Performance prediction of switched reluctance generator with time average and small signal models

Abstract

This paper presents the complete mathematical model and predicts the performance of switched reluctance generator with time average and small signal models. The complete mathematical model is developed in three stages. First, a switching model is developed based on quasi-linear inductance profile. Next, based on the switching behaviour, a time average model is obtained to measure the difference between the excitation and generation time in each switching cycle. Finally, to track control voltage and current wave shapes, a small signal model is designed. The effectiveness of the complete multilevel model combining electrical machine, power converter, load and control with programming language is demonstrated through simulations. A PI controller is used for controlling the voltage of the generator. The results presented show that the controller exhibits accurate tracking control of load voltage under different operating conditions. This demonstrates that the proposed model is able to perform an accurate control of the generated output voltage even in transient situations. The simulation is performed to choose the control parameters and study the performance of switched reluctance generator prior to its actual implementation. Initial experimental results are presented using NI-Data acquisition card to control the output power according to load requirements.

Keywords

generator / reluctance / switching model / small signal model / time average model

Introduction

Switched reluctance machines (SRM) combine desirable features of induction motor, direct current (DC) motor and permanent magnet (PM) brushless machine. The main advantages of SRM are high efficiency, cheap low manufacturing cost, elimination of electric losses in the rotor, fast response and torque/speed characteristics can be easily tailored modifiable to meet the required application. These attractive features of SRM along with the boom in power electronics have raised the performance of SRM to competitive levels for motoring and generating applications [1,2]. SRM is compatible with demanding applications, both for high speed and high temperature applications and has been built for many industrial, commercial and domestic applications ranging from a few watts to hundreds of Kilowatts. SRM as a generator (SRG) is the dual of SRM as a motor. The engineering challenges of this powerful electric machine depend on the inductance profile and the current pulse of the rotor position. Exciting the stator windings during the ascending part of the inductance profile results in positive torque and hence the machine acts as a motor. Conversely, if the current pulse is positioned in the decreasing part of the inductance profile, it will result in a negative torque as shown in Fig. 1. The prime mover drives the rotor in the opposite direction and hence voltage is generated in the stator coils to produce power.

Prior research proposes various on-line and off –line modelling techniques [1-9]. In Ref. [3] a non-linear magnetic field model of SRM drive with boundary element analysis was developed. The non-linear model requires finite element analysis or the measured flux-linkage data for analysis method. Essah and Scott [4] set forth a non-linear average value model of SRM by introducing the machine variables with an inner product of a vector of basic functions and time varying co-efficient vectors. This model requires a derivation of set of nonlinear differential equations governing the behaviour of the coefficient vectors. Besmi et al. [5] presented an analytical algorithm based on flux-tubes in four regions to obtain SRM characteristics and flux-density waveforms. This algorithm emphasizes the use of finite element analysis to find reluctances in various sections of the motor. The equivalent circuit is a series-parallel combination of reluctances. Xue et al. [6] presented non-linear magnetic characteristics of SRM drive using two-dimensional least square. The method involves the founding of polynomials and co-efficient for the fitting the function. Khalil and Husain [7] presented a novel invertible, generalized flux-current model of SRM based on Fourier-series expansion and the model is simplified for real time controller applications. The expressions contain sine and cosine terms with limited terms only. A pi-sigma neural network is employed to model the SRM drive based on the non-linear mapping of rotor position and phase currents to obtain flux-linkages in Ref. [8]. This method requires an additional knowledge of complicated neural-networks. A Lyapunov function based direct torque controller with non-linear state-space model of SRM drive is reported in Ref. [9].

A comprehensive review of publications reveal that not enough work was conducted in modelling switched reluctance generator (SRG). Skvarenina et al. [10] described a detailed SRG model written in advanced control system language (ACSL) and studied the performance of SRG with various loads. Torrey [11] reviewed the energy conversion process with electro mechanics of speed and power control applications to serve as starter/alternator in automotive applications. Chen and Ji [12] discussed the design principle of a 12/8 SRG system based on Visual Basic 6.0 and FORTRAN 5.0 software. The dynamic behaviour of SRG was described with a coupled electric and magnetic circuit model based on the SPICE simulation program in Ref. [13]. Ping et al.[14] established a linear and non-linear model of SRG in self-excited and separately excited mode of operation. To study the different working zones, the SRG model was analyzed for non-linear analysis using MATLAB/SIMULINK environment in Ref. [15]. Turker and Kuyumcu [16] elucidated a dynamic model using magnetic circuit analysis of a new design SRG with untraditional pole construction and small coupling on an exercise-bike. Liptak et al. [17] discussed the equivalent circuit of single phase SRG and investigated through asymmetrical power converter based on DC series generator scheme. The whole system was controlled with DSP under various speeds and load conditions. Abelardo et al. [18] proposed a simple AC self-excited SRG used as a battery charger in isolated locations based on the non-linear oscillator model. Hao and Jason [19] explained the major components of a dual motor SR drive—one for electric locomotive traction and the other for the variable speed generator system for wind power application. Jyoti et al. [20] explained the switching behaviour of SRG using MAGNET7.0 software. Accordingly, with its power and control circuit, models were established in a MATLAB/SIMULINK environment. From the developed plant dynamic model parameters, voltage feedback control of the SRG system was designed. Theoretical bases of the proposed control approaches were derived, and their effectiveness was demonstrated by some simulation results. Godinho et al. [21] addressed the problem of sea-wave energy conversion at low speeds by developing an analytical design and optimization based on FEM model of a linear SRG.

This paper reports the behaviour of self-excited SRG dynamics with a complete mathematical model. It focuses on the basic structure of SRG, discussed the energy-conversion process, and explains the dynamic model of SRG and the switching converter model to study the variations in input voltage, phase current and the load current. The magnetization characteristics of SRM are modelled piece-wise linear giving a quasi-linear behaviour, with current as the undetermined factor and only few details of flux-linkages at arbitrary rotor position. The time average model is discussed with the effects of dynamics introduced by the inductors and capacitors on the converter in each switching cycle. The model preserves all the relevant dynamics of the power source and the load, which is particularly useful in the systems in which a detailed small-signal study is needed to achieve the overall system stability while tracking voltage and current. Moreover, it demonstrates the effectiveness of the multilevel model of SRG using time step integration through numerical simulations. The successful and efficient operation of a switched reluctance generator depends on the choice of different control parameters. A PI controller is used for controlling the voltage of the generator. The results presented show that the controller exhibits accurate tracking control of load voltage under different operating conditions. This demonstrates that the proposed model is able to perform an accurate control of generated output voltage even in transient situations. The study is very helpful for the accurate prediction of performance, simulation and control of the SRG system. As it is based on various conduction and generation periods, the mathematical derivations is more accurate than those of previous models. The simulation is performed to choose the control parameters and study the performance of switched reluctance generator prior to its actual implementation. Initial experimental results are presented using NI-Data acquisition card to control the output power according to load requirements.

System description

The switched reluctance machine under study comprises of four rotor poles where in each of the pole protrudes cross-wise around the shaft and six-pole stator arranged around the rotor. Each of the protruding poles of the stator having a concentrated winding is shown in Fig. 2. The current passing through the each winding is unidirectional. The on-off cycle of the magnetic attractive forces causes the rotor to rotate. The details of SRM coupled to DC motor are given in Appendix A.

SRG requires a converter to drive the machine. A three-phase, unidirectional, classical power converter having three legs, each phase of which consists of two switching devices and two freewheeling diodes is illustrated in Fig. 3. The switches allow the current in the a windings to be controlled in the pre-determined manner. The reversal of the stator field is achieved by transferring current to the next winding. The current is determined by the motor winding resistance, applied voltage and excitation angles.

Proposed mathematical model

The SRG model, a three phase, 6/4 configuration, lab prototype model as presented in Fig. 2, is mathematically modelled to understand the important physical phenomenon. The non-linear model is approximated to a quasi-linear model to gain physical insight into system behaviour as demonstrated in Fig. 1. Thus, a switching model is developed first to study the dominant behaviour of SRG with the input voltage, the load current and the impact of the dynamics introduced by the inductors and capacitors on the converter. The complete structure of the machine with power converter is explained in Ref. [20].

Switching model

The phase voltages and currents of a 6/4 SRG system energized by asymmetrical power inverter in self-excited mode are given by

(1)S_{\mathrm{a}}V=i_a{R}_a+L_a{\displaystyle \frac{d{L}_a}{dt}}+{\displaystyle \frac{\partial L_a}{\partial \theta }}\omega i_a,

(2)S_{b}V=i_b{R}_b+L_b{\displaystyle \frac{d{L}_b}{dt}}+{\displaystyle \frac{\partial L_b}{\partial \theta }}\omega i_b,

(3)S_{c}V=i_c{R}_c+L_c{\displaystyle \frac{d{L}_c}{dt}}+{\displaystyle \frac{\partial L_c}{\partial \theta }}\omega i_c,

where i_a, i_b, i_c are the currents passing through the three phases a, b and c with identical winding parameters. The switching functions S_a, S_b, S_c are given by

(4)S_{\mathrm{p}\mathrm{h}}=\{\begin{array}{lll}+\mathrm{1},\hfill & bothswitchesON,\hfill & \hfill \\ -\mathrm{1},\hfill & bothswitchesOFF,\hfill & i_{ph}>\mathrm{0},\hfill \\ 0,\hfill & oneswitchON,\hfill & i_{ph}=\mathrm{0.}\hfill \end{array}

(4)

The equations describing the capacitor dynamics are given by

(5){\displaystyle \frac{dV}{dt}}={\displaystyle \frac{I_c}{C}}and{\displaystyle \frac{V}{R_L}}=i_L,

with I_c the capacitor current, i_L the load current and V the capacitor voltage across the resistive load R_L. The algebraic sum of the currents at the capacitor terminals is given by

(6)I_c=-(S_a{i}_a+S_b{i}_b+S_c{i}_c)-i_L.

Time average model

The inductance profile under study is as depicted in Fig. 4. Let T_ON, T_OFF and T_EN be the period of conduction, generation and no excitation periods of the stator phase windings.

With

(7)T_{ON}={\displaystyle \frac{{\theta }_{dwell}}{\omega }},T_{OFF}=\left[{\displaystyle \frac{i_{av}{L}_{av}}{-V+{(\partial L/\partial \theta )}_{av}\omega }}\right]andT={\displaystyle \frac{\pi /\mathrm{2}}{\omega }}.

Then

(8)T_{EN}=\left[{\displaystyle \frac{\pi /\mathrm{2}}{\omega }}\right]-T_{ON}-T_{OFF}with\omega ={\displaystyle \frac{d\theta }{dt}}.

Taking the ratio of ON and OFF time periods with respect to total time T, equation (9) can be obtained.

(9){\displaystyle \frac{T_{ON}}{T}}=\left[{\displaystyle \frac{\mathrm{2}}{\pi }}\right]{\theta }_{dwell}and{\displaystyle \frac{T_{OFF}}{T}}={\displaystyle \frac{\mathrm{2}}{\pi }}\left[{\displaystyle \frac{i_{av}{L}_{av}}{-V+{(\partial L/\partial \theta )}_{av}\omega }}\right].

Averaging of machine variables is carried out over a cycle of π/2 electrical radians with turn-on angle θ_ON and turn-off angle θ_OFF. Figure 5 demonstrates the developed time average model with machine variables. Different values of parameters obtained by time average values are given in Appendix B. With ‘V_e’ the excitation voltage applied during T_ON and ‘V ’the capacitor voltage appearing across the machine windings during T_OFF period, Eqs. (8) and (9) can be replaced by

(10){\displaystyle \frac{T_{\mathrm{O}\mathrm{N}}}{T}}{V}_{\mathrm{e}}-{\displaystyle \frac{T_{\mathrm{O}\mathrm{F}\mathrm{F}}}{T}}V=i_{\mathrm{a}}{R}_{\mathrm{a}}+L_{\mathrm{a}\mathrm{v}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{a}\mathrm{v}}}{\mathrm{d}t}}+{\displaystyle \frac{\partial L}{\partial {\theta }_{\mathrm{a}\mathrm{v}}}}\omega i_{\mathrm{a}\mathrm{v}},

(11)C{\displaystyle \frac{dV}{dt}}=-\mathrm{3}{i}_{av}-{\displaystyle \frac{V}{R_L}}.

The torque produced by SRG acts as load torque for the DC motor. The prime mover dynamics are

(12)J_m{\displaystyle \frac{d\omega }{dt}}+B_m\omega =K_t{i}_m-{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{i}_{av}^2{\left({\displaystyle \frac{\partial L}{\partial \theta }}\right)}_{av},

(13)V_m=R_m{i}_m+L_m{\displaystyle \frac{d{i}_m}{dt}}-K_b\omega .

Small signal model

Time average model equations are non-linear, involving switching ripples in the inductor current and capacitor voltage waveforms, hence a small signal model is developed for stability analysis, Perturbing with perturbations around the steady state operating point. Let

p=T_{ON}/T,q=T_{OFF}/T,

i=I+\widehat{i},v=V+\widehat{v},p=P+\widehat{p},q=Q+\widehat{q},

where I, V, P and Q represent steady state values and ‘^’ represent the perturbed values of phase current, capacitor voltage, conduction and generation periods, respectively.

(14){\displaystyle \frac{d\widehat{i}}{dt}}={\displaystyle \frac{\mathrm{1}}{L_{av}}}\left[-\widehat{i}{R}_a+\widehat{p}{V}_e-{\left({\displaystyle \frac{\partial L}{\partial \theta }}\right)}_{av}\omega \widehat{i}+P{\widehat{v}}_e-V\widehat{q}-Q\widehat{v}\right],

(15){\displaystyle \frac{d\widehat{v}}{dt}}=-{\displaystyle \frac{\mathrm{1}}{C}}\left[\mathrm{3}\widehat{i}+{\displaystyle \frac{\widehat{v}}{R}}\right].

And the prime mover model is given by

(16){\displaystyle \frac{d{\widehat{i}}_m}{dt}}={\displaystyle \frac{\mathrm{1}}{L_m}}\left({\widehat{v}}_m-{\widehat{i}}_m{R}_m-\widehat{\omega }{K}_b\right),

(17){\displaystyle \frac{d\widehat{\omega }}{dt}}={\displaystyle \frac{\mathrm{1}}{J_m}}\left[K_t{\widehat{i}}_m-\mathrm{3}{\widehat{i}}_m{\left({\displaystyle \frac{\partial L}{\partial \theta }}\right)}_{av}-\widehat{\omega }{B}_m\right].

The complete mathematical model of SRG dynamics is given by

\left[\begin{array}{l}{\displaystyle \frac{d\widehat{i}}{dt}}\\ {\displaystyle \frac{d\widehat{\mathit{v}}}{dt}}\\ {\displaystyle \frac{d\widehat{\omega }}{dt}}\\ {\displaystyle \frac{\mathrm{d}{\widehat{i}}_{\mathrm{m}}}{dt}}\end{array}\right]=\left[\begin{array}{llll}{\displaystyle \frac{-\mathrm{1}}{L_{av}}}\left[R_a+{\left({\displaystyle \frac{\partial L}{\partial \theta }}\right)}_{av}\omega \right]& \mathrm{0}& -{\displaystyle \frac{\mathrm{1}}{L_{av}}}{\left({\displaystyle \frac{\partial L}{\partial \theta }}\right)}_{av}I& {\displaystyle \frac{-Q}{L_{av}}}\\ {\displaystyle \frac{-\mathrm{3}}{C}}& {\displaystyle \frac{-\mathrm{1}}{R}}& \mathrm{0}& \mathrm{0}\\ {\displaystyle \frac{-\mathrm{3}}{\mathrm{2}{J}_m}}I{\left({\displaystyle \frac{\partial L}{\partial \theta }}\right)}_{av}& \mathrm{0}& {\displaystyle \frac{-B_m}{J_m}}& {\displaystyle \frac{K_t}{J_m}}\\ \mathrm{0}& \mathrm{0}& {\displaystyle \frac{-K_B}{L_m}}& {\displaystyle \frac{-R_m}{L_m}}\end{array}\right]\left[\begin{array}{l}\widehat{i}\\ \widehat{v}\\ \widehat{\omega }\\ {\widehat{i}}_m\end{array}\right]+\left[\begin{array}{llll}{\displaystyle \frac{V_e}{L_{av}}}& {\displaystyle \frac{V_e}{L_{av}}}& {\displaystyle \frac{P}{L_{av}}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& {\displaystyle \frac{\mathrm{1}}{L_m}}\end{array}\right]\left[\begin{array}{l}\widehat{p}\\ \widehat{q}\\ {\widehat{v}}_e\\ {\widehat{V}}_m\end{array}\right],

(18)\widehat{v}=\left[\begin{array}{llll}\mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}\end{array}\right]\left[\begin{array}{l}\widehat{i}\\ \widehat{v}\\ \widehat{\omega }\\ {\widehat{i}}_m\end{array}\right].

Simulation results and discussion

The complete mathematical model developed is simulated in a Matlab/Simulink environment to provide accurate predictions of the system behaviour in reality with a desired level of accuracy, thus serving as a prototype model and control system design as displayed in Fig. 6. Some of the simulation results like the waveforms of the switching inverter, rotor angle dependent inductance of the stator, excitation voltage, currents produced, torque developed by the SRG are presented in Figs. 7 and 8.

Results and discussion

SRG has highly non-linear characteristics due to its non-linear flux behaviour. When the rotor speed is low, SRG operates in a PWM mode and when the rotor speed is high, it operates in a single pulse mode. The control issues regarding the several applications for SRG are different. The SRG can be controlled to supply constant power or vary the output power according to load requirements. For example, in automotive applications, the demand for generated power tend to oscillate abruptly with the connection and disconnection of loads, giving rise to large transients, thus precise control on load bus becomes necessary. This justifies the difficulty in controller design for achieving robust voltage tracking response. SRG systems invariably require a feedback, which is accomplished by a PI controller that adjusts the reference current i_ref such that the converter controls input changes. A hysteresis control with fixed frequency chopping control is used as the current controller. The design of a PI controller is based on the Ziegler-Nichols tuning technique. With fine tuning the proportional and integral gain are chosen as K_p = 11.8 and K_i =8.15. The tuning of PI samples with integral absolute error (IAE), integral time absolute error (ITAE) are given in Appendix C.

Under no load conditions, the output voltage of SRG is maintained constant at 100 V as validated in Fig. 9. A load transient test is performed, when the step load is applied and removed. The voltage reference is set at 100 V and 1000 r/min. In Fig. 10(a) the DC-bus voltage overshoots by 20 V and lasts for 2 s during load removal, while in Fig. 10(b) the DC-bus voltage undershoots by 8 V and lasts for 1.5 s during load application. In both cases, the DC-bus voltage is successfully restored to the reference value with no steady-state error. The closed-loop controller requires larger phase current to supply load and charge the DC-bus capacitor. In order to completely observe the performance of the SRG system, the reference DC-bus voltage is made variable in square, ramp and sine function as shown in Fig. 11. The DC-bus voltage is tracking the square-wave stepping between 95 V and 105 V. Fig. 11(b) shows, the DC-bus voltage trying to track the reference saw-tooth ramping up from 95-105 V. The DC-bus voltage ramps down when the capacitor discharges with the time constant of its RC circuit. The electromagnetic torque increases and decreases immediately to sustain the load demand, and the output voltage is kept constant. It can be seen that in all cases, the controller reacts very fast and the voltage is obtained without overshoot.

Experimental implantation and results

A snapshot of the laboratory setup consists of the components illustrated in Fig. 12. The drive system is made up of a motor, a personal PC, the driver circuit and a 6259 NI-DAQ card. The motor is a 1.2 kW, 6/4, three phase lab prototype model. The 6259 NI-DAQ card has a sampling frequency of 1.25 MS/s with 16 differential or 32 single ended analog inputs and 24 digital input and output channels working for+/-11 V. The ADC has a resolution of 16 bits. The switching logic is implemented by Matlab/Simulink software and executed using the NI-DAQ Card.

The control algorithm is written and loaded in the DAQ card using PC. A totem pole gate driver is used as shown in Fig. 13 to turn ON the MOSFET. When the input to the driver is OFF, the NPN transistor is turned ON, providing a positive gate voltage to the MOSFET and to turn OFF. When the input to the driver is ON, the NPN transistor is turned OFF, providing a low gate voltage to the MOSFET. The driver circuit requires isolated power supply. The input-output isolation is achieved using high speed HCPL-4503 opto-coupler to transfer the control signal from the input stage to the gate driver stage. The inputs to the driver circuits are rotor position, the direction of the desired rotation and the magnitude of the control voltage. The output of the switching logic section is a sequence of gating signals that are pulse width modulated. These signals are used to drive the power converter.

The power converter is a dc/ac asymmetrical inverter utilizing 6 MOSFETs (IRFP460) and 6 freewheeling diodes (MUR3040) as shown in Fig. 3. The output of the power converter is a three-phase ac waveform. The control architecture consists of a three-layer structure. In the upper level of the control system, the controller design is based on state feedback strategy and generates the required voltage for each phase. In the middle level, the current controller generates pulses of required duty cycle to achieve the desired current demanded by the upper level. The phase voltages and currents are sensed and fed back to the ADC channel of the DAQ The current controller, in turn, depends on the rotor position information to trigger the gate pulses of the three-phase inverter connected to the generator phases. The firing angles are programmable and can be varied easily to control the generated voltage. The real no load state is defined as SRG generates the energy consumed by the control, the logic circsuits, the leakage capacitor resistance, the core, the windings, and the mechanical losses. Figure 14 gives the experimental results of a SRG when connected to a load of 140 W and 200 W respectively. The output voltage ripple is small under no-load and varies as the load increases with the generator running at a speed of 1000 r/min, connected with a capacitance of 4800 µF. Increasing the value of the filter capacitance can decrease the ripple voltage but it degrades the SRG dynamic performance. As listed in Table 1, the SRG is made to run at 1000 r/min, with different loads and its response is studied on the output of the SRG. The phase current is successfully regulated by soft chopping, as the back EMF is lower than 100 V. Figure 15 presents the current in one of the phases with an excitation voltage of 30 V, supplying a load of 40 W.

Conclusions

Some important findings on the operation and control of SRG are obtained in this study, which provides theoretical basis for further design and experimental study. Following the principle of operation, a complete mathematical model is proposed for experimental prototype model of a self-excited switched reluctance generator, which is a valuable tool for large- and small-signal analysis and control system design. The machine parameters obtained from the time average model can be used for a variety of the system’s operating conditions. The model preserves all the relevant dynamics of the power source and the load. This is particularly useful in the systems in which a detailed small-signal study is needed in order to achieve the overall system stability. A capacitor partially charged, and in parallel with the load is used as a means to provide the necessary amount of magnetic flux and start to initiate the generator’s voltage built-up. The arrangement is such that the generated voltage and current is delivered to the load in D.C. mode and tracks the reference input voltage, while the dc-link voltage is built-up, load current increases and changes according to the output. Simulation results have validated the mathematical model. The results presented show that the controller exhibits accurate control of load voltage under different operating conditions. This demonstrates that the proposed strategy is able to perform an accurate control of generated output voltage even in transient situations. It is shown that the salient features of the suggested analytical algorithm include accuracy, simplicity, convergence, and short run-time of the simulation.

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