1. Introduction
Switched reluctance generators (SRG) has been applied to many fields because of its merits such as low manufacturing cost, low inertia, fault tolerance [1, 2] high efficiency and reliability [3]. However, an SRG requires position sensor and produces acoustic noise and vibration [2, 3]. Torque production and energy conversion process of the SRG are described [4], and the SRG control systems for regulating speed and output power use a DSP controller as processor with excitation angles from a ROM-based table of switch positions. SRGs have proved for some applications, for example, starter/generator for gas turbine of aircrafts running at constant speed and connected to utility line [3] generating maximum output power depends on optimal excitation angles due to dc bus voltage and shaft speed are constant, wind generator connected with a fixed voltage unity line [5] so maximum output power depends on optimal excitation angles at each shaft speed, starter alternator in car [6] generating maximum output power depends on optimal excitation angles at each shaft speed and dc bus voltage due to its voltage may change during charging or discharging periods. To control generating maximum output power of SRG we must know relationship of control variables included dc bus voltage, shaft speed, and excitation angles. However there is no analytical equation for output power in terms of design parameters and control variables due to highly nonlinear characteristics of an SRG, therefore iterative simulation and experiments of an SRG have been used for finding output power profile [7]. Almost simulation models used to study generating maximum output power of SRG are based on lookup-table techniques [8], magnetic equivalent circuit analysis [9], cubic-spline interpolations [10], and finiteelement analysis (FEA) [11] which are very accurate, however these models require either numerous values of flux-linkage and current position or information magnetic properties about an SRG. A simple model of the nonlinear magnetization characteristics of an SRG has been proposed [12, 13] which is easy to build, its accuracy, and the machine geometry is unknowable. Finding the output power requires the knowledge of the current waveform that the expression of phase current based on magnetic field energy proposed [14] for applying to optimize method of firing based on simulations is reliable but complicated. The closed loop power control algorithm for the SRG which relies on experimental characterization at only four operating points, turn-on and turn-off angles, speed, and power, is presented [16]. Analysis of the phase inductance and the phase current divided into five periods during the excitation period and commutation period of an SRG for creating the triggering signals of the main switches are described [17]. The relationship between commutation angles and output power is described [18] that the impact of changes in excitation angles on the output voltage or power had been examined by applying various combinations of turn-on and turn-off angles. Phase current and phase flux linkage of an SRG with optimal excitation angles obtained from a nonlinear SRG model based on MATLAB/SIMULINK are analyzed to find the relationship between the conventional and freewheeling excitation patterns through SF-transform for output power maximization and optimal symmetric freewheeling excitation [19].
From those described above, there is no analytical equation for output power in terms of control variables due to highly nonlinear characteristics of an SRG, and phase current is significant parameter to find optimal excitation angles of an SRG for maximum output power. In this paper the mathematical expression of a nonlinear magnetization curve used is simple that depends on the two magnetization curves in aligned and unaligned positions of the rotor, phase current waveform used to determine optimal excitation angles of an 8/6 SRG is applied from a nonlinear magnetization curve in term of control variables included dc bus voltage (u), shaft speed (𝜔), and excitation angles ((turn-on (θon)and turn-off (θoff) angles) and its effectiveness is validated through simulated and experimental results.
2. Analysis of SRG Operation
SRG is a machine which excitation energy is supplied in every stroke. During the conducting period θon – θoff (Fig. 1), the excitation energy is converted to electrical energy after the aligned angle. No energy is supplied to the load during the conducting period. During the de-fluxing period θoff – θext (Fig. 1), the store field energy is released as output energy through the freewheeling diodes (Fig. 2). The electrical energy produced during the defluxing period exceeds the excitation energy. The phase voltage has just θon and θoff switching angles as shown in Fig. 1.
Fig. 1.Phase of voltage and current, and flux linkage
Fig. 2.Generator circuit for a phase
Figs. 1(a) - (c) show idealized current waveforms with single pulse control that the peak current occurs at θoff – θpeak. Fig. 1(a) shows the case that the current increase after turning off the switches at θoff, when the back emf in the coil is larger than the dc bus voltage (e > u). Fig 1(b) shows the case that the constant current after turning off the switches at θoff until θpeak, when the back emf and the dc bus voltage balance (e = u). Fig. 1(c) shows the case that the current decrease after turning off the switches at θoff, when the back emf in the coil is smaller than the dc bus voltage (e < u) . The control method which regulates u with speed in order to maximize the output power by keeping the condition of u = e is proposed [1, 15].
In this paper, the control scheme of an 8/6 SRG for maximum output power which controls the current waveform like the case of e = u is proposed. The analytical model and its accuracy are validated through simulated and experimental results.
The voltage equation for a phase of the switched reluctance machine by neglecting the mutual inductance between the phases is given as:
Where u is the dc bus voltage, i is the phase current, R is the phase resistance, L is the phase inductance, e is the back emf, θ is the rotor position, and 𝜔 is the shaft speed. The back emf is defined as
Phase resistance variation is small and ohmic drop on phase resistance is usually negligible compared to dc bus voltage, so we don’t consider phase resistance variation in our analysis, flux linkage equals to:
Where extinct angle(θext) = 2θoff – θon, The circuit diagram of a generator with one phase leg has been proposed [1] as shown in Fig. 2. The integral of the currents in Fig. 2 can be defined as:
The net generated current (Io) = Iout - Iin, and phase output power (Pout) = u · Io.
3. Phase Current Formulation
In earlier contribution of magnetization curve in Fig. 3 proposed [12, 13]; at unaligned position (θu) is linear to phase current, and at aligned position (θa) approximated by two curves composes of a straight line from coordinate origin O to point S and a curve from point S to point M. An analytical model proposed [12, 13] can describe magnetization characteristics of machines with sufficient accuracy. This analytical model described by flux linkage depends on phase current and rotor position as shown in (5).
Fig. 3.Flux linkage curves in θu and θa
When
Where Lu is inductance of the coil for the unaligned position, La is inductance of the coil for the aligned position, Nr is pole number of rotor, θ is rotor angle in radian. Note that ΨS, iS and ΨM, iM are the values of the flux and current taken at points S and M respectively.
Table 1 shows parameters of the candidate SRG used in this paper which has eight poles on the stator and six poles on the rotor as shown in Figs. 4(a), and (b) shows idealized inductance versus rotor position.
Table 1.Parameters of the candidate SRG
Fig. 4.The candidate SRG used in paper: (a) Structure of the 8/6 SRG; (b) Idealized inductance versus rotor position
Fig. 5 shows the magnetization curves of the candidate SRG at different rotor positions, both from measurement and the analytical model proposed [12, 13]. Parameters of the analytical model used to calculate in (5) are Lu = 40μH, La = 490μH, is = 25A, iM = 45A, Ψs = 0.0125Wb , and ΨM = 0.017Wb.
Fig. 5.Magnetization curves of the candidate SRG
The significant variable to control an SRG for maximum output power is phase current. The phase current equation in this paper obtains from substituting (2) into (5). Since the magnetization curve in Fig. 3 approximated by two curves composes of a straight line from coordinate origin O to point S therefore phase current in this period (i ≤ is) can be obtained from (6) and a curve from point S to point M therefore phase current in this period (i > is) can be obtained from (7).
Where
4. Optimal Excitation Angles
The case of Pout = f(θon, θoff) when dc bus voltage (u) and shaft speed (𝜔) are constant, therefore output power will be a constant function of just switching angles proposed [7]. In Fig. 6 shows the relationship of phase current and flux linkage waveforms, when conducting period is equals to de-fluxing period. Flux linkage at θoff and θpeak positions are defined as Ψf and Ψk respectively, and their equations are given as:
Fig. 6.Idealized of phase current and flux linkage.
To simplify the analysis, xis defined as a ratio of Ψk and Ψf that it can be expressed as:
Then (10) can be rewritten as:
Phase current depends on dc bus voltage, shaft speed, and excitation angles that peak phase current (ipeak) at θpeak position can occur in during generation period.
Then θpeak can be obtainedas follow:
An SRG operates in single pulse mode, the phase current waveforms of the SRG limited the peak value to be equal can occur in three cases as shown in Fig. 1, and when the SRG is controlled with the constant of dc bus voltage and shaft speed therefore the peak phase current can be known by; to simplify the analysis, y is defined as a ratio of the dc bus voltage and the shaft speed and yopt is defined as a ratio of the dc bus voltage and the shaft speed at peak phase current, in (12) implies that the peak phase current can exist either at turn-off position or after, the value of in (12) never exceeds 1. Those mean a ratio of the dc bus voltage and the shaft speed at peak phase current can be expressed as:
To find the optimal turn-on angle (θon.opt) the analytical model is simulated for three cases of y ; y = 0.042, y = 0.044, and y = 0.046 that all cases are fixed u at 27V when the excitation angles were adjusted to limit the peak value of the phase current to 45A, and the output power (4-phase) can be known by Pout = 4 × (u · Io). There are many combinations of turn-on and turn-off angles as shown in Fig. 7. Apparently the maximum output power exists at turn-on angle at -15° for all cases as shown in Fig. 7(a). This simulation result has been also confirmed [7]. Also in this paper the value of optimal turn-on angle (θon,opt) is -15°. Fig. 7(b) shows output power versus turn-off angle.
Fig. 7.Output power at different excitation angles: (a) Output power versus turn-on angle (b) Output power versus turn-off angle.
Fig. 8(a) shows the phase current waveforms with single pulse control at y = 0.038, y = 0.042, and y = 0.048 with u = 27V for all cases when excitation angles were adjusted to limit the peak value of the phase current to 45A, Fig. 8(b) shows their flux linkage waveforms for three cases based on (5) with Lu = 40μH, La = 490μH, is = 25A, iM = 45A, Ψs + 0.0125Wb, and ΨM = 0.017Wb, and Fig. 8(c) shows their energy conversion loops. For peak phase current at 45A, consequently the yopt based on (13) is 0.042. Simulation results; the case of y = 0.048 or y > yopt back emf in the coil is smaller than dc bus voltage (e < u) and the current decreases after θoff, the case of y = 0.042 or y = yopt back emf in the coil and dc bus voltage balance (e = u) and the current stays constant from θoff until θpeak that the SRG generates maximum output power confirmed [1, 15], and the case of y = 0.038 or y < yopt back emf in the coil is larger than dc bus voltage (e > u) and the current increases after θoff.
Fig. 8.Phase current and flux linkage waveforms, and energy conversion loops in single pulse mode of operation
To find the optimal value of x(xopt) that the θpeak needs to be resolved using the relation of θpeak with ipeak based on (12). Phase current in case of e = u in Fig. 8(a), the xopt can be known by calculating in (10) that is 0.266.
The optimal turn-off angle (θoff.opt) can be known by substituting xopt and θpeak into (11).
To validate the value of xopt, Fig. 9 shows three phase current waveforms based on (6) and (7) that limit the peak value of the phase current to 35A, 40A, and 45A. The values of θon.opt, xopt, and u for three cases are fixed at -15°, 0.266 and 27V respectively. θpeak and θoff.opt can be known by calculating in (12) and (11) respectively. Simulation results, waveforms of the phase current for all three cases are similar to phase current waveform in case of e = u.
Fig. 9.Phase current waveforms with optimal excitation angles
5. Experimental Results
The proposed model is verified via comparison with laboratory measurements, the schematic layout of the experimental system is shown in Fig. 10(a), and the test-bed in laboratory is established shown in Fig. 10(b). The 3-phase induction motor drives the 8/6 SRG which is excited through a 4-phase asymmetrical converter. This converter uses the same dc source for excitation through the IGBTs and demagnetization through the diodes. The converted energy supplies to a sufficiently stiff dc source to prevent the dc bus level running away during generation period. The rotary encoder provides rotor position as pulse train (3,600count/rev) to TMS320F2812 DSP controller. The values of excitation angles which drive the gate of converter are processed by the controller. The y ratio can be controlled by adjusting dc bus voltage and shaft speed. The output power has been measured on a test-bed of variations of ratio of dc bus voltage and shaft speed, and excitation angles.
Fig. 10.Experimental setup, converter, load and SRG
For experiment to validate the value of xopt; Fig. 11(a) shows the measured waveforms of phase of voltage and current and average output of voltage and current when the SRG is controlled to limit ipeak to 45A with the optimal excitation angles. The u and 𝜔 used for experiment are 27V and 642 rad/s respectively that the values of θon,opt and xopt obtained from the simulation results of analytical model are -15° and 0.266 respectively. For θoff.opt is 6.34° based on (11), Fig. 12(a) shows the measured waveforms of phase of voltage and current and average output of voltage and current when the SRG is controlled to limit ipeak to 45A with u and 𝜔 used for experiment are 27V and 717 rad/s respectively that the values of θon and x are -15° and 0.253 respectively. For θoff is 6.75° based on (11), Fig. 13(a) shows the measured waveforms of phase of voltage and current and average output of voltage and current when the SRG is controlled to limit ipeak to 45A with u and 𝜔 used for experiment are 27V and 558 rad/s respectively that the values of θon and x are -15° and 1 respectively. For θoff is 4.40° based on (11), and Figs. 11(b), 12(b), and 13(b) show phase current waveform both from measurement and analytical model.
Fig. 11.Optimal excitation angle for x = xopt (IL = 39.86A, UL = 27.6V, Pout = 1100.14W)
Fig. 12.Excitation angle for x < xopt (IL = 31.97A, UL = 27.52V, Pout = 879.81W)
Fig. 13.Excitation angle for x > xopt (IL = 20.6A, UL = 27.68V, Pout = 570.21W)
Experimental results; the SRG controlled with optimal excitation angles in Fig. 11 generates maximum output power that the optimal excitation angles are θon = -15° and θoff = 6.34°.
For experiment to validate the value of xopt, Fig. 14 shows the measured phase current waveforms for all three cases of ipeak ; 35A, 40A, and 45A. Table 2 shows the optimal control variables for all three cases that the values of θon,opt, xopt and u for three cases are fixed at -15°, 0.266 and 27V respectively and θoff.opt can be known by calculating in (11). Experimental results, waveforms of the phase current for all three cases are similar to phase current waveform in case of e = u.
Fig. 14.Phase current waveforms with xopt = 0.266 for all three cases of ipeak; 35A, 40A, and 45A
Table 2.Parameters for three cases of phase current
For experiment to validate the optimal excitation angles for three cases when u is fixed; Figs. 15, 17, and 19 show output power versus different excitation angles when excitation angles were adjusted to limit ipeak = 45A for all three cases with the values of u and 𝜔 shown in Table 3. The values of optimal excitation angles based on the proposed method are shown in Table 3 that the SRG generates maximum output power and the waveforms of phase current for case 1 to case 3 are shown in Figs. 16, 18, and 20 respectively.
Fig. 15.Output power at different excitation angles; case 1
Fig. 16.Optimal excitation angles (θon=-15°θoff=6.34°) (IL = 39.86A, UL = 27.6V, Pout = 1100.14W)
Fig. 17.Output power at different excitation angles; case 2
Fig. 18.Optimal excitation angles (θon=-15°θoff=6.13°) (IL = 35.97A, UL = 27.6V, Pout = 992.77W)
Fig. 19.Output power at different excitation angles; case 3
Fig. 20.Optimal excitation angles (θon=-15°, θoff=5.91°) (IL = 31.78A, UL = 27.6V, Pout = 877.13W)
Table 3.Parameters for three cases
For experiment to validate the optimal excitation angles with different u and 𝜔, all dashed lines obtained from measurements in Fig. 21 show output power versus different excitation angles when excitation angles were adjusted to limit peak phase current for three cases; 30A, 45A, and 60A, with u and 𝜔 as shown in Table 4. Experimental results; maximum output power apparently exists at turn-on angle at -15° for all cases as shown in Fig. 21(a), three points; (a), (b), and (c), of noted by asterisk in Fig. 21(b) are the θoff.opt based on (11) as shown in table 4 and Figs. 22(a)-(c) show phase of voltage and current, and average output of voltage and current when the SRG controlled with the optimal excitation angles generates maximum output power.
Fig. 21.Output power at different excitation angles
Table 4.Optimal control variables
Fig. 22.Optimal excitation angles at (a) ipeak = 30A, (b)ipeak = 45A, and (c)ipeak = 60A
6. Conclusion
Optimal excitation angles of an 8/6 SRG analyzed from phase current waveform are presented in this paper. The phase current equation is significant factor for determining the optimal excitation angles that depends on optimal values of excitation angles (θon,opt and θoff,opt), a ratio of flux linkage at turn-off angle and peak phase current positions (xopt). The analytical model is applied from a nonlinear magnetization curve in terms of dc bus voltage(u), shaft speed (𝜔), and excitation angles (θon and θoff). The experimental results have confirmed the accuracy of the analytical model for determining the optimal excitation angles of the SRG for maximum output power. All cases for the SRG controlled with the optimal excitation angles in single pulse mode of operation generate maximum output power and phase current waveforms are similar to the case of e = u. Simulation and experimental results; the value of θon,opt is -15°, the value of θoff,opt can be known by substituting xopt = 0.266 and θpeak into (11).
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