Analysis of Open-Loop Current Estimators for Current Sensor Fault Compensation in PMSM Motor Drive

The mathematical model of the current estimator is derived directly from Eqs (1) and (2). Using the flux linkages Eq. (2) and substituting them into the voltage Eq. (1), the following are obtained after transformations: (4) diddt=1Ldud−Rsid+ωeLqiqdiqdt=1Lquq−Rsiq−ωeLdid+Ψf \left\{ {\matrix{ {{{d{i_d}} \over {dt}} = {1 \over {{L_d}}}\left( {{u_d} - {R_s}{i_d} + {\omega _e}{L_q}{i_q}} \right)} \hfill \cr {{{d{i_q}} \over {dt}} = {1 \over {{L_q}}}\left( {{u_q} - {R_s}{i_q} - {\omega _e}\left( {{L_d}{i_d} + {\Psi _f}} \right)} \right)} \hfill \cr } } \right.

The above system of equations constitutes the basic model of an open-loop current estimator and will be referred to as VCS (Adamczyk and Orlowska-Kowalska, 2022). The formula contains parameters which, due to simplifying assumptions, are considered constant: resistance R_s, inductance L_dq and permanent magnet flux Ψ_f. This model is therefore sensitive to the accuracy of their determination and does not have any correction terms, as is the case with the Luenberger observer. The software implementation of this model can be achieved by using a modified Euler method with a sampling time Δt: (5) i^dk=i^dk−1+ΔtLdudk−1−Rsi^dk−1+ωek−1Lqi^qk−1i^qk=i^1k−1+ΔtLquqk−1−Rsi^qk−1−ωek−1Ldi^dk+Ψf \left\{ {\matrix{ {\hat i_d^k = \hat i_d^{k - 1} + {{\Delta t} \over {{L_d}}}\left( {u_d^{k - 1} - {R_s}\hat i_d^{k - 1} + \omega _e^{k - 1}{L_q}\hat i_q^{k - 1}} \right)} \hfill \cr {\hat i_q^k = \hat i_1^{k - 1} + {{\Delta t} \over {{L_q}}}\left( {u_q^{k - 1} - {R_s}\hat i_q^{k - 1} - \omega _e^{k - 1}\left( {{L_d}\hat i_d^k + {\Psi _f}} \right)} \right)} \hfill \cr } } \right.

In the above formulas, notation is used in which the superscript k denotes a sample at time t = kΔt, where k ∈ ℤ and k − i denotes a historical sample from i sampling periods ago. In the above equations, attention should be paid to the component Ldi^dk {L_d}\hat i_d^k appearing in the second equation, which depends on the value i^dk \hat i_d^k determined in the current calculation step.

The current estimator model presented in the previous subsection is derived from equations that were developed taking into account simplifying assumptions regarding time-invariant parameters, omitting phenomena related to saturation and hysteresis of the magnetic circuit and considering only the fundamental component of the spatial field distribution in the gap. The actual dependence of the flux linkages is generally expressed by a non-linear function of currents i_d and i_q: (6) Ψd=fdid,iqΨq=fqid,iq \left\{ {\matrix{ {{\Psi _d} = {f_d}\left( {{i_d},{i_q}} \right)} \hfill \cr {{\Psi _q} = {f_q}\left( {{i_d},{i_q}} \right)} \hfill \cr } } \right.

By determining the derivatives composed of the above relationships and substituting them into the voltage Eq. (1), the following system of equations is obtained after appropriate transformations: (7) diddt=1∂Ψd∂idud−Rsid−∂Ψd∂iqdiqdt+ωeΨqdiqdt=1∂Ψq∂iquq−Rsiq−∂Ψq∂iddiddt−ωeΨd \left\{ {\matrix{ {{{d{i_d}} \over {dt}} = {1 \over {{{\partial {\Psi _d}} \over {\partial {i_d}}}}}\left( {{u_d} - {R_s}{i_d} - {{\partial {\Psi _d}} \over {\partial {i_q}}}{{d{i_q}} \over {dt}} + {\omega _e}{\Psi _q}} \right)} \hfill \cr {{{d{i_q}} \over {dt}} = {1 \over {{{\partial {\Psi _q}} \over {\partial {i_q}}}}}\left( {{u_q} - {R_s}{i_q} - {{\partial {\Psi _q}} \over {\partial {i_d}}}{{d{i_d}} \over {dt}} - {\omega _e}{\Psi _d}} \right)} \hfill \cr } } \right.

In the above equations, the inductance parameters and the value of the flux from permanent magnets have been replaced by the values of the associated stator flux and its derivatives. This model, therefore, takes into account the actual characteristics of the flux linkages. The open-loop estimator model described above requires knowledge of the relationships Eq. (6) that will be determined by experimental measurements, and the appropriate measurement procedure will be described later in this article. The values obtained from the measurements can then be stored in a look-up table (LUT) and used during the operation of the electric drive controller algorithm. The discrete implementation of this model after applying Euler’s method is as follows: (8) i^dk=i^dk−1+Δt∂fdk−1∂idudk−1−Rsi^dk−1−∂fdk−1∂iqi^dk−1−i^dk−2Δt+ωek−1fqk−1i^qk=i^qk−1+Δt∂fqk−1∂iquqk−1−Rsi^qk−1−∂fqk−1∂idi^dk−1−idk−2Δt−ωek−1fdk \left\{ {\matrix{ {\hat i_d^k = \hat i_d^{k - 1} + {{\Delta t} \over {{{\partial f_d^{k - 1}} \over {\partial {i_d}}}}}\left( {u_d^{k - 1} - {R_s}\hat i_d^{k - 1} - {{\partial f_d^{k - 1}} \over {\partial {i_q}}}{{\hat i_d^{k - 1} - \hat i_d^{k - 2}} \over {\Delta t}} + \omega _e^{k - 1}f_q^{k - 1}} \right)} \hfill \cr {\hat i_q^k = \hat i_q^{k - 1} + {{\Delta t} \over {{{\partial f_q^{k - 1}} \over {\partial {i_q}}}}}\left( {u_q^{k - 1} - {R_s}\hat i_q^{k - 1} - {{\partial f_q^{k - 1}} \over {\partial {i_d}}}{{\hat i_d^{k - 1} - i_d^{k - 2}} \over {\Delta t}} - \omega _e^{k - 1}f_d^k} \right)} \hfill \cr } } \right. where ^ indicates estimated values and fdk=fdi^dk,i^dk−1 f_d^k = {f_d}\left( {\hat i_d^k,\hat i_d^{k - 1}} \right) , fdk−1=fdi^dk−1,i^qk−1 f_d^{k - 1} = {f_d}\left( {\hat i_d^{k - 1},\hat i_q^{k - 1}} \right) , fqk−1=fqi^dk−1,i^qk−1 f_q^{k - 1} = {f_q}\left( {\hat i_d^{k - 1},\hat i_q^{k - 1}} \right) .

Further development of the estimator model involves taking into account the dependence of the stator flux not only on currents but also on the rotor position: (9) Ψd=Fdid,iq,θΨq=Fqid,iq,θ \left\{ {\matrix{ {{\Psi _d} = {F_d}\left( {{i_d},{i_q},\theta } \right)} \hfill \cr {{\Psi _q} = {F_q}\left( {{i_d},{i_q},\theta } \right)} \hfill \cr } } \right.

The estimator equations currently take the following form: (10) diddt=1∂Ψd∂idud−Rsid−∂Ψd∂iqdiqdt−∂Ψd∂θdθdt+ωeΨqdiqdt=1∂Ψq∂iquq−Rsiq−∂Ψq∂iddiddt−∂Ψq∂θdθdt−ωeΨd \left\{ {\matrix{ {{{d{i_d}} \over {dt}} = {1 \over {{{\partial {\Psi _d}} \over {\partial {i_d}}}}}\left( {{u_d} - {R_s}{i_d} - {{\partial {\Psi _d}} \over {\partial {i_q}}}{{d{i_q}} \over {dt}} - {{\partial {\Psi _d}} \over {\partial \theta }}{{d\theta } \over {dt}} + {\omega _e}{\Psi _q}} \right)} \hfill \cr {{{d{i_q}} \over {dt}} = {1 \over {{{\partial {\Psi _q}} \over {\partial {i_q}}}}}\left( {{u_q} - {R_s}{i_q} - {{\partial {\Psi _q}} \over {\partial {i_d}}}{{d{i_d}} \over {dt}} - {{\partial {\Psi _q}} \over {\partial \theta }}{{d\theta } \over {dt}} - {\omega _e}{\Psi _d}} \right)} \hfill \cr } } \right.

The difference between the LUTE estimator Eq. (7) and the above equation concerns the presence of an additional term that takes into account changes in the flux from the position. As before, these values are tabulated, and the discrete equation after applying the Euler method is as follows: (11) i^dk=i^dk−1+Δt∂Fdk−1∂idudk−1−Rsi^dk−1−∂Fdk−1∂iqi^qk−1−i^qk−2Δt−∂Fdk−1∂θωmk−1+ωek−1Fqk−1i^qk=i^qk−1+Δt∂Fqk−1∂iquqk−1−Rsi^qk−1−∂Fqk−1∂idi^dk−1−i^dk−2Δt−∂Fqk−1∂θωmk−1−ωek−1Fdk \left\{ {\matrix{ {\hat i_d^k = \hat i_d^{k - 1} + {{\Delta t} \over {{{\partial F_d^{k - 1}} \over {\partial {i_d}}}}}\left( {u_d^{k - 1} - {R_s}\hat i_d^{k - 1} - {{\partial F_d^{k - 1}} \over {\partial {i_q}}}{{\hat i_q^{k - 1} - \hat i_q^{k - 2}} \over {\Delta t}} - {{\partial F_d^{k - 1}} \over {\partial \theta }}\omega _m^{k - 1} + \omega _e^{k - 1}F_q^{k - 1}} \right)} \hfill \cr {\hat i_q^k = \hat i_q^{k - 1} + {{\Delta t} \over {{{\partial F_q^{k - 1}} \over {\partial {i_q}}}}}\left( {u_q^{k - 1} - {R_s}\hat i_q^{k - 1} - {{\partial F_q^{k - 1}} \over {\partial {i_d}}}{{\hat i_d^{k - 1} - \hat i_d^{k - 2}} \over {\Delta t}} - {{\partial F_q^{k - 1}} \over {\partial \theta }}\omega _m^{k - 1} - \omega _e^{k - 1}F_d^k} \right)} \hfill \cr } } \right. where the derivative of the angular position is replaced by mechanical speed ω_m and Fdk=Fdi^dk,i^qk−1,θk−1 F_d^k = {F_d}\left( {\hat i_d^k,\hat i_q^{k - 1},{\theta ^{k - 1}}} \right) , Fdk−1=Fdi^dk−1,i^qk−1,θk−1 F_d^{k - 1} = {F_d}\left( {\hat i_d^{k - 1},\hat i_q^{k - 1},{\theta ^{k - 1}}} \right) , Fqk−1=Fqi^dk−1,i^qk−1,θk−1 F_q^{k - 1} = {F_q}\left( {\hat i_d^{k - 1},\hat i_q^{k - 1},{\theta ^{k - 1}}} \right) .

The verification of each of the three estimators was carried out in the FOC structure with a fixed power angle δ = π / 2, which is equivalent to forcing the current i_d = 0. In this case, the equation for electromagnetic torque Eq. (3) is simplified and, assuming a constant flux value, indicates a linear relationship between electromagnetic torque and motor current: (12) Te=32pbΨdiq {T_e} = {3 \over 2}{p_b}{\Psi _d}{i_q}

The control structure using previously developed current estimator models is presented in Figure 1. The tests were first conducted for control using measurements from CS, and then switch was changed to check the operation of the system using estimated current values. During the operation of the system on the physical measurements from the CS, estimated signals were also recorded. In this way, it is possible to compare the quality of estimation under default drive operating conditions, i.e., using measurements from sensors, and under closed-loop control conditions, when the estimated values affect the operation of the drive.

Figure 1.

Scheme of control structure under study. CS, current sensors; ELUTE, extended lookup table-based estimator; FOC, field oriented control; LUTE, lookup table-based estimator; VCS, virtual current sensor.

Each current estimator model presented in the previous chapter includes voltage and velocity measurements in its equations. The accuracy of measuring these physical quantities will directly affect the estimator’s performance, assuming that the other parameters have been correctly identified. This subsection will discuss the problem of voltage measurement.

The electric drive system includes a voltage inverter, which is treated as an ideal voltage amplifier when designing the control algorithm. However, the output voltage characteristic of the inverter is non-linear and is affected by dead time effects, commutation delays, voltage drops across power devices, and equivalent resistance and capacitance in the inverter legs. The voltage used in calculations related to the estimation of any physical quantity or in the drive parameter identification procedure can be reproduced on the basis of inverter duty-cycle commands from the control structure and DC bus voltage measurement. If this solution is chosen, the control structure must include an appropriate algorithm for compensating for the inverter’s dead time. Compensation for this phenomenon is important for drives with sensorless control, as voltage distortions have a significant impact on the operation of the estimation or identification algorithm, especially at low drive speeds.

Compensation of non-linear inverter phenomena is based on determining the parameters of the mathematical model of the inverter (identification stage) and then using these parameters to calculate the compensating voltage added to the reference voltages in the control structure (compensation stage).

The selected process for identifying inverter parameters involves performing a series of measurements in a system with deactivated compensation using a vector drive control structure (Bedetti et al., 2015). The voltage needed to maintain the reference current value is the result of the action of the controllers, and in the steady state, it corresponds to the sum of the voltage drop across the resistance of the wires and the voltage distorted by the inverter. Thus, by forcing a step current waveform, the corresponding value of the reference voltage from the regulators can be read. Based on the obtained voltage characteristics as a function of current, it is then possible to determine the total resistance of the motor stator windings and wires R_s, as well as the parameters required for compensation, which include the dead time T_DT and the equivalent capacitance in the inverter leg C_out. The process involves comparing the obtained characteristics with the mathematical model of the inverter, and the resulting parameters can then be used in the dead time compensation process.

Voltage and current measurements in the identification stage and in the compensated system, taken on the test bench, are shown in Figure 2.

Figure 2.

Measurements of current vs. voltage before and after compensation of inverter distortions.

The determined parameter values in the identification process are: RsIC=1.32Ω R_s^{{\rm{IC}}} = 1.32\Omega , CoutIC=1.47 nF C_{{\rm{out}}}^{{\rm{IC}}} = 1.47\;{\rm{nF}} and TDTIC=3.91μs T_{{\rm{DT}}}^{{\rm{IC}}} = 3.91\mu {\rm{s}} , where superscript IC stands for inverter commissioning. Figure 2 shows an example point (i_A,u_A), for which the resistance value is RsiA,uA=uAiA=5.4V4A=1.35Ω R_s^{{i_{A,}}{u_A}} = {{{u_A}} \over {{i_A}}} = {{5.4V} \over {4A}} = 1.35\Omega . This value is comparable to the identified resistance value R_s at the identification stage RsIC=1.32Ω \left( {R_s^{{\rm{IC}}} = 1.32\Omega } \right) , which means that the reference voltage from the control structure needed to force the set current value is only the voltage drop across the resistance. The estimator models do not take into account the inverter, ignoring voltage drops across power electronic devices, so any non-linearities in the real system will degrade the performance of the estimator. Inaccurate estimation of dead time will manifest itself in non-linear current-voltage characteristics, reducing the performance of the estimator. This confirms the validity and effectiveness of the inverter dead-time compensation method used in the context of developing an accurate estimator of the stator currents of the motor under study.

Estimator models require speed measurements in addition to voltage measurements. During the course of the work, it turned out that the use of any type of speed filter reduces the quality of the estimator’s performance. Regardless of how the speed is determined for the control structure and how it is filtered, the estimator model requires knowledge of the exact speed value, which does not show any phase shift. It is therefore proposed to determine the speed as a derivative of the position calculated using numerical methods. In this case, Savitzky–Golay filter was used (Savitzky and Golay, 1964). The method involves a local approximation of m consecutive signal samples with an nth-degree polynomial using the least squares method, followed by analytical determination of the derivative value at the centre point of the window. The following method with a sampling time Δt was used in the conducted research, which uses a quadratic polynomial (n = 2) and m = 7 samples: (13) dθkdt=ωmk≈128Δt−3θk−3−2θk−2−θk−1+θk+1+2θk+2+3θk+3 {{d{\theta ^k}} \over {dt}} = \omega _m^k \approx {1 \over {28\Delta t}}\left[ { - 3{\theta ^{k - 3}} - 2{\theta ^{k - 2}} - {\theta ^{k - 1}} + {\theta ^{k + 1}} + 2{\theta ^{k + 2}} + 3{\theta ^{k + 3}}} \right]

The formula above uses only six samples because the corresponding coefficient for θ^k at the centre point of the window is equal to zero. In calculations related to estimators, instead of the values of velocity ωmk−1 \omega _m^{k - 1} and ωek−1 \omega _e^{k - 1} in formulas (5), (8) and (11), a derivative calculated for a sample delayed by three computational steps was used, thus compensating for the unknown future measurements. Such a small delay is of little significance given the considerable inertia of the system. However, in the case of a very long measurement acquisition period, this should be taken into account.

It should also be mentioned that the Savitzky–Golay filter can be designed for a time window length other than the one presented, and the point for which the derivative is calculated does not have to be in the centre of the time window, as is the case here. Shifting this point allows only historical time samples to be used. However, this approach amplifies measurement noise in relation to the method presented above, which results directly from the interpretation of the differentiating operator Eq. (13) as an FIR filter (Hamming, 1989).

In the equation of the mathematical model of the VCS estimator appear the values of resistance R_s and inductance L_dq, as well as the value of the permanent magnets’ flux Ψ_f. The purpose of the ongoing research is to develop an algorithm that will provide an accurate reproduction of the current values. For this reason, it was decided to select the parameters of the mathematical model of the estimator through an optimisation process. Measurements were carried out on a test bench with a PMSM motor for 25 different operating points (10%, 25%, 50%, 75% and 100% of the rated speed and 0%, 25%, 50%, 75% and 100% of the rated torque). The parameters R_s, L_dq and Ψ_f of the VCS model were then determined by minimising the objective function E using the PSO algorithm. One of the publications on the use of PSO for estimating PMSM motor parameters can be found in Liu et al. (2008). More recent publications with numerous improvements also confirm the effectiveness of the PSO method (Liu et al., 2018). The objective function was defined as the mean squared error between the currents recorded on the test bench and the estimated currents: (14) E=1n∑k=1ni^dk−idk2+i^qk−iqk2 E = {1 \over n}\sum\limits_{k = 1}^n {\left[ {{{\left( {{{\hat i}_d}\left( k \right) - {i_d}\left( k \right)} \right)}^2} + {{\left( {{{\hat i}_q}\left( k \right) - {i_q}\left( k \right)} \right)}^2}} \right]} where n denotes the number of all recorded measurements. The calculations performed for each particle in the swarm were based on the determination of the value of the estimated current according to Eq. (4), followed by the calculation of the error E for the entire set of measurement data. Minimisation of the error E was done by updating the positions of individual particles in the swarm, which represented the set of parameters R_s, L_dq and Ψ_f. The PSO algorithm from the MATLAB environment library was used, in which the swarm size was set to 50 particles and the maximum number of iterations was set to 100. The remaining parameters were left as default. During the optimisation, the assumption of equal inductance L_d = L_q was made. The parameters obtained from the optimisation process were: RsPSO=1.38Ω R_s^{{\rm{PSO}}} = 1.38\Omega , LdPSO=LqPSO=39.7 mH L_d^{{\rm{PSO}}} = L_q^{{\rm{PSO}}} = 39.7\;{\rm{mH}} , ΨfPSO=0.3484 V⋅s \Psi _f^{{\rm{PSO}}} = 0.3484\;{\rm{V}} \cdot {\rm{s}} . The algorithm was run three times, and each time the results were identical, with the calculation time taking <4 min on a computer with an Intel i5-12500H processor. The parameters were then used in the estimation algorithm of the currents on the test bench. It should be noted that the resistance values obtained from optimisation RsPSO=1.38Ω R_s^{{\rm{PSO}}} = 1.38\Omega and from the inverter identification process RsIC=1.32Ω R_s^{{\rm{IC}}} = 1.32\Omega are roughly the same, which indicates that both processes are correct.

Determining the dependence of flux as a function of current is among the basic methods of identifying synchronous motors. The mathematical model obtained in this process, which takes into account the magnetic characteristics of the motor, is used in the design process and accurate control. Experimental methods for determining this motor model can be divided into locked-rotor and constant-speed methods. The method that makes it possible to determine the magnetic characteristics of any synchronous motor involves running the machine under test to a constant speed and then forcing currents to flow in a synchronous d–q coordinate system using a vector control structure. Based on the measurements of currents, voltages and speed, the value of the flux at the tested operating point can be determined.

The chosen method for identifying a non-linear model of a PMSM motor is to test the machine at a fixed operating point (Armando et al., 2013). In this case, the fluxes for a given current i_dq can be determined from Eq. (1) by neglecting the flux derivatives, thus obtaining: (15) Ψd=uq−RsiqωeΨq=−ud−Rsidωe \left\{ {\matrix{ {{\Psi _d} = {{{u_q} - {R_s}{i_q}} \over {{\omega _e}}}} \hfill \cr {{\Psi _q} = {{ - \left( {{u_d} - {R_s}{i_d}} \right)} \over {{\omega _e}}}} \hfill \cr } } \right.

Calculating the flux-linkages using above equation requires accurate voltage measurements and determination of resistance R_s. These conditions are met by using dead-time compensation according to the algorithm presented in subsection 3.1. of this article. However, accuracy of the resistance determination is less important in this case because procedure is performed in such a way that uncertainties of R_s are compensated. This requires recording data in motoring and braking modes of operation of the drive by forcing current vector of idref+jiqref \left( {i_d^{{\rm{ref}}} + ji_q^{{\rm{ref}}}} \right) then jdref−jiqref \left( {j_d^{{\rm{ref}}} - ji_q^{{\rm{ref}}}} \right) and again idref+jiqref \left( {i_d^{{\rm{ref}}} + ji_q^{{\rm{ref}}}} \right) . During the measurements, it was assumed that the working point grid would consist of 522 points ( idref i_d^{{\rm{ref}}} from −8 A to 0.5 A with 0.5 A step and iqref i_q^{{\rm{ref}}} from 0 A to 14 A with 0.5 step). Since average value of flux in steady-state is of interest, then all measurements must be averaged over at least one mechanical period, thus eliminating unwanted signal components. During testing, the motor maintains a constant speed which should be about 1/3 of the rated speed. Such a speed is a compromise between low iron losses and high back-electromotive force. Taking into account all these requirements the flux-linkages can be calculated with: (16) Ψd=12uq1+uq32+uq2ωeΨq=−12ud1+ud32−ud2ωe \left\{ {\matrix{ {{\Psi _d} = {{{1 \over 2}\left( {{{{u_{q1}} + {u_{q3}}} \over 2} + {u_{q2}}} \right)} \over {{\omega _e}}}} \hfill \cr {{\Psi _q} = {{ - {1 \over 2}\left( {{{{u_{d1}} + {u_{d3}}} \over 2} - {u_{d2}}} \right)} \over {{\omega _e}}}} \hfill \cr } } \right. where indexes 1, 2 and 3 correspond to motoring, braking and motoring mode, respectively. The derivatives of flux with respect to current required in the non-linear estimator model Eq. (7) can be easily determined using numerical differentiation methods and are also functions of current. The magnetic model obtained in this way (Figure 3) is stored in the controller memory using a lookup table (LUT) and used in the LUTE estimator calculations.

Figure 3.

Magnetic model of the PMSM under study used in LUTE. LUTE, lookup table-based estimator.

The flux maps obtained in the previous subsection are average values determined in a steady state. However, the actual flux contains higher harmonics of the spatial distribution of the field in the gap. Taking into account the additional dependence of the flux on the angular position of the rotor requires the use of the procedure described by Bojoi et al. (2024), which is an extension of the procedure described in the previous section. Based on measurements in a steady state, the flux value is determined for different rotor positions, making it possible to accurately reproduce the flux-current relationships characterising the tested motor. During the calculations, it was assumed that the resolution with which the fluxes were calculated was 1°, so 360° values were obtained for each operating point. As with the previous method, the determined relationships are recorded and stored in a LUT table for future use in ELUTE calculations. Figure 4 shows example flux maps for a position angle of 0°, while Figure 5 shows the waveforms of the determined values for a selected operating point depending on the position angle. The curves of the analysed quantities in Figure 5 show relatively high variability across the entire range of rotor position angles, which manifests itself in irregular surfaces in Figure 4. By computing the mean values of the waveforms presented in Figure 5 and subsequently plotting these values as a function of the operating point, the characteristics similar to the shown in Figure 3 are obtained. The variations in the mean values are less pronounced across different operating points; therefore, the resulting surfaces exhibit a smoother profile.

Figure 4.

Extended magnetic model for PMSM under study for rotor position θ = 0 deg.

Figure 5.

Changes in the flux and its derivatives depending on the rotor position for idref=0 A i_d^{ref} = 0\;{\rm{A}} and idref=4.5 A i_d^{ref} = 4.5\;{\rm{A}} .

The experimental test bench (Figure 6) consisted of a Lenze MCS14H15 PMSM-controlled motor with a power output of 2.5 kW, a rated speed of 1,500 r/min and a rated current of 6.6 A supplied from a frequency converter of custom design and a 4 kW induction motor operating as a load and supplied from a TWERD 5.5 kW industrial inverter. The PMSM motor operated according to the idea of FOC with a constant power angle, meaning that the reference current i_d was zero. The control system compensated for the inverter’s dead time according to the algorithm presented in subsection 3.1. The measurement system consisted of a set of LEM LA 55-P sensors and a resolver mechanically coupled to the PMSM motor rotor.

Figure 6.

Experimental setup. (a) PMSM motor under study, (b) load motor (induction machine), (c) frequency inverter (control of PMSM), (d) TWERD frequency inverter (control of the load) and (e) measurement system for stator currents and DC-bus voltage of the PMSM drive.

The measurements used in the studies on the optimisation of VCS parameters with PSO and the verification of the operation of all estimators were carried out for 25 different operating points (10%, 25%, 50%, 75% and 100% of the rated speed and 0%, 25%, 50%, 75% and 100% of the rated torque). Waveforms of the reference speeds and load torques are shown in Figure 7.

Figure 7.

Reference speed and torque used during research.

Two scenarios were considered in this study. In the first scenario, the control structure relied on measurements from CS, while the estimators did not influence the control. Nevertheless, the estimated values by all three methods were recorded, and this mode of operation is designated as the current sensor control loop (CSCL). In the second scenario, one of the estimators was incorporated into the control loop, and the structure used only estimated currents to determine the control signals. The measured currents, therefore, had no effect on the operation of the drive. In this case, both the current estimated by the selected estimator and the current measured by the sensors were recorded, even though the sensors had no influence on the control structure. This mode of operation is referred to as the sensorless control loop (SCL).

The example of recorded current waveforms for CSCL and SCL is shown in Figures 8 and 9. The waveforms show that the average values of the VCS for the SCL differ significantly from the measured currents. For rated speed and load, VCS and LUTE average the current value, thus not reflecting the actual waveform. The ELUTE appears to be the most accurate, as it reflects the local current dynamics.

Figure 8.

Waveforms of i_q current for different estimation methods (10% of nominal speed and 25% of nominal load). CSCL, current sensor control loop; ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; SCL, sensorless control loop; VCS, virtual current sensor.

Figure 9.

Waveforms of i_q current for different estimation methods (100% of nominal speed and 100% of nominal load). CSCL, current sensor control loop; ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; SCL, sensorless control loop; VCS, virtual current sensor.

For both scenarios (CSCL and SCL), steady states were extracted and the root mean square error (RMSE) of the estimation for every operating point was calculated according to the formula: (17) RMSE=1N∑k=1Nid,k−i^d,k2+iq,k−i^q,k2 RMSE = \sqrt {{1 \over N}\sum\limits_{k = 1}^N {\left[ {{{\left( {{i_{d,k}} - {{\hat i}_{d,k}}} \right)}^2} + {{\left( {{i_{q,k}} - {{\hat i}_{q,k}}} \right)}^2}} \right]} }

By using the above formula, the estimation error is characterised by a single factor, which further describes the accuracy of the estimation depending on the operating point of the drive. The RMSE values for all 25 operating points for CSCL and SCL are shown, respectively, in Figures 10 and 11. It can be seen that VCS has the largest error of all three estimators, while ELUTE has the smallest. The improvement in the quality of ELUTE estimation compared to LUTE is small but noticeable. For the LUTE and ELUTE estimators, an increase in RMSE is also visible with increasing drive speed. RMSE for VCS appears to be smallest for a 50% load, which will be discussed in more detail later in the article. A summary of the estimation quality for all methods is provided in Table 1, which shows the average RMSE calculated from the RMSE values for all tested drive operating points, together with the median, minimum and maximum values, range calculated as maximum minus minimum and standard deviation.

Figure 10.

RMSE across operating points of the drive for estimation methods in CSCL. CSCL, current sensor control loop; ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; RMSE, root mean square error; VCS, virtual current sensor.

Figure 11.

RMSE across operating points of the drive for estimation methods in SCL. ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; RMSE, root mean square error; SCL, sensorless control loop; VCS, virtual current sensor.

The quality of current reproduction was determined based on the correlation between measured and estimated currents in SCL (Figure 12). Based on Pearson’s correlation coefficient denoted as r, the ELUTE estimator can be selected as the best, but the difference in the coefficient for LUTE is not significant. The VCS estimator performs by far the worst, as its fixed model parameters do not work across the entire range of speeds and loads. However, referring to previous conclusions, it can be noted that in the area corresponding to 50% load, the estimated currents best fit the currents measured for VCS. This is due to the optimisation process of the parameters of this model, which were selected in such a way as to minimise the error between currents across the entire range of speeds and loads. This corresponds to the averaging effect of the model parameters and thus explains why the RMSE of the estimation is smallest at this operating point (see Figures 10 and 11).

Figure 12.

Correlations between measured and estimated current in the SCL for estimation methods under study. ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; SCL, sensorless control loop; VCS, virtual current sensor.

The impact of the selected estimator operating in the control loop was examined by determining the total harmonic distortion (THD) coefficient for the current measured by CS in phase A. The results are presented in relation to the drive operating point in Figure 13, where CSCL is the reference point of THD, and VCS, LUTE and ELUTE refer to the THD calculated for the SCL. Taking into account speeds in the range up to 75% of the rated speed, regardless of the load, the use of the ELUTE estimator was characterised by the lowest THD value among the three estimators tested. The superiority of ELUTE over other estimators is clearly visible with increasing load. Often, the THD for the SCL was comparable to that of the CSCL. Interesting observations concern increasing THD with increasing speed in the CSCL. However, the use of either estimator reduces the THD at nominal speed in the SCL compared to the CSCL.

Figure 13.

THD calculated for phase A current for CSCL and SCL with different estimation methods. CSCL, current sensor control loop; ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; SCL, sensorless control loop; THD, total harmonic distortion; VCS, virtual current sensor.

An important aspect of the estimators discussed is their applicability in the FTC system for detecting CS faults. Fault detection methods are usually based on determining the residuum between the measured and estimated currents in CSCL mode. In this case, the control structure uses the measurements from the CS in the drive system for control, and the current estimator serves as a reference model for the process. If the residuum calculated in the simplest case as the difference between the currents exceeds the set detection threshold, then the system should be switched to SCL mode. For this reason, it is necessary for the estimator to reproduce the measured currents as accurately as possible. A detailed examination of the estimation quality for changing drive operating points is illustrated in Figures 14 and 15, which shows histograms of the estimation error (residuum) calculated as: (18) error=id2+iq2−i^d2+i^q2 error = \sqrt {\left( {i_d^2 + i_q^2} \right)} - \sqrt {\left( {\hat i_d^2 + \hat i_q^2} \right)}

Figure 14.

Histograms of the estimation error for CSCL for different PMSM operating points. CSCL, current sensor control loop; ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; VCS, virtual current sensor.

Figure 15.

Histograms of the estimation error for SCL for different PMSM operating points. ELUTE, extended lookup table-based estimator; LUTE, lookup table-based estimator; SCL, sensorless control loop; VCS, virtual current sensor.

In the case of the ELUTE estimator, the data spread is the smallest regardless of the operating point, which suggests that the estimation accuracy is therefore the best among the estimators tested. Based on the histograms shown in Figure 14, it can be seen that when using VCS, it is very difficult to define the detection threshold because, depending on the operating point of the drive, the average error is not zero. The use of LUTE or ELUTE estimators in the context of fault detection would be possible, but the adaptive detection threshold would be preferable. This conclusion is a future research direction that should be investigated.