Acausal Modelling and Dynamic Simulation of the Standalone Wind ...

2013 UKSim 15th International Conference on Computer Modelling and Simulation

Acausal Modelling and Dynamic Simulation of the Standalone Wind-Solar Plant using Modelica Arash M. Dizqah, Alireza Maheri, Krishna Busawon

Peter Fritzson

Faculty of Engineering and Environment Northumbria University Newcastle Upon Tyne, UK [email protected]

Department of Computer Science Link¨oping University Link¨oping, Sweden

Abstract—In order to design model-based controllers applicable to hybrid renewable energy systems (HRES), it is essential to model the HRES mathematically. In this study, a standalone HRES, consisting of a photovoltaic (PV) array, a lead-acid battery bank, a pitch-controlled wind turbine, and a three-phase permanent magnet synchronous generator (PMSG), supplies a variable DC load demand through two boost- and buck-type DC-DC converters. It is shown that the mathematical model of the HRES can be represented by a system of nonlinear hybrid differential algebraic equations (hybrid DAEs). The developed model in this paper employs the Modelica language that allows object-oriented and acausal modelling of the multimode systems. The OpenModelica environment is utilised to compile the model and simulate the system. It is shown that the simulation provides a sufficiently accurate prediction of all the differential and algebraic states including mode transitions. The results of the simulation show a good match with the information available in the components datasheet. Index Terms—photovoltaic (PV); battery; wind turbine; hybrid renewable energy system (HRES), acausal modelling; hybrid DAE; Modelica.

I. INTRODUCTION Advances in renewable energy technologies have increased the share of the sustainable energy in the growing electricity market. A conventional HRES consists of solar and wind generators supplying DC, AC, or mixed electrical load demands. The system also consists of a battery bank to overcome the power fluctuation introduced by uncertainty in the energy resources. Figure 1 illustrates the HRES topology in this study that, without loss of generality, supplies a DC load. The solar branch consists of a PV array connected to a DC-DC converter that boosts up the generated DC voltage to match the load characteristics [1]. Meanwhile, the variable speed wind branch topology, consisting of a wind turbine, three-phase generator and rectifier, supplies the electrical energy to the DC load through a buck-type converter. The DCDC converters may be equipped with the maximum power point tracker (MPPT) algorithms to harvest the maximum available power (e.g. see [2; 3]). While the solar irradiance, wind speed, and the load demand are the non-manipulated variables of the system, the operation of the HRES can be controlled by three manipulated variables, namely, the pitch angle β of the wind turbine, the duty cycles of the wind- and 978-0-7695-4994-1/13 $26.00 © 2013 IEEE DOI 10.1109/UKSim.2013.145

Fig. 1.

The HRES topology in this study.

solar-branch, i.e. Dw and Ds ,respectively. In order to study the behavior of the HRES as well as to design model-based controllers, it is essential to model and simulate the system accurately. However, there are two major challenges, i.e. the algebraic constraints introduced by the PV module, wind turbine, and the battery, as well as the multimode operation of the battery. HRES introduces algebraic as well as discrete states (causing discontinuity) [1; 4] leading the system to be modeled with hybrid DAEs [5] and hence requires being simulated employing an acausal approach. The causal approach, decomposes a system into a chain of causal interacting blocks, requires the system being only composed of ordinary differential equations (ODE). On the other hand, the acausal modelling is a declarative approach, described with a modelling language such as Modelica [6], in which individual parts of the model are directly described as HDAE equations [5]. Modelica, as an object-oriented and equation-based language to describe complex systems using an acausal approach, provides the capability to model the hybrid systems [5]. Among a number of available implementations of Modelica compiler, the open-source OpenModelica environment [7], which provides a Modelica compiler and integrated DAE solvers, supports more features of the Modelica language comparing with the others. In this paper, a mathematical model of the HRES, which is applicable to develop model-based strategies, is proposed as

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HDAEs (Section II) [8]. Modelica is employed to develop an acausal model of the system that is solved using a general purpose DAE solver (Section III). The results of the simulation have been compared with information available in the PV, battery, and wind turbine datasheets provided by the manufacturers that indicate good accuracy (Section IV). II. HRES MATHEMATICAL MODEL Fig. 2.

Applying the model-based control strategies requires simple and still accurate mathematical modelling of the HRES. Due to existence of simultaneous fast and slow dynamics within the HRES, its simulation is becoming time-consuming and memory expensive. However, the model-based controllers are normally the outer low speed (of the order of several seconds) strategies, hence the fast dynamics can be replaced with the relevant steady-state equations.

q K Ns Tc Isc,stc kI kV S Sstc Tc,stc Voc,stc

A. The solar branch The authors in [1] presented a mathematical model of the solar branch consisting of a PV array, boost-type DC-DC converter, and a lead-acid battery bank. The presented model in [1] is based on the equivalent circuits of the PV module [9; 10] and the battery [11] as well as the average model of the converter [12]. However, the average model of the converter can be replaced with the steady-state equations (1) in the continuous conduction mode (CCM) [12] as follows: Vpv 1 − Ds = Ipv × (1 − Ds )

Vbstack =

(1a)

Ipvdc

(1b)

Isc,stc + kI (Tc − Tc,stc ) I0 = . Voc,stc + kV (Tc − Tc,stc ) q exp( )−1 nd Ns KTc

(2c)

where the experimental parameters indicated in Fig. 2 require being identified for each type of the PV module, and all others are as follows:

electron charge (1.6021810− 19) Boltzman constant (1.3806610− 23) number of the PV cells in series as the PV module (-) current amount of the PV cell temperature (K) short-circuit current of the PV module at the STC (A) temperature coefficient of the short-circuit current (A/C) temperature coefficient of the open-circuit voltage (V/C) current amount of the solar irradiance (W/m) amount of the solar irradiance at the STC (W/m) amount of the cell temperature at the STC (K) open-circuit voltage of the PV module at the STC (V)

Vbstack = Nb ×

( M odebat =

SOC = 1 −

                   

V0 − RIbat + Vexp − P1 Cmax Qact − Cmax − Qact P1 Cmax If Qact + 0.1Cmax

                  

V0 − RIbat + Vexp + P1 Cmax Qact + Qact − Cmax P1 Cmax If Qact − Cmax

charging, (3a)

charging

If ≤ 0,

discharging

If > 0,

discharging,

(3b)

Qact , Cmax

dQact 1 (t) = Ibat (t), dt 3600  P2   |Ibat |(P3 − Vexp ) dVexp 3600 (t) = P2  dt  |Ibat |Vexp − 3600



Vpv + Rs Ipv q Ipv = Iph − I0 exp( )−1 − n d Ns KTc Vpv + Rs Ipv , (2a) Rsh   Rs + Rsh S Iph = Isc,stc + kI (Tc − Tc,stc ) , (2b) Rsh Sstc

The The The The The The The The The The The

The authors in [11] introduced a model of the lead-acid battery as the following hybrid DAEs which present two modes of operation, namely, charging and discharging:

where Ds is the switching duty cycle of the converter. Fig. 2 illustrates a single-diode equivalent circuit of a PV module. The performance of different PV modules are measured at the standard test condition (STC) [10]. Applying the Kirchhoff current law (KCL) to the junction point of the series and shunt resistors gives the characteristic equation (2a) of the PV module. The photocurrent Iph and the diode reverse saturation current I0 are calculated with (2b) and (2c) [10]. 

The single-diode equivalent electrical circuit of a PV module.

dIf 1 (t) = − (If − Ibat ). dt Ts

(3c) (3d) charging, (3e) discharging, (3f)

where Vbstack , Ibat and SOC, respectively, are the voltage, current, and the state of charge of the Nb batteries in series, constructing a battery stack. The parameters P1 -P3 are the experimental parameters require being identified for each type of the battery and Vexp smooths the battery voltage during the mode transition period. The Cmax is the maximum amount of the battery capacity, R is the internal resistor of the battery, Qact is the actual battery capacity, and V0 is the

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Fig. 3. The power coefficient curve, with respect to the tip speed ratio, for three different values of the pitch angle.

battery constant voltage. If is also a filtered value of the battery current with the time constant of Ts [11]. B. Wind branch 1) Wind turbine (WT): Having the nominal power of the WT, Pnom , the generated mechanical power (4a) is calculated based on the per unit (p.u.) power coefficient Cp,p.u. . The performance of a WT can be characterized with an experimental power coefficient curve [3]. Fig. 3 illustrates the power coefficient for different tip speed ratio λ, which is the weighted ratio of the angular velocity of the generator rotor ωr to the wind speed Ux (4b). The maximum power coefficient Cp,max locates at the optimum tip speed ratio value λopt and is limited to 0.593, according to the LanchasterBetz theory [3]. In this study, the experimental Cp curve is modeled by (4d) [13], chiefly because MATLAB/SIMULINK has also adopted this model. This wind turbine, with the blade radius Rad of 4.01 meter, generates the maximum power coefficient Cp,max of around 0.48 at the optimum tip speed ratio of 8.1. Pm = Cp,p.u. (

Ux )3 Pnom , Ux,base

Rad × ωr , Ux 1 0.035 −1 λi = ( − 3 ) , λ + 0.08β β +1 C2 C5 1 Cp,p.u. = (C1 ( − C3 β − C4 )exp(− ) + C6 λ). Cp,max λi λi

λ=

(4a) (4b) (4c) (4d)

The experimental coefficients C1 −C6 are defined in TABLE II and β is the blades pitch angle [3]. λi is the intermediate variable to make the equations easier to understand. 2) Permanent magnet synchronous generator (PMSG): In spite of high cost, the PMSG is the most dominant type of the direct-drive generators in the market [14], chiefly due to higher efficiency. In order to make the mathematical model of the PMSG suitable for the model-based controllers, except the angular velocity (5a) all other fast voltage and current dynamics are ignored. Moreover, it is assumed that there is no mechanical and electrical loss through the powertrain, for simplicity’s sake. As a result, the electromagnetic power, as

Fig. 4.

The electrical circuit of the full-bridge three-phase rectifier.

the product of the electromagnetic torque Te and the angular velocity ωr , is equal to the output electrical power of the wind branch (5b) (Fig. 1). In order to apply the PMSG to the direct-drive topology (5c), it requires increasing the number of the pole pairs P [14]. The electrical frequency ωe , as a result, is P times faster than the mechanical angular velocity ωr (5d). 1 dωr (t) = (Te − Tm − F ωr ), dt J

(5a)

− Te × ωr = IW T DC × Vbstack ,

(5b)

− Tm × ωr = (Cp,p.u. (

Ux )3 Pnom , Ux,base

ωe = P ωr ,

(5c) (5d)

1 VLL = √ kE ωr , 2 √ kE = 3P ψ.

(5e) (5f)

The shaft inertia J (Kg.m2 ) and the combined viscous friction coefficient F (N.m.s) of the PMSG are given by the manufacturers. The VLL , i.e. the r.m.s. value of the line-to-line output voltage of the generator, depends on the mechanical angular velocity (5e). Having the number of the pole pairs P and the flux linkage ψ (V.s) of the generator (TABLE II), the voltage constant kE is calculated (5f). 3) 3-Phase rectifier: Fig. 4 illustrates the electrical circuit of a full-bridge three-phase rectifier where Iwt is the equivalent DC current of the dc side. Due to the inductance Ls on the ac side, there is non-instantaneous current commutation [12]. The average output DC voltage Vwt in presence of the non-instantaneous current commutation is calculated as follows: Vwt = 1.35VLL −

3 ωe Ls Iwt , π

(6)

4) Buck-type DC-DC converter: The hybrid or average models of the buck-type DC-DC converter can be replaced with the relevant steady-state CCM equations (7) [12], which is applicable to model-based outer controllers.

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Vbstack = Vwt Dw ,

(7a)

Iwt = Iwtdc Dw .

(7b)

TABLE I A N OUTLINE OF THE DEVELOPED FLAT M ODELICA MODELS FOR HRES. class HRES ...; discrete Boolean Mode bat(start = true) ”true: charging”; Real soc(start = 0.6) ”State of charge of the battery”; RealInput Ux ”The wind speed (m/s)”; RealInput RL ”The load demand (ohm)”; RealInput beta ”The pitch angle (degree)”; RealInput Ds ”The boost converter duty-cycle [0,1]”; Fig. 5.

The HRES overall model.

RealInput Dw ”The buck converter duty-cycle [0,1]”; equation

The Dw is the switching duty cycle of the converter.

Ipv = Iph - I0 * (exp((Ipv * rs Tc + Vpv) / a Tc) - 1) - (Ipv * rs Tc + Vpv) / rsh Tc Sx;

C. The overall HRES model Equations (1-7) in addition to an extra algebraic constraint (8) provide the HRES mathematical model as an implicit hybrid DAE functional (9). The power balance constraint (8) is extracted by applying the KCL to the junction point of the DC load RL , battery, and the converters (Fig. 1). The structural analysis indicates that this HDAE model of the HRES is of differential index 1 at each mode of operation. Ipvdc + Iwtdc + Ibat −

Vbstack = 0. RL

f (x, x, ˙ z, q, u, v) = 0.

(8) (9)

Fig. 5 shows the overall model of the HRES plant as well as the manipulated and non-manipulated variables, i.e. u and v. It also illustrates the differential, algebraic, and boolean states, i.e. x, z, and q.

Iph = (Npvp * ((Rsh + Rs) / Rsh * Isc stc + Ki * (Tc - Tc stc)) * Sx) / Sx stc; I0 = (Npvp * (Isc stc + Ki * (Tc - Tc stc))) / (exp(((Voc stc + Kv * (Tc - Tc stc)) * q) / (nD * Ns * K * Tc)) - 1); Vpv = Vbstack * (1 - Ds); Ipvdc = (1 - Ds) * iPV; Mode bat = if If ≤ 0 then true else false; der(If) = -1 / Ts * If + 1 / Ts * Ibat; der(Qact) = 1 / 3600 * Ibat; der(V exp) = if M ode bat then P 2/3600 ∗ abs(Ibat) ∗ (P 3 − V exp) else −(P 2 ∗ abs(Ibat))/3600 ∗ V exp; when change(M ode bat) and pre(M ode bat) then tmp = if not M ode bat then pre(V bat) − V 0 − R ∗ pre(Ibat)− (P 6 ∗ Cmax)/(Cmax − pre(Qact)) ∗ pre(Qact)−

III. THE HRES MODELICA MODEL The proposed acausal HRES model, as HDAE equations, covers both the continuous- (either differential or algebraic) and discrete-time behaviors [5]. In order to simulate the HRES by solving the HDAE functional (9), the HRES equations require being declared with a modelling language, i.e. the Modelica [6]. The OpenModelica platform [7], as an integrated Modelica compiler and DASSL [15] general purpose DAE solver, is selected chiefly because it is a free software and it supports more features of the Modelica language comparing with the others. Table I summarizes the developed Modelica models representing the HRES. In order to minimize the number of equations in this study and unlike the usual object-oriented approach, it proposes a flat HDAE model of the HRES. From Table I, it can be seen that the boolean mode (M odebat ) transition of the battery can be implemented using change(.), pre(.), and reinit(.) facilities of Modelica.

(P 6 ∗ Cmax)/(pre(Qact) + 0.1 ∗ Cmax) ∗ pre(If ) else 0; reinit(V exp, tmp); end when; soc = 1 − charge/Cmax; V bstack = if M ode bat then N bat * (V 0 − R ∗ Ibat − (P 1 ∗ Cmax)/(Cmax − Qact) ∗ Qact− (P 1 ∗ Cmax)/(Qact + 0.1 ∗ Cmax) ∗ If + V exp) else N bat * (V 0 − R ∗ Ibat − (P 1 ∗ Cmax)/(Cmax − Qact) ∗ Qact− (P 1 ∗ Cmax)/(Cmax − Qact) ∗ If + V exp); -Te * ωr = Iwtdc * Vbstack; -Tm * ωr = Cp pu * (Ux / 12)ˆ3 * Pnom; der(ωr) = (Te - Tm - F * ωr) / J; Vwt = (1.35 * P * psi * sqrt(3) * ωr) / sqrt(2) (3 * Lst * P * ωr * Iwt) / pi; lambda = (R * ωr) / Ux; lambda i = 1 / (1 / (lambda + 0.08 * beta) - 0.035 / (beta ˆ 3 + 1)); Cp = (C1 * (C2 / lambda i - C3 * beta - C4) * exp(-C5 / lambda i) + C6 * lambda) / 0.48;

IV. SIMULATION RESULTS, VALIDATION, AND DISCUSSION

Ipvdc + Ibat + Iwtdc = Vbstack / RL;

The proposed Modelica model is used to simulate a HRES consisting of an array of the Kyocera KC200GT PV 583

end HRES;

TABLE II D IFFERENT PARAMETERS IN THIS STUDY ( DIMENSIONS AS GIVEN IN THE TEXT ). Wind turbine C1 0.517 C2 116 C3 0.4 C4 5 C5 21 C6 0.007 λopt 8.1 Pnom 10K Rad 4.01 Ux,base 12 Cp,max 0.48

PMSG J 35 F 0.2 P 8 ψ 0.8 Pr 10K Ls 0.01 − − − − − -

Battery Cmax R V0 P1 P2 P3 Ts Nb − − −

stack 12.02 0.021 12.37 0.9 20.73 0.55 0.726 4 -

PV module Rs 0.221 Rsh 405.4 nd 1.3 Ns 54 Isc,stc 8.21 Voc,stc 32.9 kI 0.003 kV -0.12 − − − -

modules and a bank of the Panasonic LC-R127R2PG leadacid batteries. The authors in [10] and [11], respectively, presented the identified electrical parameters of the employed PV module and the lead-acid battery. TABLE II summarizes all parameters and their values in this study. Fig. 6 compares the simulated current-voltage (I − V ) curve of the KC200GT PV module at the STC condition with the experimental curve available by the manufacturer in datasheet (the circle markers). It is observed that the proposed model predicts the curve very close to the empirical data provided by the manufacturer. Also, Fig. 7 illustrates the simulation results of a single LC-R127R2PG lead-acid battery for a full operating cycle including charging, over-charging, saturation, discharging, over-discharging, and exhaustion zones. While the battery is being charged for 100 minutes, it is discharged afterward. It also indicates that after 25 minutes it enters into the over-charging zone. Discharging with the current of 7.2A in average, it takes around 35 minutes for the battery, which matches with the information available in datasheet, to reach the cut-off voltage that is around 10.2V . The stability of the developed model is examined by simulating the following sample scenario that applies different step changes to the manipulated and non-manipulated variables: • • • •

• • •

Simulation period is 700 seconds. The solar irradiance is 1000W/m2 and the cell temperature is 25◦ C. The PV array consists of 10 connected KC200GT PV modules in parallel arrangement. The battery bank consists of 20 battery stacks in parallel. Each battery stack consists of 4 connected LCR127R2PG lead-acid batteries in series arrangement. After 100 seconds there is a step change in load resistor from 0.3 to 1.1Ω. There is a step change in wind speed at time 300 seconds from 12 to 20(m/s). A step change in the buck converter duty cycle at time 500 seconds makes the wind turbine to be fully stalled.

Fig. 8 illustrates the manipulated variables applied by the controller to harvest the maximum power of the solar and

Fig. 6. The simulated I − V (dashed line) and P − V (solid line) curves of the KC200GT PV module against the experimental points (the circle markers).

Fig. 7.

The simulated (a) battery voltage, and (b) battery current.

wind branches (Fig. 9), i.e. 2KW and 10KW respectively. In Fig. 8, it can be seen that at t = 350 seconds, which the wind speed increases by 8(m/s), the pitch angle goes up to 23 degrees and promotes pitching to feather [3]. In effect, although the wind speed reaches to 67% above the rated value, the generated power by the wind turbine remains stable at the rated value (Fig. 9), i.e. 10KW . At t = 500 seconds, it is necessary to promote stalling and so to remove the wind branch share of the power generation. Fig. 8 and 9 depict that a sharp decrease of Dw makes the wind turbine fully stalled at t = 500 seconds and later at t = 600 seconds a rise of 11% in Dw increases the share of the wind branch to around 3.5KW . In Fig. 9, it can be seen that the wind branch needs a few seconds (i.e. 7.5 seconds) to reach the point to be able to inject energy to the system. During this period of time the solar branch and the battery supply the load demand and the battery is being discharged. However, once the rotational speed remains steady, the wind branch starts injecting energy that changes the mode of the battery operation to

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Fig. 8.

Moreover, the developed mathematical model is declared by the Modelica language that makes it applicable for simulation. The OpenModelica environment, as an integrated Modelica compiler and the DASSL general purpose DAE integrator, is utilised to simulate the system. The PV array and the lead-acid battery stack are separately simulated and validated against the experimental information available by manufacturers that show a very good accuracy. The stability of the model is also examined by simulating a test scenario that applies different step changes to the variables. The simulation results indicate accurate prediction of all the system behaviors including different currents and voltages as well as mode transitions. An outline of the developed Modelica models is also presented.

The variation of the manipulated control signals.

ACKNOWLEDGMENT The authors would like to thank the Synchron Technology Ltd. for their financial support of this research. R EFERENCES

Fig. 9.

The simulation results for battery, load, wind and solar powers.

Fig. 10. The simulation results for the SOC and the voltage of the battery stack.

the charging. The discharging to charging mode transition increases the number of equivalent cycles of the battery by 1 − SOC (Fig. 10). The number of equivalent cycles of the battery indicates the life span of the battery. Battery starts absorbing the excess of energy at t = 100 seconds after a substantial reduction in load demand. It also partially compensates the energy deficit of the system during the period of time between t = 500 and t = 600 seconds when the wind branch is fully stalled. V. CONCLUSION This paper proposes an acausal hybrid DAE model for a combined solar and wind power generation plant. The proposed model represents the different modes of the battery operation in addition to the system dynamics and nonlinear algebraic constraints, which are introduced by the PV array, wind turbine, PMSG, DC-DC converters and the battery.

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