SYNCHRONOUS MACHINE WIND TURBINE MODELLING FIDELITY

Jarrod Esau Mohammed, Mike Barnes*

*School of Electrical and Electronic Engineering, University of Manchester, PO Box 88, Manchester, M60 1QD, UK Email: j-mohammed@neal-and-massy.com,, Mike.Barnes@manchester.ac.uk

Keywords: Synchronous Machine, Modelling, Wind Power, Quality

AbstractThis paper examines prevalent modelling problems for dynamic Permanent Magnet Synchronous Generators (PMSG) coupled to the power network via a back-to-back PWM converter. A control strategy is employed using d-q analysis of both the machine and the grid-side converter, allowing a straight-forward control strategy for regulating machine speed, electromagnetic torque and DC-link voltage. The control strategy is implemented in simulation, together with a linear time-invariant model of the machine and converters. A full switching model is developed for comparison against the linear or averaged model. Reformulating the system equations to increase simulation speed is discussed.

Transient test cases are developed to study the performance of the system. The control strategies are shown to be effective as the DC-link voltage is regulated well. In addition, a close correlation is demonstrated in the response characteristics of both linear and switching models. This result allows the use of the less resource-intensive model in studies if harmonics may be neglected, thus significantly improving simulation speed without the loss of accuracy.

1 Introduction The wide-spread application of offshore wind-power will result in a gradual increase in wind turbine rating to minimise the cost of foundations, civil engineering and maintenance (all of which scale with turbine number). For higher-power turbines the use of synchronous generators with a full back-to-back rated converter has been proposed. Doubly-Fed Inductor Generators are at present the preferred technology for the majority of installations. However for large power ratings, as the connection voltage is increased and as well-defined fault behaviour becomes more important, it is likely that a gradual shift to synchronous machines will occur.

Recently, significant research effort has gone into the application of Permanent Magnet Synchronous Generators in Wind Energy Conversion Systems (WECS). Traditionally, the cost of permanent magnet materials like Neodymium-Iron-Boride (NdFeB) has restricted the use of permanent magnets in large synchronous generators. As these costs have substantially dropped in the past decade, more attention is being given to applications using permanent magnets.

Modelling this kind of turbine will require differing levels of fidelity. For manufacturers studying a single turbine, detailed switching models of all the system components will be required, to enable a close approximation to reality to be achieved. For larger studies of a wind-farm, to assess control system interaction between turbines, such models could result in an unacceptably slow simulation runtime. A reduction in fidelity is therefore permissible, as long as key dynamics are not neglected. This paper addressed these key issues of simulation time and fidelity.

1.1 Modelled System Layout

Figure 1 illustrates a block diagram of the wind energy conversion system. Elements within the dashed boxes have been excluded from the implemented model since only electrical network transients are considered. The mechanical transients, which are not dominant within the period of interest, are neglected and the shaft is represented as a constant torque fed to a machine with inertia and damping.

Figure 1 Modelled System Layout

The subsequent sections describe the mathematical model of this system in Figure 1, as well as the two methods of implementation in Matlab/Simulink (discretised, switch-averaged model) and PLECS (discretised, full switching model).

2 Physical System Model The method of d-q analysis in modelling the grid-side converter of wind energy systems is well-known in the literature, for example [1]. The convention is to use the Park and Clarke transformation matrixes to transform quantities from the three-phase stationary reference frame to the d-q synchronous frame. Ultimately, d-q analysis leads to a very simple method of controlling the real and reactive power delivered by the converter.

In a similar manner, the d-q model of the synchronous machine, which was developed based on the work of Blondel and Park, is commonly used to control both machine speed and developed electromagnetic torque. One of the main benefits of the method is the treatment of stator flux linkages, which are transformed from time varying quantities into fixed quantities [2, 3] leading to a simpler representation of the machine. The authors in [4] have adapted the method to study the performance of the Permanent Magnet Synchronous Machine (PMSM). Vector control is applied to the simplified machine model, whereby developed torque is shown to be proportional to the stator quadrature-axis current.

Developments in multi-phase direct-drive PMSG machines [5] and multi-phase induction machines [6] offer further potential reliability and hence these machines have been suggested for offshore wind turbine duty. Similar control strategies may be developed by mapping these more complex machine control topologies into similar dq or extended-dq quasi-dc control spaces.

2.1 Grid-Side Converter

A three-phase voltage source inverter is connected to an infinite bus via a transmission cable as illustrated in Figure 2. The infinite bus (the electricity grid), is modelled as a balanced three-phase voltage source with fixed magnitude and fixed frequency, while the cable is modelled as a series combination of resistance and inductance. Sinusoidal Pulse Width Modulation (SPWM) is used to control the power switches (IGBT), thereby converting the DC supply into an AC voltage. Neglecting higher order harmonics, a sinusoidal voltage is synthesised and appears at the output terminals of the converter. Power delivered by the machine side converter, which is independently controlled, is shown as a constant current source.

Figure 2 Grid-side Converter fed by a constant current source

The dynamics of the system can be described mathematically as given in Equation (1), where p denotes the differential operator.

(1)

Application of the Park and Clarke transformations to (1) yields:

(2)

It is important to note that an intermediary step has been skipped, where the three-phase system is first transformed into a two-phase system, commonly known as the - model. Vector rotation is then applied to quantities in the -reference frame, to obtain corresponding quantities in the d-q reference frame. By first computing the - quantities, the angle of the grid voltage vector may be obtained as given in (3).

(3)

The purpose of the analysis becomes clear by deriving an equation for complex power. In the d-q reference frame, the grid voltage is represented by a vector vdq = vd + jvq and the current into the grid by idq = id + jiq. Assuming that vd, vq, idand iq are per unit quantities, complex power is given by

The imaginary component of vdq is eliminated by aligning the grid voltage vector with the d-axis. Alignment is achieved by substituting for g in the Park transformation with computed by (3). With vdq = vd + j0, the per unit real and reactive power injected into the grid are given by p = vdid and q = vdiq,which can be regulated by setting and controlling id and iqrespectively.

Regulating the DC voltage is achieved by equating the instantaneous power delivered by the capacitor with the instantaneous power delivered by the converter, neglecting harmonics and switching losses. In d-q components, the output power of the converter is vdid and the output power of the capacitor is Vdcig. By making the appropriate substitutions and manipulations, the capacitor voltage may be described by (4).

(4)

2.2 Machine-Side Converter

The stator coils of a permanent magnet synchronous generator are connected to a three-phase converter as illustrated in Figure 3. The rotor is connected to a wind turbine which captures kinetic energy from the wind and converts it into mechanical energy.

N

S

Figure 3 Permanent Magnet Synchronous Generator

To simplify the analysis of the generator, the following assumptions are made.

1. Space harmonics in the air-gap magnetic field are neglected

2. Leakage effects due to stator slots are neglected 3. Magnetic hysteresis is negligible 4. Magnetic saturation effects are negligible

In matrix form, an expression for the three-phase stator voltages may be written as given in (5). The convention assumed here is that currents are directed out of the generator.

(5)

The flux linkage terms [a, b, c]T in (5) are functions of the rotor angle r and are reproduced below [2]. Note that the equations are based on a salient pole machine, thus accounting for the second harmonic variation of the inductance terms.

Laa0 = Average value of self inductance per phase, including leakage flux

Lab0 = Average value of mutual inductance between two phases, including leakage flux note that Lab0 Laa0

Laa2, Lab2 = Peak value of double frequency variable component of self and mutual inductances

r = Electrical angle between the magnetic axis of the rotor and the magnetic axis of the phase awinding

m = Flux produced by permanent magnets

In [2] the author develops the flux linkage equations for a synchronous generator with additional field and damper windings. A permanent magnet generator contains neither field nor damper windings, which are instead replaced with a fixed flux component m, whose contribution to the flux linkage of each stator coil varies with rotor position.

Application of the Park and Clarke transformation to (5), together with a suitable substitution for the stator self, mutual

and variable inductances as given by (7), yields the d-q model of the permanent magnet synchronous generator (6).

(6)

Ld and Lq are referred to as the direct axis and quadrature-axis inductances and provide a convenient means of modelling the machine. They can be considered as the inductance of two fictitious coils in space quadrature, which represent the stator windings of the generator, and which are located on a reference frame fixed to the rotor.

(7)

For a balanced three-phase system, instantaneous stator power may be expressed in d-q components as given in Equation (8).

(8)

Substitution of ed and eq from (6) into (7), yields the following expression for stator power, which comprises (from left to right) air-gap power, armature resistance losses and the rate of change of stored magnetic energy.

The air-gap power, when divided by the mechanical speed of the rotor (and noting that the ratio of electrical speed to mechanical speed is equal the number of machine pole-pairs pp), results in an expression for the electromagnetic torque Teas given in (9).

(9)

The machine description is now completed by including expressions for the dynamics of the mechanical system.

(10)

3 Control System Design An overview of the grid-side converter and control is illustrated by the block diagram in Figure 4. Transformations from three-phase to synchronous d-q quantities, as well as the key signals required and generated by each controller are illustrated. DC-link voltage is regulated by setting a reference value, which is then compared with the actual capacitor voltage. The error signal is then sent through a proportional plus integral controller which generates a reference direct-axis current.

Feedback nulling is employed to decouple the current loops, which allows independent control of real and reactive power. The switching inverter will typically operate at a moderate frequency (~2kHz).

Figure 4 Overview of Grid-Side Converter and Implemented Control Scheme

iabcabcdq

PI

PI ed'*

eq'*

abcdq SPWM

vabc*PI

iq

iq*

id*

rm*

id

+

+

+

+

++

+-

--eq*

ed*63

3

PMSG

rmr pp

r

rmfrom speed transducer

Lq

Ld

r rm

r

+

-

Figure 5 Overview of Machine-Side Converter and Implemented Control Scheme

4 Software Implementation

A potential problem with some simulation packages for certain cases employing large, complex models (such as wind farms consisting of many un- or partly aggregated wind-turbines) is if such packages employ a continuous-time variable-step solver. This type of solver computes and adjusts the duration of the each simulation time step, during the current time step, depending on the rate of change of state variables within the model. Models which contain state variables that change slowly can benefit from this feature since the time step can be made larger and the simulation can be completed quickly. However, models which contain switching devices can give rise to rapidly changing dynamics.

The high accuracy of the variable step solver means that the time step is made very small during and after switching instants, which leads to an increase in simulation time. This is particularly a problem when running large models, or run-time restricted packages (such as student versions of some popular software).

In such circumstances, a method for reducing the simulation duration is necessary. The obvious choice is to discretize the model, so that a discrete-time, fixed-step solver can be used. These solvers are better suited to non-linear system models, such as those containing switching converters, and can result in a significant speeding up of the simulation.

The models used here were discretized. In developing the discrete-time models, the controllers and plants were first represented in the continuous-time domain. Controller gains were computed based on a specified undamped natural frequency and an ideal damping ratio, while system stability was determined via Bode plots. Once the design was considered acceptable in the s-domain, a discrete-time or z-domain equivalent was then made. The method used here simply involved replacing the s-domain integral functions within each control loop with the z-domain approximation of integration, which Simulink implements using a Forward-Euler integration algorithm. Additionally, the software was configured to build discrete state-space models of the physical switching circuits, which it does by transforming the continuous-time state-space equations to the z-domain using the Tustin transformation.

Figure 6 Implementation of continuous-time (top) and discrete-time (bottom) proportional plus integral controllers in Simulink

As an example, both continuous-time and discrete-time models are constructed for the current control loop of the full switching inverter. The proportional plus integral controllers implemented in Simulink are illustrated in Figure 6. The switching frequency of the inverter is set to 2250Hz, which requires a small simulation step time in order to accurately capture the system dynamics. As such, the maximum step size for the continuous-time model is set to 1/50th of the switching

period, which is also taken as the sampling period for the discrete-time model.

Figure 7 Comparison of step response characteristics of the grid direct-axis current when using continuous-time and discrete-time models of the grid-side switching converter

Figure 7 illustrates the step responses of both models where very little difference can be observed. However, the simulation duration is reduced by almost a factor of four, from 1.31s for the continuous-time model to 0.35s for the discrete-time model. For larger system models, which include more state variables, switching devices and non-linear elements, the time savings is much greater, with very little loss of accuracy. This technique was especially useful when performing simulations of the machine-side converter in speed control tests, due to the large time constants associated with the mechanical system.

5 Simulation Results Initially the performance of the inner current loops was tested. Using parameters shown in the appendix a variety of tests were applied. Figure 8 shows a step change in d-axis reference current. Four simulations are shown - the discretized model with a converter switching modelled (labelled switching), a switch-averaged model with d and q axis control decoupled (labelled decoupled), a second order approximation to the switch-averaged model (labelled appoximate) used to for initial tuning of the controller (based on an approximation of the system closed-loop transfer function to a second order system).

As can be seen, while the switched averaged model tracks the switching model lower-frequency response well, significant higher frequency current harmonics are not represented. This is more evident in the q-axis current.

In Figure 9 a 0.01pu step decrease in speed (from rated speed) is commanded at time t = 0.5s. The initial input torque and DC-link voltage are at nominal values. Note that these results are displayed in per unit for easier comparison with the step command.

The DC-link voltage exhibits a sudden sharp decrease (within 1% of the initial voltage), before rising again and then settling to the reference value. This phenomenon can be explained by a sharp rise in machine quadrature-axis current (not shown).

Figure 8 Inverter response to step increase in reference current id* (top trace direct-axis current, bottom trace quadrature-axis current)

Figure 9 DC-link voltage, machine speed and electromagnetic torque response characteristics due to a step decrease in reference speed rm*

The speed controller generates an increase in the reference quadrature-axis current so that an electromagnetic torque may be developed that exceeds the input torque. This causes the rotor to decelerate as desired. The sudden increase in current results in a large change in stored magnetic energy. The temporary disruption in the flow of energy to the DC link results in an imbalance which causes the DC-link voltage to fall. After the initial transient, the quadrature-axis current begins to fall and, as the stator fields collapse energy is released to the DC link. By this time however, the output power to the grid has fallen in accordance with the initial drop in DC-link voltage. This new imbalance causes the DC-link voltage to rise. Eventually, the power balance is restored and the DC-link voltage returns to the reference value.

It should be noted that the transient effect witnessed can be mitigated to a certain extent by reducing the gains of both the machine speed and current controllers. From the base selected gains, it can be seen that the integral term of the current controller is indeed vary large, and should be reduced by lowering the bandwidth of the current loop.

6 Summary The paper has presented a model for a PMSG connected through an AC/DC/AC inverter stage to a utility network. The system modelling and controller design has been discussed. Means to increase computational speed for very large systems, such as might be found in some large wind farm representations, have been discussed. Simulation results have been presented and the impact of some of the trade-offs in controller gain selection have been discussed.

References

[1] R. Pena, J. C. Clare, and G. M. Acher, "Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation," Electric Power Applications, IEE Proceedings, vol. 143, no. 3, pp. 231-241, May 1996. [2] Prabha Kundur, Power System Stability and Control, 1st ed., Neal J Balu and Mark G Lauby, Eds. New York, United States of America: McGraw-Hill, Inc., 1994. [3] A E Fitzgerald, Jr., Charles Kingsley, and Stephens D Umans, Electric Machinery, 5th ed., Stephen W Director, Ed. London, United Kingdom: McGraw-Hill, Inc., 1992. [4] P. Pillay and R. Krishnan, "Modeling, simulation, and analysis of permanent-magnet motordrives. I. The permanent-magnet synchronous motor drive," Industry Applications, IEEE Transactions on, vol. 25, no. 2, pp. 265-273, March 1989. [5] H. Polinder, F. van der Pijl, G. de Vilder, P. Tavner, Comparison of Direct-Drive and Geared Generator Concepts for Wind Turbines, IEEE Trans. Energy Conversion, vol. 21, no. 3, pp.725-33, Sept. 2006. [6] J.M. Apsley, S.Williamson, A.C.Smith, and M.Barnes, Induction Motor Performance as a Function of Phase Number, IEE Proceedings Electric Power Applications, vol. 153, no. 6, pp. 898-904, Nov. 2006.

Appendix - System Parameters

Description Value Units

Three-phase nominal volt-ampere rating 2 MVA

RMS Line Voltage 4000 V

Nominal DC-link voltage 7.8 kV

Frequency of grid voltages 50 Hz

Frequency of grid voltages 314.2 rad/s

RMS Line current 288.7 A

Base impedance 8

Impedance of transmission cable 0.05 pu

Inductive to Resistive impedance ratio 10 -

Transmission cable inductance 1.27 mH

Transmission cable resistance 39.8 m

DC-link capacitance 20000

F

Inverter Switching Frequency 2250 Hz

Discretization sample time 22.22 s

Table A1 Base grid-side converter system parameters

Control Loop n (rad/s) Kp Ki

Current 0.707 691 1.20 V/A 605 V/A.s

Voltage 0.707 17.3 -0.778 A/V

-9.51 A/V.s

Table A2 Base grid-side converter controller gains