Mariam Ahmed

Ahmad Hably

Seddik Bacha

Andres Ovalle

Kite Generator System: Grid Integration and Validation

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

In the ongoing research to decarbonize the electric grid without losing its reliability, grid integration of renewable energy resources is an important issue. During the last ten years, a high altitude wind turbines'concept has emerged and is heavily investigated by many research groups and emerging startup companies (see [START_REF] Ahmed | High altitude wind power systems: A survey on flexible power kites[END_REF] for an overview of the international landscape). The main idea is that energy is produced from high altitudes by controlling tethered airfoils (mainly kites) to fly at high altitudes with a high crosswind speed. This develops a large pulling force that turns an onground generator, thus generating electricity. However, due to limitation in the tether's length and the power zone, the kite should always be redrawn to its initial position by using the onground machine as a motor, consuming energy as doing so. While a lot of research is being done to optimize the kite orientation control using different control techniques such as predicitve control [START_REF] Canale | High altitude wind energy generation using controlled power kites[END_REF], [START_REF] Canale | Fast nonlinear model predictive control using set membership approximation[END_REF], [START_REF] Houska | Optimal control of towing kites[END_REF], observer-based control [START_REF] Hably | Observer-based control of a tethered wing wind power system : indoor real-time experiment[END_REF], virtual constraint control destined for periodic systems [START_REF] Ahmed | Kite generator system periodic motion planning via virtual constraints[END_REF], and other control approaches [START_REF] Novara | Direct data-driven inverse control of a power kite for high altitude wind energy conversion[END_REF], [START_REF] Baayen | Tracking control with adaption of kites[END_REF], the grid connection part is yet to be treated. The kinetic energy, captured from high altitude wind by the kite generator system (KGS), needs to be transformed into an electric power that can be injected in the electric grid or used to supply a certain load. Among the proposed power transformation systems associated to renewable energy grid integration, the one shown in Fig. 1 offers a suitable solution for the relaxation-cycle nature of the studied system. The traction force of the kite is transformed 1 Corresponding author -Gipsa-lab, ahmad.hably@grenoble-inp.fr into a torque applied to an onground permanent magnet synchronous machine (PMSM). This leads to producing an alternative electrical energy with variable frequency. The machine is coupled with the grid, or with a certain load, via a power electronics interface that consists of two bidirectional AC/DC converters. An energy storage should be added in the case of a load or an isolated grid connection, in order to provide the necessary energy during the system's recovery phase. It is installed on the DC-bus level relating both converters.

The paper is organized as follows. Section II is dealing with the power transformation unit. Section III proposes the control scheme used to drive the later system for each connection mode: Infinite grid-connection and stand-alone operation. The validation results are presented in section IV where the problem of implementing the KGS on a Power Hardware-In-the-Loop (PHIL) simulator is addressed. The paper ends with conclusions.

II. POWER TRANSFORMATION SYSTEM

To insert the kite's generated mechanical power into the electrical grid, a power transformation system is needed to transform this power into an electrical one that respects to grid's codes. This system is presented and modeled in the coming paragraphs.

A. Torque Transmission between the Kite and the PMSM

The movement of the tether is transformed to a rotation by means of a drum coupled to the PMSM through a gearbox. Thus, the traction force F trac is translated to a torque C R applied on the machine. Eq.1 and Fig. 2 represent the mechanical connection:

C G -C R -DΩ S = J dΩ S dt (1)
with Ω S = V L K is the rotation velocity, K combines the gearbox factor and the drum diameter R. J is the total inertia of the kite, the drum, and the machine's rotor. C G is the generator torque. D is the damping factor estimation. Notice that the transmission chain elasticity is neglected. existence of a regular air gap, linear characteristics of the magnetic circuit with no saturation, and a balanced sinusoidal three-phase current behavior. To visualize the three phases at the same time, vector presentation is used, and is expressed in Park coordinates (p, q) by

v sd = R s i sd + L s di sd dt -ωL s i sq v sq = R s i sq + L s disq dt + ωL s i sd + ωφ f sd φ sd = L s i sd + φ f sd φ sq = L s i sq C G = pφ f sd i sq (2)
where

• vs = v sd + j.v sq is the stator voltage vector.

• īs = i sd + j.i sq is the stator current vector.

• φ f s = φ f sd + j.φ f sq is the induced flow vector.

• p is the number of poles pairs.

• ω = pΩ s is the electric pulsation.

C. Power Electronics Interface

The interface ensures frequency and voltage isolation between the PMSM and the electric grid or the connected loads, and at the same time it offers the possibility of power flow to and from the PMSM. This interface is composed of two converters AC/DC & DC/AC (Fig. 4 and Fig. 5) [START_REF] Munteanu | Hardware-in-the-loop-based simulator for a class of variable-speed wind energy conversion systems: Design and performance assessment[END_REF] that transform the variable frequency/voltage electric power generated by the PMSM, into a standard frequency/voltage electric power that can be injected into the grid after a filtering stage. The nature of the filter depends on the connection type. The converters are controlled using vector pulse width modulation (PWM). Both converters function as a rectifier and an inverter depending on the system's phase (generation or recovery phase), insuring a bidirectional transfer of energy. For the machine-side converter, supposing that switches and voltage sources are perfect, and the passive elements are linear and constant, Park representation of the converter's average model is written as follows: where U DC and I DC are the DC bus voltage and current respectively. β sd and β sq are the average vector PWM duty cycle. The same modeling approach is applied for the grid-side converter (Fig. 5), which results in:

L s di sd dt = β sd U DC 2 + ωL q i sq -R s i sd L s disq dt = β sq U DC 2 -ωL d i sd -R s i sq -e sq C DC dU DC dt = -I DC + (β sd i sd 2 + β sq isq 2 ) (3)
L f di Gd dt = -β Gd U DC 2 + ω G L f i Gq -R f i Gd + √ 3V G L f di Gq dt = -β Gq U DC 2 + ω G L f i Gd -R f i Gq C DC dU DC dt = I DC REC -(β Gd i Gd 2 + β Gq i Gq 2 ) (4) where:
• ω G is the electric grid pulsation.

• V G is the grid RMS voltage.

• ī G = i Gd + i Gq is the grid currents' vector.

• I DC REC is the machine converter output current. As noticed, an average model is adapted in order to have a continuous time model without switching, which allows the usage of relatively large sampling time in simulations.

III. CONTROL SCHEME

The control objective is to control of the tether radial velocity to generate the desired average power taking into account wind speed variations; as well as the control of electronics power interface electrical variables. Fig. 6 shows the general control scheme of the power transformation system. As shown, the control scheme is divided into three levels: Low, intermediate and high. Each level functions in accordance with the system operation status: Grid-connected or stand-alone operation and are presented and discussed in the following paragraphs. Fig. 6. General control scheme of the KGS power transformation system. Two control tracks applied depending whether the system is grid connected or in a stand-alone operation.

A. Grid-connected operation

When the KGS is connected to an infinite electric grid, the control strategy aims at extracting the maximum available energy and injecting it in the grid. Meanwhile the grid is responsible of supplying the necessary energy during the system's recovery phase. Briefly, the machine-side converter is driven to control the machine rotation velocity, and the gridside converter controls the DC-bus voltage and maintains the injected currents in phase with the grid voltages in order to preserve grid reliability.

1) Low Level Control:

The low level control deals with the direct control of the converters switches. It translates higher level control laws into pulse width modulation PWM commutation rates that command the converter's switches. This conversion is done via control of the converters' output currents or voltages according to the converter operation. In the case of infinite grid connection, both are current controlled. Considering a PMSM and taking into account that the machine and converter are working within their nominal limits, controlling C G is equivalent to controlling the current i sq with asserting i sd = 0 (See eq.2). This allows having a maximum torque per ampere (MTPA). To control the currents, a PI controller acting in the p -q rotating frame is used. The transfer function of such a controller takes the form:

H cim (p) = K p + K i p = K p (1 + 1 T i p ) (5)
with T i = K p /K i , where: K p , K i are, respectively, the corrector proportional and integral gains. These parameters are chosen in a way to have a response time much faster than that of higher control loops, as well as, a limited overshoot that does not exceed the converters maximum currents. Fig. 7 shows the resulted control scheme in this case based on the converter's model given by eq.3, with: E = pΩΦ f sd . The control is done numerically in the p-q space and a vector PWM is built and transmitted to the a-b-c space to be injected in the switchs. The grid-side converter's currents must be in phase with the grid voltages and have a low harmonic distortion (THD). They are controlled in the fixed coordinates a-b-c via a resonant PI controller that acts on the current harmonics desired to be eliminated. Such a controller take the following form:

H ciq (p) = K p + h n=1 2K i p p 2 + ω 2 n (6)
where the proportional K p gain affects all the current harmonics equally, and the integral part K i affects the h harmonics specified by their resonant pulse ω n . The gridside converter current control scheme is shown on Fig. 8. 8. Low level control scheme for the grid-side converter.

                      Fig.

2) Intermediate Level Control:

The intermediate level control loops generate the reference signals needed for the current control in the lower level via control of the PMSM rotation velocity and the DC-bus voltage. The rotation velocity is controlled by the machine-side converter using a classic PI controller similar to that of eq.5. It generates, according to the mechanical equation (eq.1), the generator torque reference, hence i * q . Considering that the current inner loop is much faster than that of the velocity, the resulted transfer function is a second order one from which the parameters of the PI corrector are determined by choosing a suitable response time that is much smaller than the KGS cycle period, and a suitable overshoot. Fig. 9 shows the rotation velocity control loop, where K E = K G = pΦ f sd . The gridside converter converts the direct power to a fixed-frequency alternative power, or vice versa, according to the KGS phase (traction/recovery), hence, it is driven to control the DC-bus voltage. The DC-bus is assumed to have a resistance R DC and a capacitor C DC , and once again a vector P I controller is used to generate the i * Gq . Fig. 10 shows the resulted DC-bus voltage loop. 3) High Level Control: In addition to generating the needed reference signals for the lower control levels, the high level control supervises the functioning of the system by controlling the switches that determine the power flow through it. As observed in lower control layers, the PMSM's rotational velocity control and the generator torque control are guaranteed by the PMSM-side converter. The kite applies a resistive torque C R that gives, when inserted in the mechanical equation, the rotation velocity Ω S . The velocity is corrected using a PI controller that yields a reference generator torque C Gref . The reference rotation velocity is obtained by applying a maximum power "cycle" tracking (MPCT) algorithm that seeks the kite trajectory that guarantees a maximum average generated power.

B. Stand-alone Operation

In a stand-alone operation, the KGS cannot insure a continuous deliverance of power to the connected load or isolated grid without the support of a storage unit or additional energy sources. Instead of using a storage unit to filter the KGS output power, this can be done by supplying the load by more than one KGS whose orbits are suitably chosen in order to smooth the output power. For instance, Fig. 11 shows the output power profile resulting from using 4 kites flying in T /4 delay. Different kite generator systems are connected on the DC-bus level and share the same DC/AC inverter to connect the load or the micro grid. In this case, the control goal becomes generating the needed power for the load, hence the load-connected converter's output voltages are controlled to have a constant amplitude and frequency. As a result, the machine-side converters are driven to control the DC-bus voltage, while the grid-side converter controls the AC output voltage. On the low level of control, the machine-side converter is driven by currents as in the gridconnected operation case. Meanwhile, the grid-side converter is controlled by voltages. By adding an LC filter on the output of the load-connected inverter, the converter and the load can be presented by:

G(p) = 1 p 2 + ( 1 RC f + R f L f )p + 1 L f C f (1 + R f R ) (7)
where the load is supposed to be resistive. R and R f , L f , C f are the filter components. In the intermediate control level, as in the case of grid connection, the machine-side converter controls the machine velocity, but via the DC-bus voltage control layer. The control scheme, shown in Fig. 13, results from eq.2 and using a PI corrector to regulate the DC-bus. The high level control determines the sinusoidal 50Hz-frequency voltage reference for the load voltage loop and the DC-bus reference voltage.

           

IV. KGS CONTROL VALIDATION

To validate the control scheme presented previously a Power-Hardware-In-the-Loop (PHIL) of G2Elab is used. It is a real-time hybrid simulator that consists of three main parts: A direct current machine (DCM) coupled to a permanent magnets synchronous machine (PMSM), two converters (DC/AC, AC/DC), and a grid emulator. These hardware elements are interconnected and are supported and driven by real-time digital simulators: RT-lab and dSPACE. In this simulator, the tethered kite behavior and its associated drum and gearbox are emulated by a direct current machine (DCM), while the rest of the system is physically present. The hardware is interfaced with the real-time simulator on which the optimization and the control strategy in addition to the kite model are implemented. Employing the PHIL-simulator instead of building a prototype is justified because the tests carried on here, focuses on the grid integration aspect, and the produced power maximization via control of the power conversion chain, and not on the kite orientation control. Furthermore, PHIL simulator requires less material and human investments, it can be modified easily to test different control strategies, and allows experimenting different test conditions in laboratory environment.

A. KGS Implementation on the PHIL Simulator

The KGS power transformation unit is physically present in the PHIL simulator while the DC-machine (DCM) replicates the behavior of the tethered kite.

1) KGS Scaling: The real-time PHIL experimental testbench in G2ELAB was not built for a single test set-up. Hence, in order to employ it to test the dynamics of the KGS, a scaling stage is needed. The testbench is characterized by a maximum power of 3kW and a maximum rotation velocity of 3000RP M . The KGS parameters are chosen to adapt to these values. Scaling the rotation velocity and the torque can be expressed by:

C bmax = nC max , Ω bmax = Ω max m ( 8
)
with n and m being the scaling factors. Accordingly, the power scaling equation

P bmax = C bmax Ω bmax = n m C max Ω max = n m P max .
Notice that n = m necessarily since the objective of the proposed real-time simulation is to test the dynamic behavior of the system and insure the functioning of the control strategy. However, they are chosen to be equal in order to observe the delivered power losses.

2) KGS Torque Emulation: The experimental testbench DCM is controlled to follow the dynamics of the KGS. As shown in Section.II-A, the connection between the tethered kite and the PMSM is expressed by the mechanical equation.

Replacing the kite by the DCM results in the mechanical equation (eq.9):

C G -C DCM -D E Ω E = J E dΩ E dt (9)
where C DCM is the DCM torque, D E is the damping factor estimation between the DCM and the PMSM and is a function of the rotation velocity Ω E , and J E is the total inertia of the DCM and the PMSM. Comparing eq.1 and eq.9 yields that in order to replicate the KGS behavior by the DCM, the Ω E 's dynamics have to follow that of Ω s and the DCM torque needs to be controlled to follow the reference in eq.10. The resulted mechanical connection is represented in Fig. 16 C

* DCM = C R + (J -J E ) dΩ s dt + (D E -D) Ω s (10)
Hence, the DCM torque reference consists of two parts:

The tethered kite traction torque and a correction torque C cor = (J -J E ) dΩs dt + (D E -D) Ω s . However, computing C cor is an issue in real-time applications since the rotation velocity gradient calculation induces noise, and determining the damping friction values is very difficult since they are functions of the rotation velocity [START_REF] Munteanu | Hardware-in-the-loop-based simulator for a class of variable-speed wind energy conversion systems: Design and performance assessment[END_REF]. In order to overcome the reference velocity is expressed by Ω E (p) Ω S (p) = J E (K P p+K I ) (J E p+D E )p . Thus the resulting closed-loop function is given in eq.11.

H cl = τ P I p + 1 p 2 ω 2 n + 2ξ ωn p + 1 (11)
with

τ P I = K P K I , ω n = √ K I and ξ = 1 2 √ K I D E
J E + K P . The corrector parameters are calculated according to the desired response and are dependent of the friction coefficient D E which is, as mentioned earlier, a function of the rotation velocity.

3) Experimental Set-up: The PHIL simulator is a hybrid semi-hardware semi-software one, in which the hardware part contains the electrical machines and the converters, while the software includes the control of those as well as the tethered kite model. The KGS testbench control scheme is shown in Fig. 18. The KGS parameters, the KGS orbit optimization, the kite model as well as the MPCT algorithm are implemented on Matlab/Simulink. The Simulink model has two outputs: The traction torque and the rotation velocity reference; and two inputs: The measured velocity and power. The Simulink model communicates with RT-lab via a TCP/IP protocol functioning under Labview environment who plays the role of a server for both. The digital real-time simulator RT-lab provides an interface with dSPACE. On dSPACE, the torque emulator controller (Section.IV-A.2) generates the DCM torque reference and send it to a digital signal processor card that controls the DCM chopper. Meanwhile the control of the PMSM and the power electronics interface is performed on dSPACE. Fig. 18 shows an abstract scheme of the testbench control.

B. Validation

In this section the test results of the proposed control schemes via simulation and on a PHIL simulator are presented. The KGS's parameters shown in Table .I are chosen in order to generate a radial velocity and traction torque that respect the limits imposed by the PHIL simulator [START_REF] Munteanu | Hardware-in-the-loop-based simulator for a class of variable-speed wind energy conversion systems: Design and performance assessment[END_REF]. After applying the optimization algorithm proposed in [START_REF] Ahmed | Power maximization of a closed-orbit kite generator system[END_REF] to find the kite's optimal trajectory, hence its optimal radial velocity V L , the next step is to transform the mechanical power produced by the PMSM into an electrical power that can be injected into the grid. This can be insured by tuning different control levels paremeters presented in the general control scheme (Fig. 6). Figs.19 shows the machine phase current I aS and the DC bus U DC voltage during one period of the rotation velocity Ω S . It can be noticed that U DC is well controlled with an error less than 0.9%. Fig. 20 shows a grid phase current I a-G and the grid phases voltages V a,b,c-G . The grid-side converter was successfully controlled to provide the grid with a standard 50Hz three-phase voltage, and currents. The setup of Fig. 14 was implemented to test shows how generated power is optimized from the machine point of view. Torque and velocity variations are represented by the machine currents whose frequency, amplitude and phase change accordingly. The frequency is related directly to the rotation speed f = ω 2π = pΩ 2π , meanwhile the current amplitude expresses the torque variations, and the phase is inverted when the rotation velocity direction changes. Notice as well that the DC-bus voltage keeps a constant value despite the variations in the rotation velocity. For the grid-side converter electrical variables, Fig. 22 shows the variations of the output current and the grid voltage following those of the rotation velocity. The current becomes zero before the rotation velocity reaches zero that is due to losses in the converters elements. This explains also why the current amplitude is higher when the velocity is negative (recovery phase) than when it is positive (generation phase).

V. CONCLUSION

This paper proposes a solution to KGS grid integratation. The solution is a power transformation unit consisting of a PMSM and a power electronics interface. For each connection type, a control scheme is proposed to insure the control of both the mechanical and electrical variables of the power transformation chain. The control laws are tested via simulation then validated on a Power Hardware-In-the-Loop Simulator. Further work will include testing under disturbances acting on the kite and will consider the case of unbalanced regimes. They will also take into account voltage dips propagating from the grid and the system's LVRT (Low Voltage Ride Through) capabilities.

Fig. 1 .

 1 Fig. 1. Kite Generator System Block diagram.

Fig. 2 .

 2 Fig. 2. Modeling of the mechanical connection between the kite and the electrical machine.

Fig. 4 .

 4 Fig. 4. Electric representation of the PMSM-side converter. PMSM is presented by Behn-Eschenburg model. C dc is DC-bus filtering capacitor.

Fig. 5 .

 5 Fig. 5. Electric representation of the Grid-side converter. R f and L f represent loss and filtering components.

Fig. 7 .

 7 Fig. 7. Low level control scheme for the machine-side converter.

Fig. 9 .

 9 Fig. 9. Intermediate level control scheme for the machine-side converter: Machine velocity control.

Fig. 10 .

 10 Fig. 10. Intermediate level control scheme for the grid-side converter: DC-bus voltage control.

Fig. 11 .

 11 Fig. 11. Average output power of a 4-kite-based system.

Fig. 12 .

 12 Fig. 12. Load-connected converter low level control scheme.

Fig. 13 .

 13 Fig. 13. Machine-connected converter Intermediate level control scheme.

  Fig. 14. PHIL Simulator

  Fig. 15. Representation of the mechanical connection in the case of the KGS and the DCM.

Fig. 16 .Fig. 17 .

 1617 Fig. 16. KGS replication using the DCM.

1 Fig

 1 Fig. 18. KGS testbench control scheme.

Fig. 19 .

 19 Fig. 19. Starting from the upper plot: The PMSM rotation velocity (Ω S ), PMSM phase current (I aS ), DC bus voltage (U DC ).

Fig. 20 .

 20 Fig. 20.Starting from above Grid voltages, grid current (I aG ), its frequency analysis.

Fig. 21 .

 21 Fig. 21. Starting from above: DCM torque (C DCM ), PMSM rotation velocity (Ωs), PMSM phase current (Isa), and DC bus voltage (U DC ).

Fig. 22 .

 22 Fig. 22. Starting from the upper plot: PMSM rotation velocity (Ωs), grid phase current (Isa), and grid phase voltage (V Ga ).

Fig. 23 .

 23 Fig. 23. Zoom into the grid voltage and current changes when the PMSM changes its rotation direction. It shows how the current is in phase with the voltage during the generation phase, and π 2 shifted during the recovery phase.

  L and the kite torque C R resulted from KGS simulations. V L and C R are scaled to match the experimental testbench characteristics. As observed on Fig.21, the resulting rotation velocity have a cyclic profile that varies in the range [-1000, 1000] rpm. Its variations are accompanied by synchronized equivalent variations in the kite torque, which

							TABLE I
		KITE GENERATOR SYSTEM PARAMETERS
	K	414		Gearbox factor * rotor diameter R (m)
	V	4			Wind speed (m/sec)
	Ωmax	210		Maximum rotation velocity (rd/sec)
	Γmax	30		Motor maximum torque (N.m)
	p	4			Pole's pairs number
	A, m	5, 3		Kite's area(m 2 ) and mass(Kg)
	ρa	1.2		Air density (kg/m 3 )
	CL	1.2		Lift coefficient
	CD	0.08		Drag coefficient
	r0	600		Initial tether length (m)
	the proposed control strategy. The radial velocity control via
	the PMSM, the torque emulation and the electrical variables
	control were tested through application of the optimized
	-200 200 radial velocity V 0 0 Ω S (rad/s)	2	4	6	8	10	12	14	16	18	20
			50
		I aS (A)	0
			0 -50	2	4	6	8	10	12	14	16	18	20
			605
		U DC (V)	600
			0 595	2	4	6	8	10	12	14	16	18	20
								time (sec)


Andres Ovalle
email: andres.ovalle-villamil@g2elab.grenoble-inp.fr

Seddik Bacha
email: seddik.bacha@g2elab.grenoble-inp.fr

Ahmad Hably
email: ahmad.hably@gipsa-lab.grenoble-inp.fr

Voltage support by optimal integration

Keywords: Plug-In Hybrid Vehicle, Smart charging, Vehicle-to-home, Linear programming, Voltage Support

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

The high level of fossil fuel consumption and the environmental issues caused by conventional vehicles have impulsed the development of several vehicle alternatives. Among those alternatives, Electrical Vehicles (EV) (with the current battery technologies) have been relegated to a second place because of the reduced autonomy compared with conventional vehicles [START_REF] Turker | Impact of plug-in hybrid electric vehicles (phevs) on the french electric grid[END_REF]. Hybrid Vehicles (HV) and PHEVs are some of the most applicable solutions available in the automotive industry. HVs employ an energy storage capacity to reduce fuel consumption during strong accelerations. Compared to HVs, PHEVs have a larger energy storage capacity which provides the capacity of working on electrical mode (zero CO 2 emissions) with certain autonomy. Also PHEVS can work in a similar mode of HVs [START_REF] Turker | Impact of plug-in hybrid electric vehicles (phevs) on the french electric grid[END_REF], [START_REF] Turker | Low-voltage transformer loss-of-life assessments for a high penetration of plug-in hybrid electric vehicles (phevs)[END_REF].

Without a dedicated infrastructure, PHEVs are expected to be charged at homes, representing a significant amount of new load to the residential grids [START_REF] Turker | Low-voltage transformer loss-of-life assessments for a high penetration of plug-in hybrid electric vehicles (phevs)[END_REF], [START_REF] Kim | Bidirectional energy trading and residential load scheduling with electric vehicles in the smart grid[END_REF]. This increment on the energy demand causes multiple problems and endangers the grid safe operation [START_REF] Nguyen | Optimal charging and discharging for multiple phevs with demand side management in vehicle-to-building[END_REF]. The availability of bidirectional chargers can represent a valuable solution to the integration of PHEVs to the grid and the solution of the resulting technical issues [START_REF] Kim | Bidirectional energy trading and residential load scheduling with electric vehicles in the smart grid[END_REF], [START_REF] Nguyen | Optimal charging and discharging for multiple phevs with demand side management in vehicle-to-building[END_REF], [START_REF] Couillet | Electrical vehicles in the smart grid: A mean field game analysis[END_REF]. In this way, PHEVs and EVs with improved energy storage systems represent not only loads but mobile energy sources [START_REF] Couillet | Electrical vehicles in the smart grid: A mean field game analysis[END_REF].

Appropriate strategies to provide optimal policies of consumption/injection of energy become very crucial for the integration of PHEVs to the grid [START_REF] Turker | Low-voltage transformer loss-of-life assessments for a high penetration of plug-in hybrid electric vehicles (phevs)[END_REF], [START_REF] Kim | Bidirectional energy trading and residential load scheduling with electric vehicles in the smart grid[END_REF], [START_REF] Nguyen | Optimal charging and discharging for multiple phevs with demand side management in vehicle-to-building[END_REF], [START_REF] Couillet | Electrical vehicles in the smart grid: A mean field game analysis[END_REF], [START_REF] Turker | Rule-based charging of plugin electric vehicles (pevs): Impacts on the aging rate of low-voltage transformers[END_REF]. This paper provides a linear approach to compute the voltages at any node of a residential test grid based on the house instantaneous load and the presence of PHEVs in the house (and the corresponding instantaneous consumption or injection). Based on this linear operation, the paper provides a detailed linear programming formulation of the problem of charging the PHEVs while providing a voltage support service to the grid. The set of restrictions is established based on the one proposed by [START_REF] Clement-Nyns | The impact of charging plug-in hybrid electric vehicles on a residential distribution grid[END_REF], [START_REF]The impact of vehicle-to-grid on the distribution grid[END_REF]. Additionally, multiple evaluation cases are included to test the strategy.

The paper is organized as follows. Section II introduces the modeling approach followed. Section III presents the set of restrictions for the optimization problem. Section IV presents the standard form for the linear programming problem. Section V presents the study cases and the obtained results. In section VI the paper is concluded.

II. MODELING OF THE GRID AND THE STATE OF CHARGE (SOC) OF BATTERIES

The Residential grid is modeled as a radial topology grid with a feeder and branches beginning at certain nodes of the feeder as it is shown on Fig. 1. Because it is a low voltage grid (R >> X), the reactive portion of the lines is neglected. Each node of the grid is considered as a house and each house is able to have one PHEV connected. The load is modeled as a unity power factor load (a resistance) and a current source representing the battery from the PHEV. The resistive load value is approximated with the nominal voltage of the grid and the corresponding instantaneous active power demand from the house, as R ≈ V 2 nom /P load . The current injected or consumed by the batteries is approximated from the desired instantaneous input/output power value and the nominal voltage as i P HEV ≈ P P HEV /V nom . The transformer node is represented as a voltage source with a tension that can be fixed at each time step. The nominal voltage is the transformer voltage and it is common to every node. Reactive power control, unbalances or harmonic can be considered but are not included in this paper. With these assumptions, the grid is modeled as a linear circuit where instantaneous voltages at each node can be computed from the voltage source (the transformer) and the current sources (the PHEVs) at each time step by a linear operation (a matrix multiplication).

Tr 0

1 i -1 i i + 1 I i1 iM P i V nom V 2 nom P load i PHEV House Load

A. Problem Formulation

The vector of voltages v t k ,n at a given instant t k and at each node n can be computed depending on the load of each home at that instant, and on the power consumed or injected by the PHEVs p t k ,n . The vectors of voltage and power of PHEVs are organized as follows.

v t k ,n = [v t k ,1 , v t k ,2 , • • • , v t k ,N ] T p t k ,n = [p t k ,1 , p t k ,2 , • • • , p t k ,N ] T v t k ,n = Ât k ĥt k + Ât k p t k ,n (1)
The information of home loads at the time instant t k and at each node is summarized in matrix Ât k . The expression (1) is employed to compute the vector of voltages v t k ,n . The first element of vector ĥt k has information about the voltage at the transformer node (node 0) at time t k . This expression is a superposition of two terms. The first term corresponds to the voltage levels contributed by the transformer node. The second term corresponds to the voltage levels contributed by the PHEVs.

In order to compute the voltages at any node for a discretized time frame, the following procedure is proposed.

 =       Ât1 0 • • • 0 0 Ât2 . . . . . . . . . 0 0 • • • 0 Ât f       (2) v T t,n = v T t1,n , v T t2,n , • • • , v T t f ,n (3)
p T t,n = p T t1,n , p T t2,n , • • • , p T t f ,n (4)
v t,n = Âĥ t + Âp t,n (5)
The block diagonal matrix  can be constructed with matrices Ât k corresponding to instants k = {1, 2, • • • , f } and the corresponding instantaneous load of each home in that time frame. The vector v t,n of voltages at any instant of that time frame and at any node is a concatenation of vectors v t k ,n as its is shown on (3). It can be computed with (5) based on the concatenation of the power consumption/injection vectors at any instant and at any node p t,n and on the voltage at the transformer node at any instant. This expression is similar to (1), the only difference is that (5) computes the concatenation of vectors v t k ,n .

Vectors v t,n and p t,n contain voltages an powers respectively, organized in blocks classified by instants and then by nodes. For the sake of simplicity of the optimization problem approach, matrix  and vectors p t,n and ĥt can be rearranged to get an organization of blocks classified first by nodes and then by instants in order to get a vector of voltages organized in the same way.

For nodes i = {1, 2, • • • N } and instants k = {1, 2, • • • f }, the desired voltage vector should have the following structure. v i,t = v i,t1 , v i,t2 , • • • , v i,t f T v T n,t = v T 1,t , v T 2,t , • • • , v T N,t
This rearrangement provides the matrix A and vectors p n,t , and h t which can be employed to compute the desired voltage vector with the following expression.

v n,t = Ah t + Ap n,t (6)

B. State of Charge

The state of charge for a PHEV corresponding to node i at any time is given by, the following expression.

c i,t = Bp i,t
Where matrix B is common to each PHEV and it has the following structure.

B =         ∆t 0 • • • 0 ∆t ∆t . . . . . . . . . 0 0 ∆t ∆t • • • ∆t 0 ∆t ∆t • • • ∆t ∆t         = Bu bT d
The state of charge for each PHEV in the grid during the charging period can be separated in two expressions representing the state of charge before the final instant (computed with Bu ) , and the final state of charge (computed with bd ). Matrix B u in ( 7) is constructed with individual matrices Bu and is employed to compute the state of charge until the time instant before the end of the charging period for every PHEV in the grid. On the other hand, B d is constructed with individual transposed vectors bd and is employed to compute the final state of charge as in [START_REF]The impact of vehicle-to-grid on the distribution grid[END_REF].

B u =    Bu 0 . . . 0 Bu    , B d =    bT d 0 . . . 0 bT d    (7) c n,{t1,••• ,t f -∆t} = B u p n,t c n,{t1,••• ,t f } = B d p n,t (8)

III. RESTRICTIONS FOR THE CHARGING OF PHEVS

The most important objective of the intelligent charging approach is to reduce the impact on the grid caused by charging the PHEVs. In this case, the approach followed is to provide voltage support to every node in the grid. On the optimization problem, this is imposed as a restriction on the voltages of every node even if there is absence of PHEV. The restrictions for voltage support are summarized as follows.

Ap n,t ≤ v max -Ah t Ap n,t ≥ v min -Ah t
Where, v max and v min are vectors including the maximum and minimum voltages allowed for each node of the grid. By default, these limits are set between 90% -110% of the nominal voltage. It is important to notice that the term Ah t is independent from the power consumption or injection from PHEVs.

The power injected/consumed by PHEVs is restricted as follows.

p n,t ≤ p max p n,t ≥ -p max

The chargers are assumed reversible. Power injected or consumed by PHEVs is restricted to the maximum input/output power which depends on the type of charger. In order to improve live expectancy of batteries, the state of charge is also restricted as follows.

     B u p n,t ≤ c maxu -c iniu B u p n,t ≥ -c iniu B d p n,t = c max d -c ini d
Where, the limits are established between zero and a maximum value c max . Here, the first two restrictions limit the state of charge to be within the established limits during the charging period. The third restriction ensures that the state of

IV. LINEAR PROGRAMMING APPROACH

The objective is to minimize the cost of the energy consumed in order to recharge the PHEVs while the restrictions of voltage support, input/output powers, and state of charge are accomplished. The following is the problem stated in standard form, where the powers consumed or injected by the PHEVs are represented in the vector p n,t = r n,ts n,t .

minimize ∆t k T -k T 0   r n,t s n,t y   subject to           +A ev -A ev -A ev +A ev +I ev -I ev -I ev +I ev I y +B uev -B uev -B uev +B uev +B dev -B dev 0             r n,t s n,t y   =           v max -Ah t -v min + Ah t p maxev p maxev c maxu -c iniu c iniu c max d -c ini d           r n,t , s n,t , y ≥ 0
A ev is the same as A by eliminating the columns corresponding to nodes without PHEV. I ev is an identity matrix with dimensions matching the number of columns of A ev . B uev is the same as B u by eliminating the columns and rows corresponding to nodes without PHEV. B dev is the same as B d by eliminating the columns and rows corresponding to nodes without PHEV. In the vector of costs, k has information of the tariff at each instant of the charging period.

V. RESULTS

A grid model for test is shown on Fig. 2. This is a residential grid with 8 nodes, each corresponding to a house. 3 PHEVs are assumed to be plugged-in during the charging period. They are connected on the nodes highlighted with a red circle. A sample of a 24 hour load profile is shown on Fig. 3. This load profile is employed as a base for each house on the grid. Additionally, different random numbers (between -25% and 25% of the minimum load in the load profile) are added at each node and at each time. The continuous portion of the profile corresponds to the charging period, between 18h and 06h. The voltage at the transformer node is 230V. Chargers are restricted to maximum input/output power of ±3kW. Batteries have a nominal 20kWh capacity, however only 80% of the capacity (16kWh) is employed to reduce the impact on the life expectancy. The line parameters of the grid are listed in Table I. Two cases are evaluated. On the first case, without considering PHEVs connected to the grid, the load profile does not cause voltage limit violation as it can be seen on Fig. 4. This figure shows the voltage profiles at each node during the charging period, without any PHEV connected. It is possible to see that each voltage profile remains between 0.9p.u. and 1p.u. during this period. The second case has a slight variation on the resistive value of two lines. However, this modification is enough to cause voltage limit violation on certain nodes at certain instants of peak demand, even without PHEVs connected. For this case, the voltage profiles for each node and at each instant (without PHEVs) are shown on Fig. 5.

In this study, the PHEVS are assumed to have a certain initial SOC: We have 10%, 5% and 50% of the maximum SOC for the first, second, and third PHEV respectively.

A. PHEVs without charging management -Grid Case 1

When PHEVs are plugged into the grid without any charging management, the chargers are supposed to consume the maximum possible input power in order to charge the batteries as fast as possible. When batteries are fully charged, the corresponding charger cuts out the power consumption. Fig. 6(a) shows the power consumption profiles described before. In this scenario, the power consumed to charge the PHEVs (3kW during 4.5h for the first PHEV, 4.75h for the second PHEV and 2.25h for the third PHEV) coincides with the peak hours of power demanded by the houses. Fig. 6(b) shows the corresponding SOCs. As it can be seen, batteries are fully charged in a little bit more than a third of the time available for the charging period.

For each node, the resulting voltage profiles can be seen on Fig. 6(c). Comparing with the voltage profiles without PHEVs, shown on Fig. 4, it can be seen that nodes 4 -8 go below the established security limit (0.9p.u.) between 18h and 21:45h because of the elevated power consumption. If this conditions are applied to the grid case 2, it is possible to conclude that the PHEVS power consumption will force the voltage profiles drop even more that those of Fig. 5 and Fig. 6(c).

In this case, a unique fictive tariff is considered: 1$/kWh. The resulting cost of charging the PHEVs is 37.6$. This is proportional to the total amount of energy required to charge the batteries which is 37.6kWh (corresponding to 90%, 95%, and 50% of 16kWh).

B. Charging management -Grid Case 1 -Unique Tariff

Applying the optimization strategy to the first grid scenario results in the power consumption profiles of Fig. 7(a). As a result, the power consumption is redistributed during the whole charging period. It is important to highlight that the maximum general consumption occurs during the low demand hours. Also, it is important to notice that during the peak demand hours, the most initially charged PHEV injects energy to the grid in order to compensate the consumption of the other PHEVs and maintain the voltages within the security limits. The SOC profiles are shown on Fig. 7(b). It can be seen that at the end of the charging period, the batteries are fully charged. The voltage profiles corresponding to every node are kept within the desired limits, even during peak demand hours, as it is shown on Fig. 7(c).

In this case, the same unique fictive tariff is considered: 1$/kWh. The optimal cost of charging the PHEVs is also 37.6$ as in the suboptimal case. The cost is the same because the total amount of energy required to fully charge the PHEVs is still 37.6kWh.

C. Charging management -Grid Case 1 -Two Tariff Scenario

Now, let us consider two tariffs: 1.5$/kWh during 18h-22h, and 1$/kWh during 22h-06h. In the suboptimal case, the consumption is concentrated during the interval of high price. As a consequence, the cost of charging the PHEVs without any management is 53.6$. Applying the optimization strategy to the two tariff scenario results in the power consumption profiles of Fig. 8(a). The power consumption is also redistributed during the whole charging period. However, the PHEVs now try to sell as much of their initial SOC as possible during the high-price hours, in order to reduce the cost of recharging their batteries. It is possible to confirm this in the SOC profiles of Fig. 8(b). Although the first and second PHEVs charge their batteries at the beginning of the charging period, at the end of the highprice hours the three PHEVs are fully discharged. They sell their initial energy and during the low-price hours, they fully recharge their batteries. As it is required, the voltage profiles corresponding to every node are kept within the desired limits, even during peak demand hours, as it is shown on Fig. 8(c).

The optimal cost of charging the PHEVs is now reduced to 32.4$. This cost corresponds to the cost of fully charging the batteries during the low-price hours (3 × 16kWh×1$/kWh=

D. Charging management -Grid Case 2

Assuming again a unique tariff, the grid case 2 is evaluated. In this case, the voltage profiles do not respect the security limits during peak hours, even without PHEVs. Given the fact that the peak hours coincide with the beginning of the charging period, if PHEVs do not have certain initial energy stored in their batteries, the restriction of voltage support cannot be respected and there is no feasible solution. The required initial amount of energy is somehow proportional to the deviation of voltages from the desired limits.

If the initial amount of energy available in the batteries is enough, the PHEVs can provide a voltage support service to the grid by injecting this energy and compensating the Here the most initially charged PHEV (in node 8) injects energy to the grid during peak hours, almost until it is fully discharged. Meanwhile, the other PHEVs charge and discharge their batteries. After 22h when the demand begins to decrease, The PHEVS begin to consume energy until they are fully charged. It is important to notice that the initial voltage issue of the grid is solved while the support service is provided to every node on the grid as it can be seen on Fig. 9(c).

In this case, the cost is still 37.6$ as in the case 1, because the total amount of energy required to fully charge the batteries is still 37.6kWh.

VI. CONCLUSION

This paper provides a linear approach to compute the voltages on a residential grid based on the house instantaneous load and the instantaneous consumption/injection of PHEVs. Based on this linear modeling, The paper proposes a linear programming formulation to manage the charging of multiple PHEVs on a residential grid under certain assumptions. Additionally, the proposed strategy employs the energy storage capacity of the PHEVs in order to provide a voltage support service to the residential grid. The proposed scheme is evaluated under multiple cases in order to test its ability to maintain voltages within safety limits and provide optimal consumption/injection policies. Then, the results are analyzed in detail.

Multiple alternative tests can be considered in the future: The inclusion of a differentiation between energy selling prices and buying prices; the analysis under stochastic conditions for the prices or loading profiles, etc.

Fig. 1 .

 1 Fig. 1. Radial Grid Modeling

Fig. 2 .

 2 Fig. 2. Radial Grid proposed in order to test the strategy.

Fig. 3 .

 3 Fig. 3. Sample Load Profile corresponding to a house consumption during 24 hours. The continuous segment corresponds to the chosen charging period for PHEVs plugged in the grid.

Fig. 4 .

 4 Fig. 4. Voltage Profiles for the test grid case 1 without PHEVs connected. Profiles do not evidence voltage issues.

Fig. 5 .

 5 Fig. 5. Voltage Profiles for the test grid case 2 without PHEVs connected. Certain profiles exceed established security limits.

Fig. 6 .

 6 Fig. 6. Power consumption of PHEVs, State of Charge of Batteries, and Voltage Profiles for the grid case 1 without charging management.

Fig. 7 .

 7 Fig. 7. Power consumption of PHEVs, State of Charge of Batteries, and Voltage Profiles for the grid case 1 with the application of the charging management approach.

Fig. 8 .

 8 Fig. 8. Power consumption of PHEVs, State of Charge of Batteries, and Voltage Profiles for the grid case 1 with the application of the charging management approach and the inclusion of two tariffs.

Fig. 9 .

 9 Fig. 9. Power consumption of PHEVs, State of Charge of Batteries, and Voltage Profiles for the grid case 2 with the application of the charging management approach.

TABLE I

 I

	GRID LINE PARAMETERS
	Line
	From To Case 1	Case 2
	0	1	0.05Ω	0.05Ω
	1	2	0.05Ω	0.05Ω
	2	3	0.10Ω	0.15Ω
	3	4	0.15Ω	0.25Ω
	4	5	0.15Ω	0.15Ω
	3	6	0.10Ω	0.10Ω
	6	7	0.05Ω	0.05Ω
	7	8	0.10Ω	0.10Ω


Student Member, IEEE Andres Ovalle
email: andres.ovalle-villamil@g2elab.grenoble-inp.fr

Member, IEEE Gustavo Ramos

Member, IEEE Seddik Bacha

Member, IEEE Ahmad Hably

Axel Rumeau

Decentralized Control of Voltage Source Converters in Microgrids based on the Application of Instantaneous Power Theory

Keywords: Microgrid, Distributed Generation (DG), Instantaneous Power Theory, Voltage Source Converter (VSC), Gridconnected inverter, Current Control, Load Sharing

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

A Microgrid can be defined as a cluster of elements such as loads, distributed generators (DG) (kW-scale technologies such as combined heat and power devices, photovoltaic modules, small wind turbines, etc.), distributed storage units and controllable loads, connected through a medium or low voltage interconnecting power grid. Also, the microgrid can be understood as a subset or a constructive unit of the distribution system. When the number of elements in each microgrid is increased, the problem of controlling them can be very challenging in a centralized way. Thus, each microgrid has the responsibility of controlling their own units locally. In this way, from the point of view of the whole distribution system, the microgrid can be seen as a controlled entity, either generator or load [START_REF] Hatziargyriou | Microgrids[END_REF], [START_REF] Kroposki | Making microgrids work[END_REF]. Security, reliability, environment impact, power quality and energy efficiency challenges for future electricity grids could meet their solution in the microgrid concept [START_REF] Hatziargyriou | Microgrids[END_REF].

In grid-connected mode, the grid provides a robust frequency reference (because of the presence of large rotating masses) and it supplies the mismatch between DG generation and demand in the microgrid. An excess of generation can be stored or supplied to the grid and DGs can provide voltage support services to the grid [START_REF] Kroposki | Making microgrids work[END_REF]. In autonomous or standalone mode, load must be shared among generators, while frequency and voltage (active and reactive powers) must be controlled locally [START_REF] Sao | Autonomous load sharing of voltage source converters[END_REF], [START_REF] Divshali | Decentralised vsc-based microgrid's general power flow[END_REF]. While in grid-connected mode, DGs can behave like constant power sources, in autonomous mode, they must share load while keeping frequency and voltage within established security limits. The frequency response of large power systems is based on rotating masses, which are essential for the stability of the system [START_REF] Kroposki | Making microgrids work[END_REF]. However, in standalone mode there is a lack of rotating masses and the microgrid is highly dominated by DGs interfaced with power electronics converters. Another interesting fact is that a generator cannot be assumed to be connected to an infinite bus, and the system cannot be studied with conventional power flow techniques because none of the generators can be considered as a slack node [START_REF] Divshali | Decentralised vsc-based microgrid's general power flow[END_REF]. The system dynamics are imposed by VSCs, power regulation controllers and the network variable parameters [START_REF] Pogaku | Modeling, analysis and testing of autonomous operation of an inverter-based microgrid[END_REF]. Although there are some important difficulties in the operation of grid-connected mode, autonomous mode has challenging problems to be analyzed as it can be stated.

Power electronics converters are key elements in the microgrid concept. They are employed to transform energy of the energy sources (DC or AC) to grid-compatible energy (AC at 50-60Hz), or grid-energy to energy compatible with the distributed storage units. Voltage Source converters (VSC) are power electronics converters employed to interface energy sources and storage units with the grid [START_REF] Hatziargyriou | Microgrids[END_REF], [START_REF] Sao | Autonomous load sharing of voltage source converters[END_REF], [START_REF] Bongiorno | Dynamic performance of vector current controllers for grid-connected vsc under voltage dips[END_REF], [START_REF] Harnefors | Input-admittance calculation and shaping for controlled voltage-source converters[END_REF].

The VSC topology definition employed in this work is shown on Fig. 1. Together the DC-AC converter and the lowpass passive filter on the right, will be considered as the DG unit. LC filters are usually employed on industrial applications to mitigate high frequency content on output current and voltage waveforms [START_REF] Monfared | Analysis, design, and experimental verification of a synchronous reference frame voltage control for single-phase inverters[END_REF]. An interconnection line between the microgrid and the DG is modeled by parameters R s and L s . Knowing that the microgrid is a low or medium voltage system, the resistive parameter of the line cannot be neglected.

v i v o v l i i i o i f C f L f L s R s Inverter Filter Line Grid Bus V dc i a i dc C dc
When the interface impedance between a generator and a grid bus has a negligible resistive component, and the angular difference θ between the generator voltage and the bus voltage is very small, some behaviors can be assumed for active and reactive powers drawn by the bus. In this case, active power P strongly depends on the angle θ, while reactive power Q depends on the voltage amplitude difference through the impedance [START_REF] Bergen | Power Systems Analysis[END_REF], [START_REF] Glover | Power Systems Analysis & Design[END_REF]. These relationships are exploited in large scale systems, where synchronous generators drop their frequencies when output power is increased. On the other hand, when the inductive part of the impedance is negligible, the roles are reversed. Reactive power Q depends on the angle θ, while active power P depends on the voltage amplitude difference through the impedance [START_REF] Bergen | Power Systems Analysis[END_REF], [START_REF] Glover | Power Systems Analysis & Design[END_REF]. A VSC controlled with a conventional power sharing droop method employs artificial droops to emulate these behaviors [START_REF] Sao | Autonomous load sharing of voltage source converters[END_REF], [START_REF] Guerrero | Output impedance design of parallel-connected ups inverters with wireless load-sharing control[END_REF]. The basic idea is to mimic the governor of a synchronous generator [START_REF] Pogaku | Modeling, analysis and testing of autonomous operation of an inverter-based microgrid[END_REF]. One important advantage of this type of methods is that they offer the option of load sharing without communication links between DGs [START_REF] Guerrero | Advanced control architectures for intelligent microgrids;part i: Decentralized and hierarchical control[END_REF], [START_REF] Majumder | Improvement of stability and load sharing in an autonomous microgrid using supplementary droop control loop[END_REF]. This is desirable because the microgrid can span over a wide area [START_REF] Majumder | Improvement of stability and load sharing in an autonomous microgrid using supplementary droop control loop[END_REF]- [START_REF] He | Flexible microgrid power quality enhancement using adaptive hybrid voltage and current controller[END_REF].

The conventional frequency/voltage droop method is a wellstudied strategy, however it has important drawbacks. Although it is useful to share a common active load between DGs, the reactive power control scheme is strongly dependent on the grid parameters [START_REF] Sao | Autonomous load sharing of voltage source converters[END_REF], [START_REF] Guerrero | Advanced control architectures for intelligent microgrids;part i: Decentralized and hierarchical control[END_REF], [START_REF] Vasquez | Modeling, analysis, and design of stationary-reference-frame droop-controlled parallel three-phase voltage source inverters[END_REF]. The power sharing accuracy is strongly affected by impedance unbalances and there exist a tradeoff between power sharing accuracy and output voltage regulation [START_REF] Guerrero | Output impedance design of parallel-connected ups inverters with wireless load-sharing control[END_REF]. The imminent frequency and magnitude deviations can strongly affect the system stability [START_REF] Guerrero | Output impedance design of parallel-connected ups inverters with wireless load-sharing control[END_REF]. High droop gains have a negative impact on stability of the system [START_REF] Majumder | Improvement of stability and load sharing in an autonomous microgrid using supplementary droop control loop[END_REF]. It is not suitable when nonlinear load is shared [START_REF] Guerrero | Advanced control architectures for intelligent microgrids;part i: Decentralized and hierarchical control[END_REF], [START_REF] Vasquez | Modeling, analysis, and design of stationary-reference-frame droop-controlled parallel three-phase voltage source inverters[END_REF]. Several works have been reported to overcome these problems and provide improved solutions to the control of stand-alone mode microgrids [START_REF] Sao | Autonomous load sharing of voltage source converters[END_REF], [START_REF] Guerrero | Output impedance design of parallel-connected ups inverters with wireless load-sharing control[END_REF], [START_REF] Etemadi | A decentralized robust control strategy for multi-der microgrids 2014;part i: Fundamental concepts[END_REF]- [START_REF] Savaghebi | Autonomous voltage unbalance compensation in an islanded droop-controlled microgrid[END_REF].

The aim of this paper is to provide the theoretical base and an experimental evaluation of a new strategy to control VSCs based on instantaneous measurements and calculations of voltages and currents and the application of Instantaneous Power Theory. The approach defines a theoretical methodology to define a grid status parameter that provides multiple alternatives to operate the VSC autonomously. One of the alternatives of the proposed technique achieves load sharing among VSCs. Contrarily to droop control based techniques, these alternative works without employing frequency fluctuation as a communication link between VSCs.

The paper is organized as follows. In section II a brief review of the Instantaneous power theory is presented, along with some important related facts. Section III describes the theoretical approach employed to formulate the new strategy. Also in this section, some important aspects of the strategy are presented along with a description of the multiple operative alternatives provided for the VSC. Section IV presents the extension of the proposed approach in order to manage nonlinear loads. Section V describes the proposed load sharing scheme. Section VI shows details of the controller of the VSCs, with the proposed strategy immersed. In section VII, the experimental validation is described along with the description of obtained experimental results. Finally, section VIII concludes the paper.

II. INSTANTANEOUS POWER THEORY

The instantaneous power theory, or pq Theory, defines a set of instantaneous powers in the time domain. Since this theory does not impose any restriction to waveforms, it is applicable to non-sinusoidal currents and voltages and not only on steady state but also during transient states [START_REF] Akagi | Instantaneous Power Theory and Applications to Power Conditioning[END_REF].

The Clarke transformation maps a three-phase system of instantaneous voltages or currents on the abc frame, into instantaneous voltages or currents on the αβ0 frame. The transformation for generic quantities is given by,

  α β 0   = 2 3    1 -1 2 -1 2 0 √ 3 2 - √ 3 2 1 √ 2 1 √ 2 1 √ 2      a b c  
On a three-phase three-wire system, the zero-sequence component of current do not exist. On a three-phase fourwire system the zero-sequence component of voltage do not exist if three-phase voltages are balanced. For example, let us assume a three-phase ideal voltage source feeding a three-phase balanced load. Applying a system of three-phase voltages containing only positive sequence on the fundamental frequency (sinusoidal and balanced) to the Clarke transformation, provides the following quantities on the αβ frame,

v α (t) = √ 3V cos (ωt) v β (t) = √ 3V sin (ωt) (1)
The constant value V is the RMS value of voltage waveforms on the three-phase system. Thus, the Clarke transformation represents a three-phase system of instantaneous quantities, in the fundamental positive sequence, as a rotating vector on the αβ frame.

Instantaneous powers (instantaneous real power p and instantaneous imaginary power q) in the p-q Theory are defined by p = v α i α + v β i β and q = v β i α -v α i β , or equivalently by the following two expressions [START_REF] Akagi | Instantaneous Power Theory and Applications to Power Conditioning[END_REF]. Additionally, the instantaneous zero-sequence power is defined by p 0 = v 0 i 0 . Applying these definitions to the case where the three-phase system of sinusoidal balanced voltages feed a linear load (which draws sinusoidal current waveforms), gives the following expressions for instantaneous real and imaginary powers.

p q = v α v β v β -v α i α i β (2)
p q = i α i β -i β i α v α v β (3) L f Ls Rs io i i vo C f V dc io Ls Rs L l R l Zo v l v l vo Zs Z load Zo = Zs + Z load
p = 3V Icos (φ) q = -3V Isin (φ) (4)
These expressions are equal to the conventional definitions of three-phase active and reactive powers. As it can be noticed, real and imaginary instantaneous powers are constant (average powers) over time when voltages and currents exhibit only positive sequence on the fundamental frequency. It should be noticed that if voltages and powers are known variables, currents can be determined by applying the inverse matrix in [START_REF] Kroposki | Making microgrids work[END_REF]. On the other hand, if currents and powers are known variables, voltages can be determined by applying the inverse matrix in [START_REF] Hatziargyriou | Microgrids[END_REF]. Expressions resulting from these procedures are given by,

i α i β = 1 v 2 α + v 2 β v α v β v β -v α p q (5)
v α v β = 1 i 2 α + i 2 β i α -i β i β i α p q (6)

III. NEW APPROACH -STRATEGY TO FIND φ ex

Let us assume that VSCs with their corresponding LC filters behave like current sources. The current source is connected to a grid through a line with parameters L s and R s as shown on Fig. 2.

In a first case, the local current source is not operating (i o = 0), while the other sources are working. Therefore there is no voltage drop over the line and v o = v l . By the superposition theorem, the voltage v l is the algebraic sum of voltages v lρ , each one contributed by the generator ρ acting alone.

In a second case, the local source is working (i o = 0), while the other sources are not operating (zero output current). Assuming steady state, if quantities are treated like phasors,

v o = Z o i o ,
where Z o is the equivalent impedance from the filter output as seen on Fig. 2. Thus, the angular displacement between i o and v o , and the magnitude of v o , will remain constant under variations on the phase angle φ io of i o .

v o = v lc + M ρ=2 v lρ = v lc + v ex (7)
Assuming each source has a corresponding index ρ and the local source is ρ = 1, v o is the sum of voltages contributed by each source operating alone, and is given by [START_REF] Bongiorno | Dynamic performance of vector current controllers for grid-connected vsc under voltage dips[END_REF]. v lc is the contribution of the local current source and v ex is the sum of the contributions from the other sources. As it was explained before, v ex is fully independent from i o , while v lc depends on i o and the equivalent impedance Z o , and is completely independent from other generators.

i oα = √ 3I o cos (ωt + φ io ) i oβ = √ 3I o sin (ωt + φ io ) (8)
v oα = √ 3V o cos (ωt + φ v ) v oβ = √ 3V o sin (ωt + φ v ) (9)
Let us assume i o and v o are vectors representing three-phase positive-sequence currents and voltages, on the αβ frame. The instantaneous representation of their components are given by ( 8) and ( 9) respectively, where I o and V o are the corresponding RMS values of the abc quantities.

v oα = v α lc + v αex v oβ = v β lc + v βex (10)
v α lc = √ 3V lc cos (ωt + φ lc ) v β lc = √ 3V lc sin (ωt + φ lc ) (11)
v αex = √ 3V ex cos (ωt + φ ex ) v βex = √ 3V ex sin (ωt + φ ex ) (12)
The αβ components of v o can be disaggregated into the local and external voltage contributions given by [START_REF] Bergen | Power Systems Analysis[END_REF]. These local and external contributions can be described in an instantaneous way. Because of the linearity of the system and the homogeneity in the frequency of all the current sources, αβ components of the external contribution are sums of sinusoidal functions which are sinusoidal functions as well. The αβ components of the local term are evidently sinusoidal because they depend only on i o . Local and external αβ components are described by ( 11) and ( 12) respectively. It is important to notice that V lc = I o |Z o |, and φ lc = φ io + φ Zo , where φ Zo is the phase angle of the equivalent impedance.

However, a practical superposition analysis is not viable because it requires turning off generators, interrupting the power supply to loads. Thus, measuring v α lc , v αex , v β lc and v βex is not possible, only quantities v oα and v oβ are available to be measured and the local and external quantities are immersed in these measurements.

A. Instantaneous power theory application

Instantaneous real and imaginary powers are defined by p = v α i α +v β i β and q = v β i α -v α i β respectively [START_REF] Akagi | Instantaneous Power Theory and Applications to Power Conditioning[END_REF]. Computing these expressions with ( 8) and ( 9) provides,

p = 3V o I o cos (φ vo -φ io ) q = 3V o I o sin (φ vo -φ io ) (13)
Instantaneous powers are constant and independent of time because voltages and currents have only fundamental positivesequence components. Real and imaginary powers can be computed with ( 8) and (10) as well.

p = 3V lc I o cos (φ lc -φ io ) + 3V ex I o cos (φ ex -φ io ) q = 3V lc I o sin (φ lc -φ io ) + 3V ex I o sin (φ ex -φ io ) (14)
Expressions on ( 13) and ( 14) are equivalent. Although p and q are the only possible quantities to be measured, it should be noticed that they can also be disaggregated in local and external contributions p = p lc + p ex and q = q lc + q ex .

p = 3V lc I o cos φ Zo + 3V ex I o cos (φ ex -φ io ) q = 3V lc I o sin φ Zo + 3V ex I o sin (φ ex -φ io ) (15)
Taking into account that φ lc = φ io + φ Zo , where φ Zo is the phase angle of the equivalent impedance, expressions on [START_REF] Chaudhary | Negative sequence current control in wind power plants with vsc-hvdc connection[END_REF] indicate that under variations of φ io , powers p lc and q lc remain constant while p ex and q ex change. Furthermore, the maximum real power transfer is achieved when φ i = φ ex , i.e., when the current is in phase with the sum of external voltage contributions.

∂p ∂φ io = ∂p ex ∂φ io = 3V ex I o sin (φ ex -φ io ) (16)
∂q ∂φ io = ∂q ex ∂φ io = -3V ex I o cos (φ ex -φ io ) (17)
As it was mentioned before, the local contributions to p and q remain constant under variations on φ io . Hence, the variations on p and q under variations of φ io are equal to the variations on p ex and q ex under variations of φ io , respectively. These behaviors are represented in [START_REF] Ottersten | Vector current controlled voltage source converter-deadbeat control and saturation strategies[END_REF] and [START_REF] Svensson | Grid-connected voltage source converter -control principles and wind energy applications[END_REF] which are the partial derivatives of p and q with respect to φ io .

V ex ≈ ∆p 2 + ∆q 2 3I o ∆φ io (18)
φ ex ≈ φ io + tan -1 ∆p -∆q (19)
It should be noticed that ∂p/∂φ io = q ex and ∂q/∂φ io = -p ex . With small finite variations in φ io , the system can be perturbed and the consequent changes in p and q can be employed to approximate the partial derivatives with ∆p/∆φ io and ∆q/∆φ io . These approximations provide an evidence of the quantities p ex and q ex which can be employed to compute an approximation to the external voltage contribution v ex . Therefore, the approximate RMS value of the external voltage contribution is given by [START_REF] He | An islanding microgrid power sharing approach using enhanced virtual impedance control scheme[END_REF], while an approximation of its angle is given by [START_REF] He | Flexible microgrid power quality enhancement using adaptive hybrid voltage and current controller[END_REF].

p lc = p -p ex q lc = q -q ex (20)
With the approximation of external power contributions, local power contributions can be computed with [START_REF] Vasquez | Modeling, analysis, and design of stationary-reference-frame droop-controlled parallel three-phase voltage source inverters[END_REF]. The obtained approximations for the local power contributions are an important output of the proposed strategy. These powers provide information of the equivalent impedance of the grid Z o , seen from the output of the generator (the generator is assumed as the ensemble inverter-filter) as it is shown on Fig. 2.

V lc = p 2 lc + q 2 lc 3I o ( 21
)
|Z o | = V lc I o = p 2 lc + q 2 lc 3I 2 o ( 22
)
φ Zo = tan -1 q lc p lc [START_REF] Lopes | Defining control strategies for microgrids islanded operation[END_REF] Once the approximations to the local power contributions are found, the approximate RMS value of the local voltage contribution is given by [START_REF] Etemadi | A decentralized robust control strategy for multi-der microgrids 2014;part i: Fundamental concepts[END_REF]. Thus, the approximate magnitude of the equivalent impedance can be computed with [START_REF] Ahn | Power-sharing method of multiple distributed generators considering control modes and configurations of a microgrid[END_REF], while the equivalent impedance angle can be computed with [START_REF] Lopes | Defining control strategies for microgrids islanded operation[END_REF].

B. Alternative calculations

Real and imaginary instantaneous powers can also be computed with components on the rotative dq reference frame as follows [START_REF] Akagi | Instantaneous Power Theory and Applications to Power Conditioning[END_REF],

p q = i d i q -i q i d v d v q
Thus, if currents and powers are known variables and voltages are unknowns, then the inverse of the matrix of currents can be applied to the expression above and the unknown voltages can be obtained as follows [START_REF] Akagi | Instantaneous Power Theory and Applications to Power Conditioning[END_REF],

v d v q = 1 i 2 d + i 2 q i d -i q i q i d p q (24)
With [START_REF] Zhang | Theoretical and experimental investigation of networked control for parallel operation of inverters[END_REF], and the discrete approximations to the external contributions of real and imaginary power p ex = -∆q/∆φ io and q ex = ∆p/∆φ io , the external voltage contribution can be computed.

v dex v qex ≈ 1 i 2 od + i 2 oq i od -i oq i oq i od -∆q ∆p 1 ∆φ io (25)
v lc = v o -v ex (26)
The external voltage contribution components on the dq frame are given by [START_REF] Savaghebi | Autonomous voltage unbalance compensation in an islanded droop-controlled microgrid[END_REF]. As it was mentioned before, the output current should be in phase with this voltage in order to get maximum real power transference to the grid. With v ex defined on the dq frame and the measurement of v o , the local voltage contribution vector v lc can be computed with [START_REF] Akagi | Instantaneous Power Theory and Applications to Power Conditioning[END_REF].

C. Controlling output current angle φ io and output powers

The vector diagram on Fig. 3(a) can be constructed as a result of the outcomes of the proposed strategy (v ex , φ ex , Z o , φ Zo , v lc ).

If the reference frame is always chosen to be aligned with the output current vector as in Fig. 3(a), the effect of rotating i o by an amount φ io is observed in this frame as a rotation of v ex by an amount -φ io . On the other hand, the local voltage

φ Zo i o v lc φex v ex v o d 1 q 1 (a) p q p ex q ex q max q min p max φex -φ io = -π 2 φex -φ io = π 2 φex -φ io = 0 φex -φ io = φ Zo {p = p lc , q = qmax} {|Vo |max } {p = p lc , q = q min } {p = pmax , q = q lc } p lc q lc 0 0 φ vo -φ io = 0 {q = 0} φ Zo p min φ vo -φ io = π 2 {p = 0} φ vo -φ io = -π 2 {p = 0} φ Zo (b)
Fig. 3. VSC control strategy: (a) d 1 q 1 reference frame with d 1 axis always aligned with output current, (b)pqφ io control diagram contribution vector v lc maintains it angular difference φ Zo with respect to i o because rotating the output current vector does not affect the equivalent impedance at all. Assuming a reference frame with the d axis aligned always with the output current vector i o , the external voltage contribution vector v ex can be rotated within this frame, in order to manipulate the instantaneous powers supplied by the VSC. The first alternative consists in obtaining maximum real output power. This alternative is achieved when φ ex -φ io = 0. In this case, the external voltage contribution vector v ex is aligned with the output current vector i o , maximizing the external real power contribution p ex , while the external imaginary power contribution is q ex = 0.

The second alternative consists in obtaining maximum output voltage amplitude. This alternative is achieved when φ ex -φ io = φ Zo . In this case, the external voltage contribution vector v ex is aligned with the local voltage contribution vector v lc , maximizing the magnitude of the output voltage vector v o = v lc + v ex and the magnitude of the vector of powers

(p lc + p ex ) 2 + (q lc + q ex ) 2 .
The third alternative consists in obtaining maximum imaginary output power. This alternative is achieved when φ exφ io = π/2. In this case, the external voltage contribution vector v ex is perpendicular to the output current vector i o (output current lagging external voltage contribution), maximizing the external imaginary power contribution q ex , while the external real power contribution is p ex = 0.

The fourth alternative consists in obtaining minimum imaginary output power. This alternative is achieved when φ exφ io = -π/2. In this case, the external voltage contribution vector v ex is perpendicular to the output current vector i o (output current leading external voltage contribution), minimizing the external imaginary power contribution q ex , while the external real power contribution is p ex = 0. If |v ex | ≥ |v lc |, it is possible to obtain a capacitive behavior (negative q min ).

The fifth alternative consists in obtaining zero imaginary output power. This alternative is achieved when φ v -φ io = 0. In this case, the output voltage vector v o is aligned with the output current vector i o , compensating the local imaginary power contribution with the external imaginary power contribution. This is possible only if

|v ex | ≥ |v lc |.
Additional alternatives can be explored. For example, there are two grayed areas on Fig. 3. The dark grayed area corresponds to the portion of the circumference to the right of the q axis, where the preceding alternatives lie. Here the VSC behaves like a generator. The light grayed area corresponds the portion of the circumference to the left of the q axis, where the generator absorbs real power, behaving like a controllable load or like a storage unit. This last portion of the circumference is defined by the angular differences φ v -φ i = π/2 and φ v -φ i = -π/2, where the output voltage vector v o is perpendicular to the output current vector i o and the real output power p = p lc + p ex is zero. This alternative is also possible only if

|v ex | ≥ |v lc |.
In the case of maximum output voltage amplitude |v o | max , it is possible to manipulate the magnitude value of the output current vector i o while maintaining the angular difference φ ex -φ io = φ Zo . This parameter can be manipulated in order to share load while regulating the output voltage magnitude and keep it within security levels. Besides, a stability limit can established when the magnitude of the output current cannot be increased enough to secure output voltage magnitude within these security levels.

IV. NONLINEAR LOAD MANAGEMENT

Let us assume there is a nonlinear load on the microgrid, then a similar analysis to the one shown on Fig. 2 and section III can be applied. If the control action is only applied to fundamental positive sequence output current i of , then the nonlinear load will be excited and will consume a portion of the fundamental current i rf . At the same time it will inject harmonic currents to the grid, represented on the quantity i rh .

A portion i ohρ of these currents will naturally flow through the grid to the capacitors of the filters (which represent small impedances at higher frequencies) corresponding to each VSC (with sub-index ρ), while the other portion i lh will flow through the linear loads. This last portion will cause the appearance of harmonic components on the voltages of the grid. This scenario is represented on Fig. 4(a).

Assuming that the harmonic components of the grid voltages are eliminated by some additional control action over the VSCs output currents, then the currents consumed by the linear loads will have only fundamental component. This implies that the harmonic currents i rh generated by the nonlinear loads are consumed by the VSCs. Then an equivalent scenario can be formulated in Fig. 4(b). Here, the nonlinear loads are equivalently represented by a controlled current source injecting harmonic currents and an effective portion of load consuming fundamental component of current (represented with parameters L r and R r ). At the same time, the equivalent linear load is represented with parameters L l and R l . This two linear loads (Z load ) and the impedance of the connection line, represent the equivalent impedance that the proposed technique is able to find without any modification. The harmonic current source is a controlled source depending on the voltages feeding the nonlinear loads. However, given the fact that it does not provide any fundamental component of current, it can be neglected from the superposition analysis, which is performed over fundamental positive sequence sources of current.

Given the fact that the X/R ratio is very small in this type of systems (R >> X), then the harmonic components of the voltage v lh on the linear portion of the load can be evidenced on the harmonic components v oh of the voltage v o on the capacitor on the filter of the VSC. On the other hand, given that inductive loads increase their impedance at higher frequencies, then it is assumed that the harmonics on the equivalent linear load are proportional to the resistive portion of Z o . Assuming a close to unity power factor on the equivalent linear load, then Output Currents

vlc 1 vlc 1 vlc 1 vlc 3 vlc 2 vlc 2 vlc 3 vlc 3 vlc 2 io 1 io 2 io 3 io 1 io 1 io 2 io 3 io 2 io 3 1 st Step 2 nd
Step 3 rd Step Start

i lh ≈ v oh |Z o | + i ohρ |Z s |
The second term can be eliminated because the impedance of the line can be neglected. As a consequence, a benefit of finding Z o is that every VSC has valuable information to be able to correct the harmonics on the voltage of the grid. To achieve that, besides the corresponding i ohρ attracted by the capacitor, the VSCs have to consume a portion of the harmonic current i lh proportional to the number of VSCs.

i dqo ≈ v oh M |Z o | -i ohρ (27)
Following the sign convention of Fig. 4, expression ( 27) is included as an additional term for the reference of the output current in order to share the consumption of the harmonic currents, and as a consequence, the compensation of voltage harmonics.

The unbalanced condition condition is not treated in this manuscript. However, as the VSC controls the fundamental positive sequence output current, the effect of unbalanced load will be observed in the voltage of the capacitor as well. The result will be zero and negative sequence components. The information to compute the equivalent impedance Z o is available in the positive sequence component of the voltage while the other to components will provide information of the unbalance. As in the case of harmonic content, the controller has to act over negative and zero sequences with the information provided by the strategy, in order to share the compensation of the unbalanced voltages.

V dc sC f ωC f ωC f sC f - i * o d i * oq i * i d i * iq vo q vo d vo dq |x| = x vo dq |vo dq | * = vo dq * - PI i * o d i * oq 0 Parameter Estimation (vex, v lc , Zo , i hc o dq ) (a) (b) (c)
p = v d i d + vq iq q = vq i d -v d iq p q |vo dq | *
Fig. 6. Block diagram of the control approach. The "Output Power Control, Parameter Estimation" block performs the proposed strategy and provides the references for the output current fundamental positive sequence i * o d q , and the dq components for compensation of harmonic currents i hc o d q .

V. MAXIMUM OUTPUT VOLTAGE -LOAD SHARING

The power sharing approach that can be followed from the proposed strategy (maximum output voltage alternative) can be described with Fig. 5. The microgrid can be simplified as a set of current sources providing its output current to a bus where an equivalent load is connected as it is shown on Fig. 5(a). Thus, each current vector and its corresponding local voltage contribution have a phase difference between them, which is equal to the phase angle of the impedance φ Zo . For the purpose of the explanation, current and local voltage vectors are assumed unitary. Fig. 5(b) shows a random disposition of the local voltage vectors and their corresponding current vectors (out of phase by φ Zo ) at a start point. Let us assume that the first generator is a reference generator which does not perform any phase displacement. Thus, looking for maximizing its output voltage, the second generator displaces its current vector in order to align its local voltage vector with the sum of the other two local voltage vectors (the external voltage contribution) corresponding to generators 1 and 3. This results in the scenario of the 1 st step. Then, generator 3 finds out that the external voltage contribution that it sees is no longer the same. Then, it performs the same action, resulting in the scenario of the 2 nd step. Then, generator 2 displaces again by the same reason, which results in the 3 rd step. If this sequence of actions continues, then after a certain amount of steps, local voltages (and output currents) will be very close to each other, so it can be assumed they are aligned, maximizing the current provided to the load. As a consequence, voltage v o will be maximized and if generators provide currents with the same magnitude, then they will share the real and reactive powers demanded by the load.

The resulting voltage vector v o is included in the diagram of each step on Fig. 5(b) in order to notice the angle differences with respect to the output current vectors of each VSC. Although it depends on the equivalent load, this provides an idea of how active and reactive powers of each VSC could behave at each step.

VI. INVERTER CONTROL STRATEGY

A first approximation to the implementation of the proposed strategy is provided in this section and Fig. 6(a). In order to control the two-level three-phase inverter on the VSC topology as a current source, an hysteresis current control technique is employed [START_REF] Milosevic | On the control of distributed generation in power systems[END_REF], [START_REF] Kazmierkowski | Current control techniques for three-phase voltage-source pwm converters: a survey[END_REF]. This controller is intended to maintain the three phase balanced currents within a maximum error band.

The current vector i s o is considered as the contribution of the local VSC to share the load of the power microgrid with the other operating VSCs. Thus, the available reference is not the inverter output current reference i s * i but the filter output current

i s * o . C f dv s o dt = i s i -i s o (28)
The current that flows through the filter capacitor is given by [START_REF] Kazmierkowski | Current control techniques for three-phase voltage-source pwm converters: a survey[END_REF]. Quantities on a stationary αβ reference frame can be transformed to a rotating dq reference frame employing v s = v r e jωt , where the superscript r refers to rotative, and v r is the representation of v s on the rotative frame [START_REF] Monfared | Analysis, design, and experimental verification of a synchronous reference frame voltage control for single-phase inverters[END_REF], [START_REF] Zhang | Theoretical and experimental investigation of networked control for parallel operation of inverters[END_REF]. ωt is the rotation angle measured from the stationary α axis. Thus, the filter capacitor current on the rotative dq frame is given by,

C f dv r o dt + jωC f v r o = i r i -i r o
Thus, the inverter current dq references can be computed from the output current dq references by,

i * id = i * od + C f dv od dt -ωC f v oq i * iq = i * oq + C f dvoq dt + ωC f v od
These computations are performed on the block Inverter current reference of Fig. 6(a) which corresponds to Fig. 6(b). Inverter Current, Output Current and Output voltage abc quantities are measured, transformed to the dq frame which rotates at a fixed frequency of 50Hz and filtered to control only the fundamental positive sequence. A PI regulator is included to compensate steady state error on the dq components of the output current and improve the data employed by the strategy.

On the other hand, the references are provided by the Output Power Control block on Fig. 6(a), whose methodology is explained here. This block performs the proposed strategy and provides the references for the output current fundamental positive sequence i * o d q , and the dq components for compensation of harmonic currents i hc o d q . The block receives as inputs the computed instantaneous powers p and q, the corresponding dq filtered components of output current i o dq and voltage v o dq and the separated harmonic content of current and voltage in the dq frame (i h o dq and i h o dq ) [START_REF] Heredia | A comparative study between the dpc-svm and the multiresonant controller for power active filter applications[END_REF]. The block modifies the angle of the output current reference with an small perturbation step ∆φ io . The resulting active and reactive power deviations are applied to expressions [START_REF] He | An islanding microgrid power sharing approach using enhanced virtual impedance control scheme[END_REF] to [START_REF] Lopes | Defining control strategies for microgrids islanded operation[END_REF] (also alternatively [START_REF] Savaghebi | Autonomous voltage unbalance compensation in an islanded droop-controlled microgrid[END_REF] and ( 26) can be applied). Thus, quantities v ex , φ ex , Z o , φ Zo , v lc are obtained (the diagram of Fig. 3(a) can be constructed). Then, the output current references are modified again in order to Fig. 8. Experimental perturbation-displacement scenario achieve the power sharing objective (aligning the local voltage vector with the external voltage vector). On the other hand, the reference vector for compensation of harmonics is established with the obtained impedance Z o as explained in section IV.

One VSC is chosen to be the reference VSC, the other VSCs will be the tracking VSCs and will perform the strategy.

With respect to Voltage regulation, once the tracking VSCs have found their equivalent impedances, their output current magnitudes can be fixed to a value inversely proportional to the magnitude of the impedance and the number of VSCs in the microgrid, and proportional to a desired magnitude of voltage

|v o dq | * . |i o dq | * = |v o dq | * M |Z o | (29)
The reference magnitude can be computed with (29) for the tracking VSCs, where M is the total amount of VSCs. Since this reference is based on the assumption that the VSCs are sharing the demanded powers (Maximum Output Voltage Amplitude), the method will have an error until the tracker VSCs have found the reference VSC. The PI regulator on Fig. 6(c) is proposed for the reference VSC in order to reduce the error in steady state for the desired regulation magnitude of voltage.

VII. EXPERIMENTAL RESULTS

In order to test the proposed strategy and its maximum output voltage amplitude alternative for power sharing, an experimental validation is proposed with the scheme shown on Fig. 7. This experimental microgrid consists of three VSCs, three linear loads and a three-phase diode bridge rectifier in a Real-Time(RT) Power Hardware-in-the-Loop (PHIL) test scheme. Two of the VSCs and two of the linear loads are RT emulated with an RT-LAB HILBox 4U platform. This portion of the grid is modeled with MATLAB/Simulink and its parameters are included in the scheme. The third VSC, the A three-phase power amplifier Puissance+ PCU 3x5000 is employed as power interface between the PHIL portion and the RT portion of the grid. The power amplifier provides a voltage while its output current is measured with OP5511 sensors to provide a power feedback loop to the RT portion of the grid (the power amplifier has an operating bandwidth from DC to 20kHz and a 5kVA/phase maximal power). There is no communication link between VSCs, the controller for the real inverter is implemented in MATLAB/Simulink and compiled for DS1005 dSpace target hardware. The RT VSCs controllers are implemented in MATLAB/Simulink as well, and the corresponding compilation is sent via TCP/IP to the RT-LAB platform [START_REF] Gaztanaga | Real-time analysis of the control structure and management functions of a hybrid microgrid system[END_REF]- [START_REF] Dufour | Review of stateof-the-art solver solutions for hil simulation of power systems, power electronic and motor drives[END_REF]. In this test, the output currents magnitudes are fixed to 7A peak to easily show the performance of the proposed technique.

Fig. 8 shows a scenario (with linear load) where the RT VSC modifies its reference output current angle by a quantity ∆φ io = 5 • . This modification of the reference angle represents a perturbation to the grid. Then it computes a desired phase displacement based on the computed parameters (v ex , φ ex , Z o , φ Zo , v lc ), and 400ms later it rotates its current vector to the desired position. As it was mentioned on section V, it the maximum output voltage amplitude alternative, each VSC tries to align its local voltage contribution vector with the external voltage contribution vector, resulting in a phase displacement relatively close to the other VSC's current vectors.

A similar test is performed with the nonlinear load. Once each tracking VSC has made its first perturbation, it can approximate its equivalent linear load. From this computation each VSC is able to compensate the harmonic content on the grid voltage. Also, with each perturbation it is able to update the value of Z o . Fig. 9 shows the phase A output currents from each VSC performing the harmonic compensation. It is important to notice that the closeness of the VSC 1 filter capacitor to the nonlinear load forces it to attract more harmonic currents than the other capacitors, however the compensation Voltage of the power amplifier (phase A) before and after compensation is performed by each tracking VSC as it can be seen on Fig. 9. A contrast between compensation and non-compensation cases can be seen on Fig. 10 and Fig. 11 for the rectifier current and the amplifier output voltage. As it can be observed, the compensation eliminates a fifth-harmonic on the voltage and improves the rectifier current.

Under a certain amount of perturbations from the tracking VSCs, they are able to find the reference which is assigned to VSC 3. As it was mentioned before, when the trackers achieve to find the reference, then the active and reactive powers are shared. This can be observed in Fig. 12 where two cases are exposed for the filtered output power profiles. In both cases, the initial angle difference between VSC 2 and the reference VSC is 45 • (this value was arbitrary chosen for the experimental evaluation). In both cases, the initial angle difference between VSC 1 and the reference VSC is unknown because it depends on the non-synchronized initialization of both portions of the grid (physical and real-time emulated). In both cases it can be observed that the VSCs get close to the alignment of their currents, each providing a third part of the maximum power that can be delivered to the grid at a fixed 7A peak current for each VSC. The impedance magnitude seen from the emulated VSCs is about 5, 74Ω, while the impedance seen from the real VSC is about 5, 98Ω. This implies an slight difference between the delivered active powers, which assuming an alignment of currents (21A peak), will be around 850W per VSC.

A similar arbitrary case is registered for the reactive power. As it can be seen from Fig. 7, the demanded reactive power is very small, thus the proposed approach should make the VSCs to get close to zero reactive output power. The reactive power sharing is shown on Fig. 13. As it can be seen, after the fifth perturbation, the VSCs get very close to the desired output. It is also important to highlight that the time for perturbations is arbitrary chosen for the illustration of the approach. As it can be seen the time between perturbations can be smaller and each VSC can decide when to perturb depending on the error between its reference and its output.

VIII. CONCLUSION

A new strategy to control microgrids highly penetrated by Voltage Source Converters (VSC) and operating in stand-alone mode is proposed. The strategy is based on instantaneous measurements and calculations of voltages and currents and the application of Instantaneous Power Theory. This approach employs each VSC along with an LC filter as a current source. The grid parameters are not known by the controller, only the filter inductance and capacitance are known. The approach is capable of giving an approximation of the equivalent impedance of the system seen from the filter output, without difficult procedures or measurements. Additionally, it offers great versatility to control the generator and make profit from the different alternatives according to output powers. Results of the first approximation show the potential of the strategy. However, a more robust strategy to perturb the grid and automatically decide when to perturb should be further studied in order to improve the performance of the proposed strategy.Two alternative methods are provided to find an approximation to the angle of the external voltage contribution φ ex . Besides, every possible application derivable from this angle is detailed, under the proposed strategy. The methodology should be further studied in order to treat unbalanced conditions, however the experimental results have shown its potential application.

Fig. 1 .

 1 Fig. 1. VSC topology

Fig. 2 .

 2 Fig. 2. VSC current source scheme proposed

Fig. 3 (

 3 b) shows some of the multiple available alternatives to control the instantaneous output powers. This Fig. shows the case of inductive equivalent impedance. These alternatives can be achieved once the external voltage contribution angle φ ex and the equivalent impedance angle φ Zo are found. With the control action of φ io , the output real and imaginary powers lie in the circumference shown in this figure.

Fig. 4 .

 4 Fig.4. VSC as harmonic compensator, Approach followed.

Fig. 5 .

 5 Fig. 5. Power sharing approach. Diagrams for explanation

Fig. 7 .

 7 Fig. 7. Experimental Test grid for validation

Fig. 9 .

 9 Fig. 9. Output currents (phase A) compensating harmonics of voltage during the experiment

  Fig. 11.Voltage of the power amplifier (phase A) before and after compensation

Fig. 12 .

 12 Fig. 12. Two experimental scenarios and the resulting active power sharing.

Fig. 13 .

 13 Fig. 13. The resulting reactive power sharing for an experimental scenario.