Optimal Operation and Control for an Electrical Micro-grid

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1 Optimal Operation and Control for an Electrical Micro-grid Fernando Ornelas-Tellez, Jose Ortiz-Bejar and J. J. Rico School of Electrical Engineering Universidad Michoacana de San Nicolas de Hidalgo Ciudad Universitaria, Morelia, 583, Mexico Edgar N. Sanchez CINVESTAV-Guadalajara Av. del Bosque 1145, Zapopan, 4519, Mexico Abstract This paper presents an optimization and control scheme for power converters in a micro-grid, which is composed of a wind energy system, an energy storage element (supercapacitor), a load and the interconnection to the utility grid. Based on the results of a dynamic optimization model, which establish the energy flow in the micro-grid, an optimal control scheme uses these results of electrical power values as set-points to efficiently integrate renewable energy to/from the utility grid through the optimal control of power converters. A case study micro-grid is used to integrate the energy from renewable resources, with the facility to storage energy, to provide of energy to loads and to provide/consume energy to/from the utility grid. Simulation results are presented to assess the performance of the proposed controller for the case study micro-grid. Index terms: Micro-grid, Optimal operation, Power converters. I. INTRODUCTION Due the interest in exploiting renewable energy sources, it is common to integrate such energy resources through distributed power generation systems. These systems usually constitute a micro-grid, which is composed of multiple generation sources (solar, wind, etc.), energy storage devices, local loads and the connection to the utility grid for exchanging energy. In general, the functionality and the interconnection between the components of the micro-grid is realized by means of power converters, which requires of adequate control techniques to provide the micro-grid of reliability and efficiency. Correctly designed, power converters and controllers ensure that the micro-grid can meet its own as well as the utilities needs in an economic way. The basic inputs to the power converter controllers are the set points or references for the exchanged power among components of the micro-grid, or best levels of local bus voltages and currents determined in accordance to energy to be transferred. For instance, to achieve energy transfer among multi-sources the authors in [1] have proposed an energy router that dynamically controls energy flow between sources, it is based on an idea described by Duindam and Stramigioli in [2]. In [1] the feedback linearization technique and the Proportional-Integral (PI) scheme are used for controlling the energy flow between different sources. A main drawback of this controller is that it does not work well when the system is nonlinear and its operating point is different from the point for which the controller was tuned /15/$31. c 215 IEEE This paper proposes an optimization and optimal control strategy for a micro-grid. The optimization strategy is used to determine the energy to be managed by power converters in the micro-grid through its respective control system. The proposed controller ensures the system stability and an efficient operation of the converters. A general scheme for integrating renewable resources is proposed, which can easily include new sources, loads, storage elements and the connection with the utility grid. Simulation results demonstrate the adequate operation of the micro-grid. The organization of this paper is described as follows. Section II presents the optimization scheme for a micro-grid. Section III presents the optimal control for state dependent coefficient factorized (SDCF) nonlinear system. In Section IV, the proposed optimal control scheme is applied for a case study micro-grid. Simulation results are presented in IV-F. Finally, Section V concludes the paper. II. THE MICRO-GRID OPTIMIZATION The proposed micro-grid to be optimized is shown in Fig. 1. The figure illustrates the energy flow between the components of the micro-grid, which is composed of a wind energy system (with electrical power P W ), an energy storage element (a supercapacitor) (with power P S ), a load (with power P L ) and the interconnection to the utility grid (with power P G ). The amount and direction of the energy is determined by an optimization algorithm, which considers the prediction of the wind energy availability, energy demand and the costs of electrical energy along a period of time T. This section reviews the results of an optimization scheme for an electrical micro-grid based on the forecast of time series, such as load energy demand, electrical energy prices and the forecast for generated wind energy in a micro-grid. The main objective of the optimization scheme is to determine the optimal amount of energy to be transferred from/to the storage element and the utility grid, such that it is minimized the cost associated to the energy consumption from the utility grid and simultaneously it is maximized the profit due to the sale of electrical energy stored and generated in the utility grid.

2 subject to: P G, k P S, k = P L, k P W, k V C,k1 = αv C,k βi k (i k1 i k ) 2 γ 2 V Cmin V C,k V Cmax (1) i max i k i max P G max P G, k P G max V C, =.7 V Cnom V C,T =.85 V Cnom Fig. 1. Optimization scheme for the micro-grid. A. Optimization Model The optimization results in this section were presented in [3], which now are revised and will be used to determine the set-points for power converters to transfer the required electrical power. The objective function to be minimized in the optimization process corresponds to a function composed of 24 future values (i.e., T = 24) for the variables, where the values are taken in periods of 1 hour. This objective function considers the maximization of the profits associated with the sale of the generated energy to the utility grid and at same time reducing the consumption from the utility grid. The future values in the optimization problem are the predicted values for each respective time series, corresponding to wind energy, buying/selling energy prices and load profile, such that an efficient use of energy generated in the microgrid can be achieved through the determination of the power to be storage and injected/subtracted to/from the utility grid. In order to determine the time series models, we use the software R, which is an open source numerical system. This software includes the package forecast, which provides means for automatic generation of Autoregressive integrated Moving Average (ARIMA) and Exponential Smoothing (ETS) models to forecast time series such as load demand, energy prices and wind energy. The corresponding model is selected in accordance with least prediction error. The models are used to perform a forecast of 24 values for each series, all in time intervals of 1 hour. For details on the respective time series forecasting, see [3]. The proposed model is given as min T 1 λ k [C B, k P L, k P G, k C S, k P W, k P S, k ] k= where < λ 1 is a discount factor, C B, k is the cost per kwh associated to the buying of electrical energy from the utility grid, C S, k is the cost per kwh associated to the selling of electrical energy toward the utility grid, V C,min =.7V Cnom and V C,max = V Cnom, V Cnom is the nominal voltage of the storage device, γ is a positive constant which limits the current flow. The decision variables are P G,k and P S,k = V C,k i k, where V C,k and i k are the supercapacitor voltage and current, respectively. Parameters α and β correspond to the dynamical model of the supercapacitor. In the optimization model, we used the value T = 24 to take the decision of charging or discharging the storage element (supercapacitor). Note that the dynamics of the storage element in the optimization model (1) establishes a close-loop between the optimization scheme and the control of the electrical power to be managed by the converters. In this sense, the consideration of the storage element dynamics allows to the optimization algorithm to maximize the micro-grid profits by evaluating in advance the amount of the energy to be stored/extracted in accordance with the energy availability and prices. B. Optimization Results The values used for the optimization model are: T = 24 hrs, λ =.995, the sampling time T s =1hr, γ =4, R C = ohms, C = 1F, V Cnom = 12 V, i max = 1 A and i max = 1 A. Fig. 2 shows the forecast-based optimization results, where in the top of the figure it is shown the power flow of the microgrid components and in the bottom figure the respective prices for the energy. The results are described as follows: For the time (1, 7) hrs: the generated energy is greater than the consumed by the load, and besides the energy is cheap, then energy is extracted from the utility grid and stored in the supercapacitor. For the time (7, 11) hrs: the energy consumption by the load is greater than the generated by the wind system, and the energy is expensive, thus the energy is taken from the supercapacitor and only a small amount of energy is taken from the utility grid. For the time (11, 18) hrs: the generated energy is greater than the consumed by the load, and the energy is cheap, hence there is energy to be sold to the utility grid and to charge the supercapacitor.

3 (a) Optimization results. (b) Buying/selling energy prices. Fig. 2. Optimization results based on forecast and energy prices. Finally, for the time (18, 24) hrs: the load energy consumption is greater than the generated, and besides the energy is expensive, therefore the stored energy is used by the load and additionally, it is required to buy energy from the utility grid in order to fully provide from energy to the load. III. OPTIMAL CONTROL FOR SDCF NONLINEAR SYSTEMS This section describes the optimal tracking control solution for a class of nonlinear systems which can be presented as a state dependent coefficient factorized nonlinear systems. This control strategy will be applied for each one of the power converters in the micro-grid. A. State-dependent coefficient factorized nonlinear systems Let us consider the input-affine nonlinear system ẋ = f(x)b(x) u, x(t )=x (2) y = h(x) (3) where x R n is the state vector, u R m is the control input and y R p is the system output; the functions f(x), B(x) and h(x) are smooth maps of appropriate dimensions. Consider that functions f(x) in (2) and h(x) in (3) can be decomposed in the state-dependent coefficient factorization as f(x) =A(x) x and h(x) =C(x) x, respectively [4], [5], [6], [7]; then system (2)-(3) results in ẋ = A(x) x B(x) u (4) y = C(x) x. (5) As established in [8], [9], the assumptions f() =, h() =, f ( ) C 1 and h ( ) C 1 guarantee that the factorization as described in (4) (5) can be carried out. This salient feature will be used in this paper to obtain an analytical solution for the optimal control via the Differential Riccati equation. Note that factorizations A(x) x and C(x) x are not unique [1]. In order to obtain well-defined control schemes, appropriate factorization for these representations should be determined such that controllability and observability properties are fulfilled for system (4) and output (5). B. Optimal Tracking Controller Synthesis For many applications, such as power converters, aerospace, electrical machines, robotics, among others, it is important that the controller could track a desired trajectory; then it is required that an output of the closed-loop system tracks a desired trajectory as close as possible in an optimal sense and with minimum control effort expenditure [11], [12]. In order to introduce the trajectory tracking, let us define the tracking error as e = r y = r C(x) x (6) where r is the desired reference to be tracked by the system output y. The quadratic cost functional J to be minimized, associated with system (4), is defined as J = 1 2 t ( e T Qe u T Ru ) dt. (7) where Q and R are symmetric and positive definite matrices. Matrix Q in (7) is a matrix weighting the performance of the state vector x, meanwhile R is a matrix weighting the control effort expenditure; hence these matrices are used to establish a trade-off between state performance and control effort [13]. Therefore, the optimal tracking solution is related to determining the control u(t), t [t, ), such that the criterion (7) is minimized. The optimal tracking solution is established as the following theorem.

4 Theorem 1: Assume that system (4)-(5) is state-dependent controllable and state-dependent observable. Then the optimal control law u (x) =R 1 B T (x) (P (x) x z(x)) (8) achieves trajectory tracking for system (4) along a desired trajectory r, where P (x) is the solution to the symmetric matrix differential equation P (x) = C T (x) QC(x)P (x) B(x) R 1 B T (x) P (x) A T (x) P (x) P (x) A(x) (9) and z(x) is the solution to the vector differential equation ż(x) = [ A(x) B(x) R 1 B T (x) P (x) ] T z(x) C T (x) Qr (1) with boundary conditions P ( ) = and z( ) =, respectively. Control law (8) is optimal in the sense that it minimizes the cost functional (7), which has an optimal value function given as J = 1 2 xt (t ) P (t ) x(t ) z T (t ) x(t )ϕ(t ) (11) where ϕ is the solution to the scalar differentiable function V CBus Fig. 3. The case study micro-grid. i Bus C Bus R Bus icbus i RBus ϕ = 1 2 rt Qr 1 2 zt B(x) R 1 B T (x) z (12) with ϕ( ) =. Proof: For proof details see [14]. IV. OPTIMAL CONTROL FOR A SINGLE-PHASE MICRO-GRID: ACASE STUDY This section presents the application of the optimal tracking control scheme for the different power converters used in the micro-grid to efficiently distribute the generated renewable energy. Without loss of generality, this paper proposes the control for a single-phase micro-grid composed of the following subsystems: a DC bus, a wind system, a load, a storage element and the connection with the utility grid. It is worth pointing out that additional resources could be integrated to the micro-grid and its respective control system can be synthesized by using the proposed control scheme. Fig. 3 shows the considered micro-grid. For the control of the micro-gird, it is considered that each subsystem can be represented as ẋ i = A(x i ) x i B(x i ) u i y i = C(x i ) x i (13) where x i is the state vector of the N-th subsystem, with x i =[x i1,x i2,...,x ij ] T, i =1,...,N and j =1,...,n i, being n i the order of the corresponding subsystem. Matrices A(x i ), B(x i ) and C(x i ) are of appropriate dimensions. The respective cost functional becomes J i = 1 2 t ( e T i Q i e i u T i R i u i ) dt. (14) Fig. 4. DC bus circuit and an equivalent representation. Note that system (13) has the form of (4) (5), therefore the optimal control technique, as developed in the previous section, can be used. Indeed, this constitutes a decentralized control scheme, which easily allows the integration of different energy resources and components, corresponding to generation, consumption, storage and connection with the utility grid. It is worth noting that the structure of (13) is rather general, many practical nonlinear systems could be modeled to achieve such mathematical representation [6], [14], [15]. The components of the micro-grid and its optimal controller design are designed as follows. A. A capacitor, as the depicted in Fig. 4, is used as the DC bus.. The continuous-time voltage dynamics of the charge/discharge of this device is given as V CBus = V C Bus R CBus C Bus 1 C Bus i Bus (15) where V C is the capacitor voltage, i Bus is the current in the capacitor, C is the capacitance and R C is the respective internal resistance of the capacitor. For easy of notation, system (15) is represented by x b ẋ b = 1 i b (16) R CBus C Bus C Bus where x b = V CBus and i b = i Bus. Current i b is calculated as the sum of the input and output currents in the bus.

5 L s x 1 R s L L x 21 V i µ 1 µ 1 µ 1 µ 1 µ 2 µ 2 µ 2 µ 2 C L x 22 R L B. Wind Energy System Fig. 5. AC-DC power converter circuit. For wind energy generation we use the circuit of the full bridge power converter as displayed in Fig 5, where it is assumed that an AC voltage is obtained by a wind-driven electrical generator. The switches formed by a diode in parallel to a transistor are controlled in its gates by a switching signal µ, which denotes the switch position taking values in the finite set {1, 1}. The notation µ denotes the logic complement of µ. It is well known that when the switching frequency is sufficiently high, the model describing the behavior of the circuit can be represented by its averaged model as [16], [17] ẋ 11 = x b u/l s R s x 11 /L s V i /L s (17) where x 11 is the inductor current, x b is the bus voltage, R s is the resistance of the inductor and the associated impedance of the source and switches and L s is the inductance, V i = V p sin (ωt) is the voltage of an AC source, with ω =2πf, where f is the frequency in Hertz and V p is the amplitude of the input voltage. For this average model, u [, 1] becomes the duty cycle, which is used to generate the switching signal µ for the circuit in Fig. 5 by means of a pulse-width modulation (PWM) strategy. Note that for system (17), variable x 11 represents the average value of the current. The generated electrical power (P W ), is considered to be known from a maximum power point tracking algorithm, then the current x 11 can be calculated from the knowledge of the voltage (V i ) as x 11RMS = P W RMS V irms which will be used as the current reference when synthesized the optimal controller. 1) Controller Synthesis: For space limitation, only the controller synthesis for this subsystem will be described, meanwhile for the other ones the optimal controllers are designed in a similar way. For i =1in (13), subsystem (17) can be presented as ẋ 1 = A(x 1 ) x 1 B(x 1 ) u 1 E 1 (18) y 1 = C(x 1 ) x 1 (19) where x 1 = x 11, A(x 1 ) = [R s /L s ], B(x 1 ) = x b /L s, C(x 1 )=1and E 1 = V i /L s. Matrices for the performance index (14) are selected as Q 1 = 1 and R 1 = 5. Fig. 6. DC-AC power converter circuit with a resistive load. Note that Q is a parameter weighting the performance of the tracking error e, meanwhile R is a parameter weighting the control effort expenditure; hence these parameters are used to establish a trade-off between the state variables performance and control effort. In addition, these parameters are used to obtain the solution of the corresponding differential equations (9) and (1) (with their respective subscript i =1). Finally the reference vector used in the control scheme (8)-(1) becomes r 1 = r 11 = x 11RMS 2sin(ωt) where r 11 is the desired value for the inductor current. The optimal controller for (18) results in u 1 (x 1 )=ū 1 R 1 1 BT (x 1 )(P 1 x 1 z 1 ) (2) where ū 1 = V i /x b, which is included to cancel the constant term E 1 in (18). C. Load in the Micro-grid To provide from energy to the subsystem corresponding to the load, we are considering a DC-AC converter [18], as displayed in Fig. 6. The averaged model in its state space representation is given as [16], [17] ẋ 21 = x 22 /L L x b u 2 /L L ẋ 22 = x 21 /C L x 21 /(R L C L ) (21) y 2 = x 22 where x 21 is the inductor current, x 22 is the output capacitor voltage, L L is the inductance, C L is the capacitance, R L is the resistive load and u 3 is the control input (duty cycle for the PWM). For this subsystem, its respective optimal control scheme is synthesized similarly as the developed in the previous section. For simulation purposes, the load R L is calculated in accordance to the knowledge in advance of the dissipated power by the load (P L ). The subsystem and controller parameters are given in Table I. D. Energy Storage Fig. 7 shows the electrical circuit used for achieving energy exchange between the bus and the storage element, which is modeled by the capacitor C St in parallel with R St. This storage component could correspond to a rechargeable battery, a capacitors bank, etc.

6 µ 3 µ 3 µ 4 µ 4 L St x 31 C St Fig. 7. DC-DC power converter. µ 4 µ 4 L i x 41 x 42 x 32 L o x 43 R St C o V S Fig. 8. Inverter and LCL filter for the utility grid connection. The dynamical average model is described by ẋ 31 = x 32 /L St x b u 3 /L St ẋ 32 = x 31 /C St x 32 /(R St C St ) (22) y 3 = x b x 31 where x 31 is the inductor current, x 32 is the capacitor voltage, L St is the inductance, C St is the capacitance, R St models the internal resistance of the capacitor and u 3 is the control input (duty cycle for the PWM). The power to be transferred in the DC-DC converter (P S ) is calculated directly from the bus voltage and the inductor current, which establishes the reference for P S as P Sref. Parameters for this system and for the controller are given in Table I. E. Utility grid connection This section focuses on the component (converter plus filter) to transfer/extract energy toward/from the utility grid through a LCL filter as shown in Fig. 8. This is achieved by controlling of the power P G calculated as P G = V S x 43. The average model for this system is given as ẋ 41 = x 42 /L i x b u 4 /L i ẋ 42 = (x 41 x 43 )/C o ẋ 43 = (x 42 x 44 )/L o (23) ẋ 44 = ωx 45 ẋ 45 = ωx 44 y 4 = x 43 (24) where x 41, x 43 are the currents across the inductors L i and L o, respectively; x 42 is the voltage in the capacitor C o ; the TABLE I SUBSYSTEMS AND CONTROLLERS PARAMETERS. R L Parameter Value Parameter Value C Bus.1 F R Bus 1 MΩ L s 1 mh V p 15 V R s 2.2Ω f 6 Hz R 1 5 Q 1 1 Vo 2/2 Ω V o 12 2 V L L ( 1 mh ) C L 5 µf P LRMS R 2 1 Q 2 2 r 1 V o sin(ωt) V ω 2 πf rad/s L St 195 µh C St 5 F R St 1 MΩ R 3 1 Q 3 1 r 3 P Sref L i 2 mh L o 833 µh C o 1µF R 4.1 Q 4 1 r 4 P Gref V S /23 2 parameters L i, L o and C o in the circuit are the inductances and capacitance, respectively; V S is the utility grid voltage, represented by an exosystem modeled by the linear oscillator in the last two equations of (23), with initial condition for x 45 () = 23 2 V. The remaining initial conditions are settled to zero values. By considering all subsystems, the bus current i b is calculated as i b = x 11 x 21 x 31 x 41. F. Simulation Results The system and controller parameters used for the simulation are given in Table I. Simulations are performed by using Mathematica R. Figures 9 and 1 displays the electrical power control for the subsystems in the case study micro-grid. The references signals for each power converter are obtained from the optimization scheme, which are the presented in Fig. 2(a). The figures present the optimal trajectory tracking in a separate way for each subsystem. It can be seen the trajectory tracking, to the reference signal (electrical power reference), by each output power of the respective converter. V. CONCLUSIONS This paper proposes an optimization model and optimal control strategy to manage in an optimized way the energy exchange in a micro-grid. The values of the amount and direction of the energy to be transfered in the storage device and in the utility grid are determined as a result of the optimization process based on an interval of 24 hrs, which considers the predicted values for energy availability, energy prices and load energy demand. Based on the optimization results, an optimal nonlinear control has been proposed to achieve trajectory tracking of the electrical power in a micro-grid through power converters, which are used to exchange energy in a micro-grid. Numerical simulations are used to evaluate the effectiveness of the proposed micro-grid, the optimization algorithm and the optimal control of power converters. The simulation results illustrate the adequate operation of the whole micro-grid to manage renewable energy.

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