Simulations and Control of Direct Driven Permanent Magnet Synchronous Generator Project Work Dmitry Svechkarenko Royal Institute of Technology Department of Electrical Engineering Electrical Machines and Power Electronics December, 2005
Contents 1 Introduction 3 2 Wind Turbine Model 4 2.1 Generator Model......................... 4 2.2 Converter Model......................... 5 2.3 Grid Representation....................... 6 3 Control of PMSG 7 3.1 Control of the Generator Side Converter............ 7 3.2 Control of the Grid Side Converter............... 10 4 PMSG Analysis 11 5 Conclusions 14
Chapter 1 Introduction One of the benefits of using a permanent magnet synchronous generator in wind power applications as an alternative to conventional generators is its higher efficiency, as the copper losses in the rotor disappear. The elimination of the gearbox and introduction of the variable speed control would even further increase the availability of the system, reduce its active weight and the need for maintenance. In this project, the wind turbine with a permanent magnet direct driven synchronous generator and power converter connected to a strong grid is analyzed. The control strategy is established in order to maintain the output characteristics of the system at the optimum level. The model is realized in MATLAB R /SIMULINK.
Chapter 2 Wind Turbine Model 2.1 Generator Model The voltage equations of a permanent magnet synchronous generator in the dq reference frame are given by v qs = R s i qs + ω m ψ ds + dψ qs dt, v ds = R s i ds ω m ψ qs + dψ ds dt. (2.1) Where v is the voltage, i the current, ψ the flux linkage, R the resistance, ω m the angular velocity of turbine rotor. The subscripts d and q stand for direct and quadrature components, respectively (see fig. 2.1), the subscript s for stator. The quantities are given in per unit (p.u.). The correlation between physical values and per unit can be among others found in (3). The flux linkages are ψ qs = (L qm + L σs )i qs, ψ qs = (L dm + L σs )i ds + ψ f. } (2.2) Where ψ f is the flux produced by the permanent magnets. In order to simplify analysis of the wind generator, the transients occurred in the stator, according to the assumptions used in PSDSs, can be neglected (1). By substituting flux linkages eq. 2.2 in system of eqs. 2.1 the voltagecurrent relationship can be obtained u qs = R s i qs + ω m (L dm + L σs )i ds + ω m ψ f, u ds = R s i ds ω m (L qm + L σs )i qs. } (2.3)
2.2 Converter Model 5 Figure 2.1: Salient-pole permanent magnet synchronous machine in dq reference frame. Where L qm, L dm are the mutual inductances in q and d axes, respectively, and L σs is the stator leakage inductance. The torque in the airgap T ag is given by T ag = ψ ds i qs ψ qs i ds = ψ f i qs + (L dm L qm )i qs i ds (2.4) The active P s and reactive power Q s of a synchronous machine are } P s = v ds i ds + v qs i qs, (2.5) Q s = v qs i ds v ds i qs. 2.2 Converter Model The power converters can according to (1) be modeled as a fundamental frequency current sources. The dynamic dc-link voltage V d can be derived as (2) I v I d = C dv d (2.6) dt Where the dc current flowing into the grid side converter I d is given by I d = v qci qc + v dc i dc P d,loss V d (2.7) The dc current flowing out of the generator side converter I v is derived as
2.3 Grid Representation 6 I v = v qsi qs + v ds i ds P s,loss V d (2.8) By substituting eqs. 2.7 and 2.8 in 2.6, the dynamic equation of the dclink voltage could be define as dv d dt = 1 [ ] (v qc i qc + v dc i dc ) + (v qs i qs + v ds i ds ) (P s,loss + P d,loss ) CV d (2.9) In eqs. 2.6-2.9 the subscript c stands for converter, and P s,loss, P d,loss are the converter losses. The active P c and reactive power Q c of the converter are given by P c = v dc i dc + v qc i qc, Q c = v qc i dc v dc i qc. 2.3 Grid Representation } (2.10) The grid side converter is connected through a step-up transformer with internal reactance x t to the stiff bus with the voltage V ex e jδ = v qex + jv dex. The common practice for the stiff bus is however to assume δ = 0 and V ex = 1 p.u. The active P ex and reactive power Q ex delivered to the grid by the generator is calculated as follows P ex = v dex i dc + v qex i qc, Q ex = v qex i dc v dex i qc. } (2.11)
Chapter 3 Control of PMSG The schematic representation of the system subject to control is depicted in fig. 3.1. As it can be observed, the generator is fully decoupled from the grid by means of the power converter; thus, the power factor of the generator is independent of the reactive power factor at the grid connection. 3.1 Control of the Generator Side Converter The terminal voltage of the generator in terms of modulation ratio m 1 and phase angle Θ is defined as } v qs = m 1 V d cos(θ), (3.1) v ds = m 1 V d sin(θ). Where the phase angle Θ is dθ dt = ω base(ω m f) (3.2) The reference power to archive the maximum efficiency of a wind turbine is given by P ref = T m ω m = K turb ω 3 m, (3.3) Where T m is mechanical torque and K turb is the turbine coefficient, which, according to (2), can be calculated as follows
3.1 Control of the Generator Side Converter 8 Figure 3.1: Schematic of the control system. K turb = 1 2 ρ C p,optπr 5 ω 3 base λ 3 opts base, (3.4) Where ρ is the air density [kg/m 3 ], C p,opt is the optimal turbine efficiency, λ opt is the optimal tip speed ratio, r is the turbine radius [m], S base is the base power of the generator [MVA], and w base is the nominal rotor speed [rad/s]. Recognize that for a synchronous generator the stator frequency is equal to the rotor speed in per unit values, the control of the generator frequency can be performed as presented in fig. 3.2. The modulation ratio m 1 of the generator side converter is controlled as shown in fig 3.3, where V s = vqs 2 + vds 2 (3.5)
3.1 Control of the Generator Side Converter 9 Figure 3.2: Control of the generator frequency f. Figure 3.3: Control of the modulation ratio m 1.
3.2 Control of the Grid Side Converter 10 3.2 Control of the Grid Side Converter The terminal voltage of the grid side converter is given by } v qc = m 2 V d cos(α), v dc = m 2 V d sin(α). (3.6) The phase angle α and the modulation ratio m 2 can therefore be controlled independently (figs. 3.4, 3.5). Figure 3.4: Control of the phase angle α. Figure 3.5: Control of the modulation ratio m 2.
Chapter 4 PMSG Analysis The control strategy for the control of the converter is to be established in MATLAB R / SIMULINK. The wind turbine parameters used in the simulations are summarized in tab. 4.1. Table 4.1: Simulation parameters of the wind turbine. Parameter Unit Value Nominal generator speed, ω base [rad/s] 2.26 Nominal generator power, P base [MW] 2.5 Mutual inductance in q-axis, L qm [p.u.] 0.7 Mutual inductance in d-axis, L dm [p.u.] 0.7 Stator leakage inductance, L σs [p.u.] 0.02 Stator resistance, R s [p.u.] 0.0025 Compensating capacitor, C [p.u.] 0.1 Rotor radius, R [m] 40 Air density, ρ [kg/m 3 ] 1.1 Optimal turbine efficiency, C p,opt [ ] 0.474 Optimal tip speed ratio, λ opt [ ] 7.5 The control of the converters was performed in such way that the following constraints were fulfilled: Generator terminal voltage V s 1 p.u. DC-link voltage V d = 1.75 p.u. Stiff bus voltage V ex = v qex + jv dex = 1 p.u.
12 Reactive power delivered to the grid Q ex = 0 p.u. To fulfill the last constraint, a certain quantity of reactive power should be produced by the grid side converter in order to compensate transformer reactance x t. Recognize that stiff bus voltage in d-axis is v dex = 0, the required greed side converter voltage V c = V c e jδ can be obtained as follows S ex S c = P ex + jq ex = P ex = V ex Ī c = P c + jq c = V c Ī c Ī c P 1 = V c V ex jx t = V V c V ex ex jx t V c = V ex 2 + jx t P ex V c = V 2 δ V ex ex + = arctan ( ( x t P ex = V ex V c V 2 ex jx t V ex x t P ex V 2 ex = V ex + j x tp ex ) ) 2 V ex To keep the reactive power exchange between the turbine and the grid equal to zero, the grid side converter terminal voltage should be equal to V c = v qc + jv dc = 1 + j0.05 p.u. and the produced reactive power should therefore be Q c = 0.05 p.u. The results of the performed simulations of the wind turbine are shown in fig. 4.1. The rotor speed was selected to be in the range of ω m = 0.9 1.1 p.u. As it can be observed, the generator terminal voltage is 1 p.u., the dc-link voltage is 1.75 p.u. The active power delivered into the grid tends to change with changing of the rotor angular speed. The amplitude is however somehow different. One of the probable reasons for it can be the coarse adjustment of the PI-controllers. Therefore, more thorough investigation of the effect of the coefficients on the behavior of the entire system is required. The grid side converter terminal voltage has an expected amplitude and changes in order to keep the exchange of reactive power with the grid at zero level.
PSfrag replacements 13 PSfrag replacements 1.1 Generator terminal voltage V s Active power Ps 1.5 PMSG production of active power P s 1.08 1.06 1.4 1.3 Rotor speed ωm Active power Ps 1.04 1.2 1.02 1.1 [p.u.] 1 [p.u.] 1 0.98 0.9 0.96 0.8 0.94 0.7 0.92 0.6 0.9 0 50 100 150 200 250 300 Time t, [s] 0.5 0 50 100 150 200 250 300 Time t, [s] PSfrag replacements 1.8 DC-link voltage V d PSfrag replacements 1.1 Grid side terminal voltage V c 1.79 1.09 1.78 1.08 1.77 1.07 1.76 1.06 [p.u.] 1.75 [p.u.] 1.05 1.74 1.04 1.73 1.03 1.72 1.02 1.71 1.01 PSfrag replacements 1.7 0 50 100 150 200 250 300 Time t, [s] 1 0 50 100 150 200 250 300 Time t, [s] Active power Pc 1.5 Active power deliverd to the grid P c PSfrag replacements 0.05 Reactive power delivered to the grid Q ex 1.4 1.3 Rotor speed wm Active power Pc 0.04 0.03 1.2 0.02 1.1 0.01 [p.u.] 1 0.9 [p.u.] 0-0.01 0.8 0.7 0.6-0.02-0.03-0.04 0.5 0 50 100 150 200 250 300 Time t, [s] -0.05 0 50 100 150 200 250 300 Time t, [s] Figure 4.1: Simulation results of the wind turbine.
Chapter 5 Conclusions The control system for the permanent magnet direct driven synchronous generator was developed and tested. The system in the selected range of the wind speed showed good correspondence with the expectations. In order to further improve the control system, the coefficients of the PI-controllers should be tuned more precise. This study would also be necessary for the testing of the system in higher wind speed range, which would be a case for the off-shore wind turbines. With the increasing of the amount of electrical power produced by means of wind turbines, an important aspect to consider will be the voltage sag ride-through ability of a generator, as it has relatively big impact on a power quality and stability of the power system in general.
Bibliography [1] Wind Power in Power Systems, edited by T. Ackermann, ISBN 0-470- 85508-8, John Wiley & Sons, Chichester, England, 2005. [2] Ian Norheim, Dynamic Modelling of Turbines (2): Generators, converters, control. Course in Wind Power held in Smøla, Norway, 6-10 June, 2005. [3] Chee-Mun Ong, Dynamic Simulations of Electric Machinery, ISBN 0-13-723785-5, Prentice Hall PTR, Upper Saddle River, New Jersey, USA. 1998.