the microgrid of the reserves of ESS2 is not respected as, due to its positioning, the second battery has been used to reduce the losses. However, with the proposed supervisor, the ESS1 is used to minimize the losses and mitigate to the fluctuations. Those two simulations demonstrate that the proposed supervisor can efficiently reduce the line losses without deteriorating the economic performance of the microgrid supervisor. This is achieved at the cost of power fluctuations to correct the misleading economical references. Finally, the proposed supervisor is improves on the conventional one, but it is to be noted that the repartition of the available reserves is incorrect regarding what is expected from the economical point of view. This point may be a challenge if different storage devices exhibit different objectives, such as hydrogen storage for long term and lead-acid battery for instance for intra-day management. This point could be improved with a sensitivity analysis of the economical layer to improve the weighting factors used to minimize the losses.

IV. CONCLUSION

We proposed an improved supervisor that preserves the common three-layer structure of microgrid supervisor. It is able to tracks the economical references and takes advantage of the model embedded in the MPC to minimize the losses without performing a non-linear OPF at each time step. The simulations presented an improvement in the minimization of the losses at the cost of changes in active power references that may lead to errors in the economical tracking problem. The main perspectives are twofold. First, due to the linearization of the network model, the losses may not be correctly estimated. A comparison with successive optimal power flows may confirm this assumption. The computation time required to solve the non-linear optimal power flows. Secondly, the droop coefficient may also be used as a third lever to reduce the losses. It is expected that the closer to the fluctuations the storage devices, the higher the droop coefficient.

V. APPENDIX

Fig. 2 :

 2 Fig. 2: Micro-grid supervisor synoptic

Fig. 3 :

 3 Fig. 3: Model Predictive Controller synoptic

Fig. 6 : 1 Fig. 7 :

 617 Fig. 6: Power profile -case 1

Fig. 8 :

 8 Fig. 8: Power profile -case 2

Fig. 9 :

 9 Fig. 9: PV and load power Profiles

Fig. 10 :

 10 Fig.10: Modified CIGRE European LV network testbench[START_REF] Strunz | Benchmark Systems for Network Integration of Renewable Energy Resources[END_REF]

TABLE I :

 I Performance comparison

	Performances	EMS PMS1 PMS2
	State of charge [p.u]	ESS1 0.220 0.218	0.201
		ESS2 0.595 0.563	0.610
	Losses [p.u]	-	0.007	0.006

TABLE II :

 II DER ParametersTypeNode P nom[kW]

	ESS	1	100
	ESS	16	100
	PV	8	100
	PV	11	100
	Genset	5	100

,(((


domain: spi phys info

text:
Mateus Carvalho
email: carvalho@laplace.univ-tlse.fr

Cédric Raibaudo

Sébastien Cazin

Moïse Marchal

Gilles Harran

Clément Nadal

Jean-François Rouchon

Marianna Braza

Dynamics of a Cambered A320 Wing by Means of SMA Morphing and Time-Resolved PIV at High Reynolds Number

Keywords: aerodynamics, turbulence, PIV, closed-loop control, morphing, wind-tunnel experiments, smart materials

come

Introduction

It is well known that presently airfoil shapes are not designed to optimize the aerodynamic performance in all flight phases, but still the drag-lift ratio of an aircraft must vary depending on the flight state. Since it has been a major matter of interest in aeronautical industry, recent studies [START_REF] Barbarino | A Review of Morphing Aircraft[END_REF] show that camber control of the trailing edge may be a solution to reduce aircraft operational costs. Adapting the shape of the wing during the flight can save several percent of fuel consumption for a commercial aircraft. Such concept is called morphing and it has been of extensive research for more than 40 years [START_REF] Barbarino | A Review of Morphing Aircraft[END_REF]. The concept of morphing aims to optimize drag-lift ratio by means of shape modification in order to adapt the flow in real time.

The main inspiration for this project is the nature. It has been proven that some species of fish are able to manipulate surrounding turbulence to reduce muscle activity [START_REF] Liao | Fish Exploiting Vortices Decrease Muscle Activity[END_REF]. Going even further, owls have an unique mechanism to reduce noise during flight [START_REF] Sarradj | Silent Owl Flight: Bird Flyover Noise Measurements[END_REF]. All of these aspects can and should be used to enhance the performance of aircrafts. It is not a simple task thought. Conventional electromechanical actuators are not the most suitable way to achieve shape optimization mostly because of cost, weight and complex mechanical integration [START_REF] Barbarino | A Review of Morphing Aircraft[END_REF]. These issues have motivated recent advances in the field of smart materials such as shape memory alloys (SMA) and piezoelectric macrofiber composites (MFC). Smart materials provide sufficient stiffness to withstand the aerodynamic loads, while being flexible enough to be deformed, showing the potential to be an alternative to conventional electromechanical actuators [START_REF] Barbarino | A review on shape memory alloys with applications to morphing aircraft[END_REF][START_REF] Rouchon | Activation dune voilure dformable par des cbles dAMF rpartis en surface[END_REF].

Considering electroactive materials, Shape Menory Alloys (SMA) and piezoelectric materials are commonly used. SMA are characterized by thermomechanical behaviors, and most of the applications use actuation by means of Joule heating. Typical applications are shape adaptation at low deformation speed. Piezoelectric materials are activated when exposed to an electric field. Piezoelectric composites are suitable from low to very high frequencies and allow easy integration with electrodes.

Electroactive morphing has been the focus of a partnership between LAPLACE and IMFT laboratories for more than 15 years in various collaborative research projects. For the Smart Morphing and Sensing (SMS) European project an electroactive reduced scale prototype has been realized. This prototype embeds both SMA and piezoelectric macro fiber composites (MFC). SMAs ensure high deformation, around 10% of the chord, for camber control while MFCs perform deformation of the trailing-edge at higher frequencies up to 400Hz [START_REF] Scheller | A hybrid morphing NACA4412 airfoil concept[END_REF]. Wind tunnel experiments show that piezoelectric MFC actuation reduces the energy of the wing's wake thanks to an interaction between the vibrating trailing-edge and the shear layer [START_REF] Scheller | Trailing-edge dynamics of a morphing NACA0012 aileron at high Reynolds number by high-speed PIV[END_REF].

Following previous work, [START_REF] Scheller | Trailing-edge dynamics of a morphing NACA0012 aileron at high Reynolds number by high-speed PIV[END_REF][START_REF] Jodin | Dynamics of a hybrid morphing wing with active open loop vibrating trailing edge by time-resolved PIV and force measures[END_REF], the first section of the article is dedicated to the design of the reduced scale prototype. The second section introduces the wind tunnel experimental setup, used to perform force measurements ans High Speed Time Resolved Particle Image Velocimetry (HS TR-PIV). The following sections show experimental obtained after hybrid morphing (SMA and MFC actuation) and present the perspectives for the SMS project.

Prototype Description

The considered prototype is an electroactive hybrid morphing wing. It embeds both camber control and Higher Frequency Vibrating Trailing Edge (HFVTE) actuators. The baseline airfoil is a wing section of an A320. The chord is 700 mm and the span 590 mm. This aspect ratio affects the flow, but it does not affect the actual results, as the experiments are dedicated to the changes in the flow due to morphing compared to a non-morphing wing. Fig. 1 presents the prototype and its three different sections. The actuators are sized, simulated and implemented on the last 30% of the chord, corresponding to usual flap positions. The hollow fixed leading edge contains electronics and tubing for all temperature, pressure and position transducers as well as actuator interfaces. The camber control actuators working principle relies on distributed structure embedded actuators: SMA wires are spread under the upper and lower aluminum skins of the wing. The actuation of the upper wires (suction side) causes bending of the trailing edge towards higher cambered shapes. Antagonistically, the actuation of SMA wires under the pressure side skin causes a decrease in camber. Figure 1b shows the SMA actuated wing section. The selected wires are made of Nickel and Titanium. Their properties are activated by a change in temperature. They are heated thanks to electric current through themselves. When warmed, the wires tend to retract by a 3% strain that have been previously trained. The modification between martensite and austenite phase in the material internal structure generates a change from plastic to elastic regime, so the SMA tend to recover the initial length. The SMA actuators are designed to support the aerodynamic loads corresponding to a Reynolds number equal to 1 million, which corresponds to a flow velocity of 21.5 m/s in IMFT wind-tunnel. They are able to generate intense stress of more than 600 MPa under these large deformation levels. These materials have been studied for decades, Lexcellent provides an accurate SMA handbook [START_REF] Lexcellent | Shape-Memory Alloys Handbook[END_REF].

The HFVTE actuators are composed of metallic substrates sandwiched between "Macro Fiber Composite" (MFC) piezoelectric patches on both sides. MFC patches are LZT piezoelectric fibers and electrode networks encapsulated within epoxy. When supplied by a voltage, the patches stretch out and generate bending. The whole is covered by a flexible molded silicon that gives the trailing edge shape. Figure Fig. 2 illustrates the actuator topology. The active length of the HFVTE is 35 mm chordwise. Push-push actuators are also seen in litterature for this purpose [START_REF] Scheller | A hybrid morphing NACA4412 airfoil concept[END_REF]. This design allows for quasi-static tip deformation peak to peak amplitude on the order of 2 mm, while able to vibrate at amplitudes large enough up to 400 Hz. The authors invite the reader to refer to the previous publications [START_REF] Jodin | Dynamics of a hybrid morphing wing with active open loop vibrating trailing edge by time-resolved PIV and force measures[END_REF][START_REF] Jodin | Dimensionnement d'une maquette pour l'investigation du morphing lectroactif hybride en soufflerie subsonique[END_REF] for more details related to the design and the electromechanical characterization of this electroactive morphing wing.

Experimental Set-up

The experiments were performed in the IMFT (Institut de Mcanique des Fluides de Toulouse) wind-tunnel. The test section is 592 mm width per 712 mm high. The prototype is mounted at an incidence of 10°. As a result a blockage ratio of 18% is obtained, which is considered acceptable in these experiments. The turbulence intensity of the inlet section is about 0.1% of the free stream velocity. Measurements are carried out at ambient temperature (22°C).

PIV campaign was performed to investigate the influence of the SMA actuation on the flow. In other to image both the upper and the lower surfaces of the wing a 400mm Nikon lens was chosen. Indeed, as shown in Fig. 3, the experimental set-up of this campaign, the 3 meters distance between the camera and the test section of the wind tunnel reduces the parallax effect. It was then necessary to use a flexible mask to avoid direct laser reflection on the prototype surface. Furthermore, to Particle images are recorded at a rate of 5 kHz using a digital high-speed camera Phanton V2012. The laser sheet is generated by a Photonics DM60-527-DH. 0.5m particles were employed to obtain an investigation window size of 170 per 260 mm. The commercial software DaVis10 from LaVision was used for post-processing.

Results

The PIV campaign was performed at Reynolds number of 1 million in which corresponds to a flow velocity of 21.5 m/s in the IMFT wind-tunnel. The high speed camera recorded a sequence of snapshots of the SMA actuation of the prototype. The trailing edge comes from static position, that means no actuation performed, to full cambered position in about 1.5 seconds.

The first step for the analysis of the experimental results was to plot the streamtraces of the snapshots obtained with the PIV campaign. It is a fast and useful tool to understand the flow dynamics. In Fig. 4 one can see the snapshots corresponding to the static position and the full cambered case. As it is expected there is an increase in the recirculation zone such as in take-off and landing configurations of commercial aircrafts.

A monitor point was chosen in the lower shear layer in the wake of the wing. The normalized coordinates in relation to the chord of the wing are x/c = 1.14 and y/c = -0.20. Then the power spectral density of the two velocity components were calculated. The results are shown in Fig. 5. They are consistent with numerical simulation results obtained [START_REF] Simiriotis | Morphing of a supercritical wing by means of trailing edge deformation and vibration at high Reynolds numbers: experimental and numerical investigation[END_REF] where the frequency of von Kármán instabilities were found to be around 120 Hz in the lower shear layer region.

POD Analysis

The Proper Orthogonal Decomposition is a mathematical method used to detect the coherent structures featured by the flow based on their wake number and frequency. It allows us to understand the flow behavior by finding the so-called POD modes that form the dynamic system. This approach is largely used in the analysis of experimental data [START_REF] Perrin | Analyse physique et modlisation d'coulements incompressibles instationnaires turbulents autour d'un cylindre circulaire grand nombre de Reynolds[END_REF][START_REF] Jodin | Hybrid electroactive morphing at real scale -application to Airbus A320 wings[END_REF] and numerical simulation. The reader can find a more detailed description of the POD in [START_REF] Sieber | Spectral proper orthogonal decomposition[END_REF].

In Fig. 6(a) we see the POD modes ranked by their energy. In the case of full camber position, higher modes have more energy than in the static case. One can see it clearly in Fig. 10 were the PSD of the temporal coefficients of the modes 3 and 5 are presented for static and cambered cases. The rise of the energy can be explained by the increase of big coherent structures during SMA actuation. In general, we need more POD modes to describe the dynamics of the flow around the cambered wing, as it is shown in Fig. 6(b). Since the recirculation zone is larger, beyond the big energetic vortices, more small structures are formed which correspond to less energetic modes (Fig. 9).

The dynamics of the flow are reined by big structures inside the recirculation zone. Fig. 7 and Fig. 8 show the modes 3 and 5 respectively. In both figures we see an augmentation of the vortices after SMA actuation which is consistent with the results obtained with PSD analysis and energy rank.

Reconstruction from POD modes

Another possibility offered by the POD is the reconstruction of the dynamic system using a reduced number of modes. This approach allows us to identify the most energetic structures by suppressing small instabilities. Fig. 11 presents the reconstruction of two different snapshots using 31 modes. Two solution times were chosen to capture both static (t = 1.4s) and cambered (t = 2.36s) configurations. The unsteady behavior of the flow in the actuated case is noticeable in both velocity components. The flow vorticity was also calculated using 5 modes to verify the increase of the turbulence in the wake of the wing (Fig. 12). Effectively, the expected result was obtained. We observe the growth of the vorticity specially inside the recirculation zone.

Conclusion

This paper presents the experimental study of electroactive morphing using a reduced scale prototype of an A320 wing through time-resolved PIV. As it was presented in the prototype description, the wing is embedded with two types of actuators. This work focuses in the POD analysis of the flow which shows the augmentation of wake's width caused by SMA actuation.

The influence of hybrid morphing (high and low frequency actuation at the same time) will be further investigated in the following experimental campaigns. Drag and lift measurements in the wind-tunnel are foreseen such as new PIV campaigns.

In the context of the SMS project, a closed-loop control system is intended. Through dynamic pressure sensors, the goal is to actuate the trailing edge at optimal frequencies to enhance aerodynamic performance. The results presented in this paper are promising in the sense that the variation in flow dynamics around the region where the pressure sensors are placed is quite perceptible.

Acknowledgment

This research is part of the EU funded project: H2020 SMS Smart Morphing and Sensing for aeronautical configurations, http://smartwing.org/SMS/EU.

Fig. 1 :

 1 Fig. 1: Reduced scale prototype developed by Laplace and IMFT.

Fig. 2 :

 2 Fig. 2: HFVTE actuator.

Fig. 3 :

 3 Fig. 3: PIV equipment. Configuration used to to image both upper and lower surfaces of the wing.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Streamtraces from PIV campaign. (a) No morphing, (b) full SMA actuation.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Comparison between: (a) energy from POD modes and (b) accumulated energy.

Fig. 8 :

 8 Fig. 8: (a) shows the u component of the velocity corresponding to the mode 5 while (b) shows the y component. (c) and (d) show the same corresponding to the actuated case.

Fig. 9 :Fig. 10 :

 910 Fig. 9: Velocity components of mode 15. Cambered position.

Fig. 11 :

 11 Fig. 11: Reconstruction of the flow using POD modes: (a) and (b) show the two velocity components of the static case (t = 1.4 seconds); (c) and (d) correspond to the full camber position (t = 2.36 seconds).

Fig. 12 :

 12 Fig. 12: Vorticity of the flow calculated from reconstructed velocity fields.


M Legry

F Colas

C Saudemont

J Y Dieulot

O Ducarme

A Two-layer Model Predictive Control based secondary control with economic performance tracking for islanded microgrids

Keywords: Microgrids, Predictive Control, Hierarchical control, Energy Management System, Power Management System I

This paper proposes a two-layer microgrid supervisor based on Model Predictive Control (MPC). The supervisor in the upper layer relies on an economical optimization that considers the cost of energy and the load and production forecasts to define the State of Charge (SoC) targets for each storage device on a timescale of 15 minutes. The lower layer displays a shorter timescale and aims to control the equipment to ensure the stability of the overall system and SoC tracking while satisfying the economic constraints specified by the upper layer. These two layers require an uniformization of the timestep and of the references in order to behave properly. The main contributions of this paper are the microgrid network modelling embedded in the optimization routine of the lower layer and a discretization for integrating upper-layer references.

INTRODUCTION

Hierarchical control structure is commonplace in the scientific and industrial communities for microgrid control. It has been studied for a long time since the renewal of the microgrid concept [START_REF] Guerrero | Advanced Control Architectures for Intelligent Microgrids; Part I: Decentralized and Hierarchical Control[END_REF]. It was first a mimic of control levels of conventional power systems and of ancillary services [START_REF] Vandoorn | Microgrids: Hierarchical control and an overview of the control and reserve management strategies[END_REF]. It was also convenient in such a way that higher-level layer could consider and optimize the stochastic nature of the renewable production in an offline optimization routine without altering the system stability by the increase of computational burden. However, it only optimizes the economic operation of the microgrid. Thus, a second layer is needed to handle the technical supervisor of the microgrid. For this layer, several control techniques may be applied such as fuzzy logic in [START_REF] Robyns | Methodologies for supervision of Hybrid Energy Sources based on Storage Systems -A survey[END_REF] in which the author developed rules-based supervisor to define the power references. Another technique exploits an optimization routine and a prediction model named Model Predictive Control in [START_REF] Dieulot | Economic supervisory predictive control of a hybrid power generation plant[END_REF]. The MPC technique allows to handle multiple states and inputs when the model is well known and in presence of accurate forecasts, which is the case for microgrid dynamics.

Most of the literature on multi-source power cells focus mainly on an accurate modeling of the dynamics of every equipment and aims to optimize an economic criterion based on operational cost, CO2 emissions. For example, in [START_REF] Dieulot | Economic supervisory predictive control of a hybrid power generation plant[END_REF], a performance index is defined as a combination of the different costs and the output powers. However, this literature generally ignores the fact that in real life, microgrids encompasses lines, loads and low-level controller which contribution cannot be neglected in the overall model. Along with the purpose of alleviating the computational burden, a multi-layer hierarchical control is preferred to a centralized supervisor when applied to microgrids. In addition, the microgrid system embeds distributed local controllers such as droop control to preserve stability that must be used as lever. Such a multi-layer supervisor of a grid-connected microgrid control is proposed in [START_REF] Cominesi | A multilayer control scheme for microgrid energy management[END_REF], in which the upper layer computes the daily exchange with the grid, and an intermediate layer translates these daily references into real time values for two clusters with similar equipment. By doing so, Cominesi et al. reduce the computational burden but require an additional central supervisor for each cluster to deal with the diversity between the assets within a single cluster. In [START_REF] Cominesi | Two-layer predictive control of a micro-grid including stochastic energy sources[END_REF], the same authors improved the hierarchical control by adding a stochastic MPC to a daily economic planning to minimize the discrepancies due to the stochastic resources. The use of the shrinking horizon MPC at the lower layer allow to integrate the dynamics of the assets and the complete model is probabilistic in its nature, so are the constraints. A similar approach has been proposed in [START_REF] Zhu | Managing microgrids with intermittent resources: A two-layer multi-step optimal control approach[END_REF] in which the upper layer aims to mitigate the severe fluctuations and the lower layer to keep constant the output power. However, in these publications, the embedded model in the fast supervisor is the same as that displayed in the upper-layer or neglect the impact of the microgrid network. Moreover, both layers are assumed to be synchronous and consider a shrinking horizon to handle the upper layer references, whereas the timescale of the layers in the actual supervisors are not the same. Different multi-layer MPC architectures have been compared in [START_REF] Würth | A two-layer architecture for economically optimal process control and operation[END_REF], one with a steady state optimization and a target correction at the lower layer side that can handle an additional task (Figure 1 (a)), a global nonlinear MPC (Figure 1 (b)) and a dynamic real-time optimizer with a tracking controller (Figure 1 (c)).

To address the issues previously pointed out, we propose an improved two-layer supervisor combining the advantages of the first architecture with the tracking controller depicted in the third architecture. Throughout the paper, the upper layer is referred to as EMS, and the lower as PMS, standing respectively for Energy Management System and Power Management System. Our supervisor embeds an economic optimization at the upper level, based on a classical modeling of the microgrid, and a trajectory tracking problem at the lower level. This optimization algorithm embeds a refined model of the network which is able to handle and anticipate the voltage and frequency variations and takes the dynamics of the grid into account.

The rest of this paper is organized as follows. In section II, the components and network models with the corresponding assumptions are detailed. The third section presents different strategies to embed the economic constraints in the lower layer. Finally, the last part displays the simulation results of the proposed strategy.

II. PREDICTIVE MICROGRID CONTROLLER

It has been chosen to develop a discrete MPC for the PMS. The sampling time is set to 1 minute and the horizon control to 15 minutes which matches one upper layer timestep. Model Predictive Control has been widely used in the control engineering community [START_REF] Qin | 8: EMS (blue line) -PMS (red line) Comparison -top: batterie output power, bottom: State of charge Figure 9: Top: node voltage profile, bottom: frequency profile control technology[END_REF]. The synoptics of this technique is shown in Figure 2 and it exhibits several advantages: i)

Design of a prediction model to accurately handle system future behavior, ii) ability to handle multiple dynamics, iii) minimization of a multi-objective cost function while respecting constraints.

A. Controller objectives

Ideally, the lower layer controller can reach the nominal state according to the references and keep it along the timestep. Due to the stochastic nature of the production and consumption that prevail in the microgrid, the upper layer only considers the global energy over the whole timestep (or constant power over a fixed timestep) and the evolution of the States of Charge. Our first objective is then to track the predicted states of charge coping with the production and load fluctuations. The levers to meet these objectives are the output powers references which are sent to the controlled equipment: storage devices, diesel generator. A second objective is to minimize the changes of references. The objective function of the microgrid supervisor is formulated as:

( ) ( ) 2 2 * 1 min ( ) ( ) ( ) Nc u k J x k x k u k   =  = - +     (1)
Where the first term represents the error between the SoC targets and the SoC reached by the lower layer, 𝑥 ̃(𝑘) and 𝑥 * (𝑘) are the predicted and reference state for timestep 𝑘 and the second minimize the effort of the levers. In this formulation, the control horizon 𝑁 𝑐 is equal to the prediction horizon.

B. Component modeling 1) Storage devices

The State of Charge dynamics for the battery can be modelled by the following discrete time state equations:

  +  - +    +=  +  - +    (2)
Subject to:

( where η ch and η dis are the charging and discharging efficiencies of the battery, P b the output power, T s the sampling time and 𝑋 and 𝑋, are the maximum and minimum bounds of the variable 𝑋.

2) Grid forming inverter

A grid forming inverter is a power electronic interface that can create and maintain the characteristics of a power system, that are voltage and frequency. This converter is mainly used along with storage devices or coupled renewable-storage. It can be modeled by the following equations representing the droop control:

* * ( 1) 0 ( 1) 0 p q k k P V k V k Q   + -+  =   + -+  =   (3)
where 𝑢 and 𝜔 are the output voltage and pulsation resp., and the superscript * denotes the reference values, 𝑘 𝑝 and 𝑘 q are the active and reactive droop coefficients resp.

In most of the grid codes, the reactive production of the converters is constrained to a predefined regulation such as Q(u) or tan(φ). For islanded microgrids, such regulations may not be applied, and the coordination of reactive power driven by the optimal supervisor. In the rest of this paper, we assume that inverters are only limited by the apparent output power:

22 max P Q S + (4)

3) Diesel generators

The diesel generators are modelled as controlled output power sources with the following constraint:

( 1) gen gen gen

P P k P  +  ( 5
)
The generator is also constrained by a minimal ON and OFF period. For the PMS prediction horizon, the state constraints are determined by the EMS output. In case of infeasibility, this is the only lever available and we remove these constraints and the cost and constraints are updated accordingly.

4) Renewable sources

The renewable energy resources are modelled as uncontrolled but predictable output powers. For the sake of simplicity, forecasts are assumed known for each time step over the prediction horizon.

C. Microgrid network and load modeling

The active and reactive power flow from node i to node k are defined by the following equations [START_REF] Kundur | Power system control and stability[END_REF]:

1 1 cos( ) sin( ) N gen load i i ij j i i j ij j N gen load i i ij j i i j ij j P P Y V V Q Q Y V V       = = - = -- - = --  
In which, |𝑉 𝑖 |, |𝑉 𝑗 |, 𝛿 𝑖 , 𝛿 𝑗 are the voltage magnitude and angle resp., |𝑌 𝑖𝑗 | and 𝜃 𝑖𝑗 are the admittance matrix magnitude and angle resp. The power flow equations are nonlinear. In order not to increase the computation burden and by assuming that the disturbances are small, the model is linearized:

i i i i i i i i dP dP dP PV V dQ dQ dQ QV V      =  +  +      =  +  +    
The compact discrete model of the network is as follows:

1 * 1 1 * 2 * 1 * ( 1) () ( 1) () ( 1) () (
i i i i i i i i i V k V k V k k k k k k k       + = +    + = +    + = +   (7)
The Jacobian is updated with each new reference from the upper-layer. It is assumed constant for the routine of the lowerlayer.

The voltage and frequency deviations are constrained to 5 % and 10% respectively. 0.95

( 1) 1.05 0.9

( 1) 1.

1 i k Vk   +     +   (8)

D. Global supervisor model

Finally, the supervisor answers the following optimization problem:

( ) ( ) 2 2 * 1 min ( ) ( ) ( ) Nc u k J x k x k u k   =  = - +     (9)
subject to (2)-( 8)

III. INTERLAYER STRATEGIES

As for many microgrid supervisor, the upper layer defines optimal operating point according to an economical optimization routine. However, these references need to be discretized in order to be embedded into a faster controller that predict the electrotechnical behavior of the microgrid, neglected in the upper layer.

A conventional shrinking-horizon MPC is illustrated in Figure 3, in which the EMS reference is kept constant during all the EMS timestep. One of the advantages to use a receding horizon and a discretization algorithm appears clearly from Figure 3 (b). Any disturbance at the end of the timestep will force the supervisor to degrade its performances to reach the set points in one lower layer timestep.

A straightforward strategy consists of an integral control of the error between the lower layer tracking and the interpolated upper layer references. Figure 4 (a) and (b) display two strategies with a discretization of the optimal economic set points, using a step shape and a linear interpolation. In the first one, the references are kept constant for the coming PMS time step. In the later, a linear interpolation of the EMS references defines the trajectory (Figure 4 (b)). This kind of discretization among others are already studied in [START_REF] Abreu | Inter-Layer Interactions in Hierarchical MPC for Building Energy Management Systems[END_REF], however, in this paper, the faster layer uses a shrinking horizon and the models are not that different as the ones used in microgrid supervisors.

Finally, the general algorithm to interface both layers is shown in Algorithm 1. It consists of the calculation of the next two EMS references and a projection on to the faster timestep to define the PMS trajectory that are integrated into the objective function.

IV. SIMULATION RESULTS

The proposed supervisor has been implemented in MATLAB with the toolbox YALMIP [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF].

Five-second real measurement data for photovoltaic and ten-minute load profile are used in these simulations. The forecast provided to the supervisor is a one-minute average of the measurement introducing fluctuations. The simulations are performed with microgrid as shown in Figure 5.

The flowchart of the proposed supervisor is depicted in Figure 10. At first, the economical references are obtained from the EMS module. Figure 6 shows the output of the receding horizon-based EMS with the load and PV production as parameters. The graph at the bottom shows optimal trajectory of the SoC that is used as reference in the PMS. From this layer it can be observed that the battery will not only be recharged by the PV production, but also by the generator set that need to be started up several time during the two days. By doing so, the state of charge of the battery will remains within its limitations (0.2-1 p.u).

The next step is dedicated to the discretization of the economical trajectory and to the construction of the prediction model based on a Newton-Raphson-based power flow algorithm. Finally, the prediction module is updated, and the MPC-based supervisor defines the optimal changes of references for the actuator.

Figure 7 displays the results of the proposed supervisor, the power profile and State of charge tracking for two consecutive days. By comparing the PV production for both layers, it is worth focusing on the compensation of fast fluctuation of the photovoltaic production by adjusting the output power of the batterie. Still, no significative effect induces a change in power references, and the PMS let the droop controller adjust the required power. This results in the voltage and frequency deviation shown in Figure 9. Different parameters and coefficients to more penalize the state of charge deviation could lead the supervisor to change the power references in case of larger SoC or frequency or voltage deviations. The node voltages and frequency remain within the boundaries with a maximal deviation of 0.018 p.u and 0.023 p.u respectively. Figure 8 displays the comparison between the expected output power from the EMS, and the actual output power of the PMS layer (top) and the comparison between SoC reference and real SoC (bottom). It can be noticed that the proposed supervisor is able to track the reference with an error less that a percent while maintaining the system stable and during fast changes in the forecasts. Finally, comparing these figures for the two different days simulated, one with an accurate forecast, and the other with fast fluctuation, it can be observed that the system is subject to significative disturbances even with a simple microgrid network. This highlights the interest of an intermediate layer that deals with the network characteristics for more and more complex microgrid topologies.

V. CONCLUSION

This paper proposes a novel supervisor for real time management of an islanded microgrid. This supervisor is based on a receding horizon two-layer Model Predictive Controller that embeds a long-term economic optimization and a faster dynamic tracking MPC. The latter layer aims to tracks the economic performance predicted while maintaining voltage and frequency stability within the microgrid. For this purpose, a discrete linearized model of the network and of the equipment is included. Simulations outlined the effectiveness of such a supervisor to track the economical references. Further works will focus on two different aspects. first one is handling severe disturbances such as a loss of production or predictive model taking into account physical model of the microgrid. In contrast with classical microgrid model, the proposed supervisor would be able to figure out precisely and in real time, the energy margin to interact with different microgrids connected through a MV or LV network to provide ancillary services and emergency responses to global network disturbances.

Figure 1 :

 1 Figure1: Multi layer architecture as proposed in[START_REF] Würth | A two-layer architecture for economically optimal process control and operation[END_REF]

Figure 2 :

 2 Figure 2: Model Predictive Controller synoptic

Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 3456 Figure 3: Conventional stair-based shrinking MPC supervisor (a) initial state of the system and control sequence -(b) control sequence and predicted trajectory at t= t p

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