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The proposed book will focus on the controls of induction machines aimed at exploiting them in the field of speed variation (electric vehicles, aeronautics, etc.). In this work, the authors utilize cascaded multi-level converters to reduce the harmonic distortion rate of the voltages at the machine’s input. The converter is controlled by pulse width modulation (PWM) technique. The use of active disturbance rejection control (ADRC) will contribute to eliminating internal disturbances (undesired variations in machine resistances) as well as minimizing external effects (inertia, resisting torque). State observers are employed to estimate mechanical speed, allowing the elimination of the speed sensor in the system. The mathematical models obtained have been simulated using MATLAB/Simulink software. Several scenarios have been tested through simulation, including comparing ADRC and PI control, and the effect of varying atmospheric conditions on PV power supply and consequently on the machine.
Systems and Applications Engineering Laboratory (LISA), University Cadi Ayyad University, National School of Applied Sciences (ENSA), Marrakesh, Morocco
*Address all correspondence to: abdououkassi16@gmail.com
1. Introduction
In recent years, the integration of renewable energy sources with advanced control techniques has sparked growing interest in the field of electrical systems and industrial applications. Such an integrated system involves the combination of an induction machine, indirect vector control, power supply from photovoltaic (PV) panels, cascaded multilevel converters, and closed-loop control using either active disturbance rejection control (ADRC) or proportional-integral (PI) control. This integrated system represents a sophisticated solution aimed at effectively harnessing renewable energy while ensuring reliable and precise control of the induction machine’s operation. By combining intermittent energy generation from PV panels with the dynamic performance of the induction machine, this system provides a sustainable energy solution suitable for various applications, ranging from water pumping to industrial processes.
Induction machines, also known as asynchronous motors, are widely used in variable-speed drives due to their excellent performance and reliability. Here are some key performances of the induction motors used in variable speed drives [1]. They operate stably over a wide load range and exhibit good tolerance to load variations, ensuring reliable operation in various conditions. They allow continuous speed adjustment over a wide range, making them suitable for a variety of applications, from fans to industrial conveyors. Compared to other types of motors, induction machines are often less expensive to manufacture and maintain, making them an attractive choice for many applications. Currently, vector control (direct or indirect) based on a mathematical model of the machine is widely used to achieve the previously mentioned performances [2, 3].
The use of renewable energies, especially solar energy, in electromechanical systems has grown considerably. Indeed, the performance of photovoltaic (PV) sources depends largely on the ability to maximize the capture and use of available solar energy. A crucial method to achieve this is the use of maximum power point tracking (MPPT) control [4]. MPPT systems optimize the performance of solar panels by continuously adjusting their operation to extract maximum energy possible, despite variations in environmental conditions such as sunlight and temperature. In the future, the focus will be on the use of photovoltaic sources to power static energy converters. These converters can be integrated into various real systems, such as electric cars, railways, and aerospace applications. Here, the focus is on the development of a variable-speed drive based on an induction machine and powered by photovoltaic sources.
Multilevel converters are increasingly used in electric vehicles due to their advantages in terms of energy efficiency and performance. They facilitate efficient conversion of DC power from PV panels into high-quality AC power, minimizing losses and harmonic distortions. Due to their design, multilevel converters can reduce electromagnetic interference [5], thereby improving the reliability and durability of electronic systems in electric vehicles. Their use in this context enables better management of electrical energy conversion, resulting in improved overall efficiency of the electric propulsion system in vehicles. Additionally, their use offers better control capability of electrical power, allowing for more precise regulation of the speed and torque of the electric motor.
The use of indirect vector control allows for precise control of the speed and torque of the induction machine, ensuring optimal performance in variable operating conditions. This control method is based on a cascade regulation loop that regulates both stator current and magnetic flux [6]. By adjusting these parameters, indirect vector control can accurately control the speed and torque of the induction machine, enabling fast dynamic response and excellent regulation accuracy. The use of indirect vector control offers numerous advantages, including improved energy efficiency, rapid response to load changes, and reduced mechanical stress on the machine. The parameters of the vector control will serve as reference signals, which will then drive the PWM control circuit of the multi-level converter.
Furthermore, the closed-loop control strategy, implemented either with ADRC or PI control, adds an extra layer of stability and robustness to the system. ADRC offers active disturbance rejection capabilities, allowing the system to adapt to changing conditions and effectively reject external disturbances [7]. This maintains high control performance even in the presence of external disturbances or load variations. Unlike some other advanced control methods, ADRC does not require a precise model of the system, simplifying its implementation in real-world applications where modeling may be difficult. Due to its ability to actively reject disturbances, ADRC can offer better reference tracking and disturbance rejection performance compared to PID control in some cases. On the other hand, PI control, a well-established method, offers simplicity and reliability, making it suitable for applications where precise modeling may be difficult [8, 9].
The use of state observers to estimate the mechanical speed of an induction machine is a crucial area in the control and monitoring of electromechanical systems. However, their mechanical speed, which is often a vital parameter for precise system control and monitoring, is not directly measurable. This is where state observers come into play. These sophisticated algorithms leverage available measurements, such as electrical currents and voltages, to estimate internal system variables like mechanical speed, which are not directly accessible. State observers are particularly useful in applications where dedicated sensors for certain variables are expensive or difficult to install. In this context, the Leunberger observer is an effective and robust method for estimating the mechanical speed of induction machines by utilizing available electrical measurements and combining them with a dynamic model of the system [10]. This approach plays a crucial role in the precise control and monitoring of these critical electromechanical systems in various industrial applications. The Leunberger observer offers several advantages, including its robustness against disturbances and model uncertainties, as well as its ability to operate effectively even under nonlinear operating conditions [11]. Additionally, it is relatively simple to implement and does not require detailed knowledge of the distribution of measurement errors, making it attractive for many industrial applications.
Asynchronous machines, or induction machines, are among the most widespread and essential electromechanical devices in modern industry. Electrically, an asynchronous machine can be represented by a set of equations delineating the interactions among magnetic fields, currents, and voltages. And from an electromechanical perspective, the equations describe the relationship between electrical and mechanical quantities, such as electromagnetic torque and rotational speed. Here are the fundamental equations used in the mathematical model of an asynchronous machine in the form of a state-space model (Eq. (1)).
The coefficients that may be affected by the variation of the machine parameters (stator resistance and rotor resistance) are: a11, a13, a42, a44. The matrixA'can be written as (Eq. (2)).
A'=A+ΔAE2
With: ΔA=Δa110Δa1300Δa220Δa24Δa310Δa3300Δa420Δa44Δa11=Δa22=−KrsTsσ+Krr1−σTrσ, Δa13=Δa24=Krr1−σTrMσ, Δa31=Δa42=KrrMTrΔa31=Δa42=KrrMTr, Δa33=Δa44=Krr−1Tr, 0≤Krs%≤100, 0≤Krr%≤100. Krr and Krs are the coefficients of resistance variation.
Photovoltaic sources are widely used in various applications, including residential, commercial, and industrial electricity production, standalone power generation systems, large-scale solar power plants, etc. In terms of equations, mathematical models of photovoltaic sources can be based on physical laws such as the current-voltage characteristic equation, equations describing temperature-dependent behavior, and equations describing efficiency variation with solar irradiance [12]. These equations allow predicting the electricity production of photovoltaic sources under different environmental and operational conditions. In our case, these sources are used to power the multi-level converter. The fundamental equations used to model a photovoltaic cell include:
The current generated by the panel can be given as (Eq. (3)):
I=Iph−ID−IPE3
The relationship between I and V is given by Eq. (4):
I=Iph−I0expV+RsIa.Vt−1−V+R.sIRpE4
The Iph is a grandeur that gives information on the spectrum of the photovoltaic cell. Practically, its evolution is related to ambient temperature and irradiation G. Its formula is Eq. (5):
Iph=GGrIscr−KiT−TrE5
Tr is the reference temperature, Iscr is the cell’s short circuit current at Tr, ki is the temperature coefficient of the short circuit, and Gr is the nominal irradiation. The diode saturation current I0 depending on temperature may be expressed by Eq. (6):
I0=IorTTr3.expqEgAk1Tr−1TE6
Where Ior is the nominal saturation current, and Eg is the band gap energy. Usually, a PV installation is broken down into several cells (series or parallel). Before operating a cell, it is necessary to know its physical parameters (e.g., Table 1 illustrates the characteristics of a panel).
Multilevel converters have emerged as a crucial technology in modern power electronics, offering advantages such as higher voltage levels, reduced harmonic distortion, and improved efficiency compared to traditional two-level converters. Several types of multilevel converters exist, each with its own unique configuration and characteristics. One common type is the neutral-point clamped (NPC) converter [13], which features capacitors connected between the neutral point and each phase to achieve multiple voltage levels. Another type is the flying capacitor (FC) converter [14], which utilizes a set of capacitors connected across the semiconductor switches to generate the desired voltage levels. Additionally, the cascaded H-bridge (CHB) converter is widely used in high-power applications, employing multiple H-bridge cells stacked in series to achieve multiple voltage levels. Other variants include the modular multilevel converter (MMC), which utilizes submodules interconnected in a series-parallel configuration, offering scalability and fault tolerance. The cascaded H-bridge (CHB) inverter topology consists of multiple H-bridge inverter modules connected in series [15]. This configuration enables the inverter to generate a higher number of voltage levels, thus improving the quality of the output waveform. In this configuration (Figure 1), the converter consists of 12 single-phase H-bridge modules arranged to power a three-phase induction machine. Each phase of the induction machine is powered by 4 single-phase H-bridge modules. Each H-bridge module can switch between three voltage states: Vdc, 0, and −Vdc.
Figure 1.
Converter structure (nine CHBs) for a single phase.
4.2 PWM control
The basic principle of the multi-carrier MCPWM is the use of multiple triangular carrier signals of different amplitudes or phases, which are then compared to a single sine waveform to generate the control pulses. The best known MCPWM classes are: phase disposition (PD), phase opposed disposition (POD), and alternative phase opposed disposition (APOD). In the technique (PD), all carriers are in phases. In the POD technique, carriers are all in phase both above and below the zero-reference value. Nonetheless, there exists a 180-degree phase shift between those above and below zero. Conversely, in the technique (APOD), carriers are alternately two-phase by 180 degrees in relation to each other [16]. A modified PWM technique to improve the total harmonic distortion of the multilevel inverter is used [17]. Table 1 shows the algorithmic organization of the proposed PWM control to drive the switches of each bridge. The advantage of this structure is to eliminate the logic gates found in other structures. Table 1 clearly indicates each phase: the parameters of the sinusoidal voltage source (peak and frequency), the parameters of each triangular wave (frequency of each triangular wave and amplitude variation interval), and the four bridges (switches used in each bridge). The proposed algorithm allows that for each bridge:
A sine wave is compared (Comp sup) to a triangular wave having a positive level. If the sine wave is greater than the positive triangular wave, then pulses are sent to two opposite switches (e.g., S1 and S4).
The same sine wave is compared (Comp inf) to a negative triangular wave having the same level as the positive triangular wave previously used. If the sine wave is less than the negative triangular wave, pulses are sent to the remaining two switches (e.g., S2 and S3). Figure 2 shows the logic control circuit proposed in Table 2.
Figure 2.
PWM control electronic circuit to generate pulses for 16 IGBT.
Sine wave
Triangular wavef = 2 KHz
Comparison between sine wave and triangular waves
H-Bridge
H-Bridge switches
Level
Volts
Comp-sup
Comp-inf
Peak 1 Volts f = 50 Hz
+1
[0 0.25]
Sin wave > level + 1
1
(S1-S4)
−1
[-0.25 0]
Sin wave < level-1
1
(S2-S3)
+2
[0.25 0.5]
Sin wave > level + 2
2
(S5-S8)
−2
[−0.5–.25]
Sin wave < level-2
2
(S6-S7)
+3
[0.5 0.75]
Sin wave > level + 3
3
(S9-S12)
−3
[−0.75–.5]
Sin wave < level-3
3
(S10-S11)
+4
[0.75 1]
Sin wave > level + 4
4
(S13-S16)
−4
[−1–0.75]
Sin wave < level-4
4
(S14-S15)
Table 2.
Organization of the PWM control logic for a 9-level converter.
Indirect field-oriented control (IFOC) is a sophisticated control approach widely employed in the field of electric drives, particularly for AC induction motors and permanent magnet synchronous motors (PMSMs). This method is designed to achieve precise control of motor speed and torque while minimizing losses and improving efficiency. At its core, IFOC aims to decouple the motor’s electromagnetic dynamics, separating the torque and flux components [1]. By utilizing the relationships between magnetic fluxes and currents in the rotor windings (Eqs. (7) and (8)):
0=Rrird+dφrddt−ωgφrqE7
0=Rrird+dφrqdt+ωgφrdE8
For decoupling control, we have φrq=0, φrd=φr. The expressions of the sliding pulsation (Eq. (9)) and the torque (Eq. (10)) are:
ω̂g=isqiφLrR̂rE9
Ce=pM2LriφisqE10
The mechanical speed estimated as a function of electromagnetic torque and load torque is given by the following expression (Eq. (11)).
JdΩ̂rdt=−KfΩ̂r+Ce−CrE11
An additional set of expressions, utilizing the concept of decoupling within the framework of IFOC, are presented by Eq. (12). These equations describe the relationships between voltages and currents in the stator windings of the machine.
ADRC stands for active disturbance rejection control, which is an advanced control strategy used in various engineering applications, particularly in the field of control systems. ADRC is designed to actively reject disturbances and uncertainties in the system by employing an extended state observer (ESO) to estimate and compensate for these disturbances in real time. ADRC is the result of proportional-integral derivation (PID), the key to its success, and this is a control based on errors rather than on a model [18]. Figure 3 provides a visual representation of how these components are interconnected and how information flows within the ADRC system, illustrating the overall architecture and functionality of the control system.
Figure 3.
Interconnection between components of the ADRC system.
This Equation (Eq. (13)) describes how the output y of the system evolves over time based on its internal dynamics (y, ε, t) and the input signal u, with b0 representing the scaling factor or sensitivity of the system to the input [19].
y=fyεt+b0uE13
Using the state-space form, equation (Eq. (13)) can be represented as follows (Eq. (14)):
y=10T.xẋ=0100.x+10.b0u+01.ḟE14
The extended state observer (ESO) is commonly represented by an equation that estimates the internal state of the system as well as the disturbances affecting the system dynamics. A generic form of the ESO equation could be (Eq. (15)).
ŷ=10T.zż=0100.z+10.b0u+2ω0ω02.y−ŷE15
with ω0 the observer bandwidth, determined by pole placement, in order to provide a fast, high-quality output [20]. According to ADRC, a law of control is then determined by (Eq. (16)):
u=u0−z2b0E16
With: z2 being a correct estimation of “f” and z1 a correct estimation of “y”.
The Luenberger observer is a crucial tool in control theory used to estimate the state of a dynamic system based on its measured outputs. The classical formulation of the Luenberger observer is given by the following equation (Eq. (21)). In your context, it is indicated that the coefficients of the matrix D are chosen so that the observed model is faster than the real model. This means that the eigenvalues of the matrix A−DC should be chosen such that they are larger in absolute value than those of the matrix A [21, 22].
ŷ=A.x̂x̂̇=B.x̂+Cu+D.y−ŷE21
The real physical model of the machine is identical to the one found in Section 2. It simply requires linking the stator reference frame to the (α, β) reference frame. The state vectors, the different matrices of the machine, and the coefficients of the observation matrix are as follows:
Lyapunov function for mechanical speed systems can be conceptualized to analyze the stability of the speed dynamics of the system. To estimate a mechanical speed (Eq. (22)), the Lyapunov function is utilized [23].
V=x−x̂x−x̂T+ωr−ωr̂2ρE22
With: ρ being a positive constant coefficient. x−x̂ is the estimation error vector of the induction motor and LO. The derivative of V in time, which should be negative, is defined by the following equation (Eq. (23)):
The expression B−ADT+B−ADx−x̂Tx−x̂ is always negative. Both last terms may be set to zero since they are insignificant in comparison with the first (Eq. (25)).
Many scientists have suggested a PI adaptation mechanism for improving estimation accuracy. It gives the following expression (Eq. (27)) for the estimated speed:
The proposed (ADRC-IFOC- PI) diagram for the sensorless induction motor fed by HCBI is depicted in Figure 4. This system comprises interconnected subsystems aimed at maximizing performance. In the power supply segment, photovoltaic generators supply energy to a 9-level inverter via a boost converter. The multilevel inverter’s output voltages are directly applied to the induction machine. A Luenberger observer is employed to estimate mechanical speed in the absence of sensors. Leveraging this observer, the estimated speed can be compared to a reference using a control law based on either the ADRC or PI approach. By integrating the indirect field-oriented control (IFOC) method with these two approaches, the system generates requisite signals for driving PWM control vectors, thus enhancing overall system efficiency and performance.
Figure 4.
Optimized operation of sensorless induction machine via nine-level converter with ADRC and PI controls.
The characteristics of the induction machine are illustrated in Table 3. Two scenarios were considered. The simulation tests conducted in the first scenario consist of not varying the parameters of the machine (stator and rotor resistances, moment of inertia). The second scenario involves considering these variations at different times.
Parameters
Value
Power
1kw
Voltage
220/380 V
Current
4.58A/2.56A
Stator resistance
6,8 Ω
Rotor resistance Rr
5,43 Ω
Stator inductance Ls
0,3973 H
Rotor inductance Lr
0,3558 H
Mutual inductance M
0,39 H
Number of pole pairs
2
Viscous friction coefficient Kf
0,0025 Nm.s/rad
Rated speed n
1440Tr/min
Rotor time constant Tr
0.0655 s
Stator time constant Ts
0.0584 s
Table 3.
Parameters of induction machine.
9.1 Scenario 1: variation of the speed reference and load torque
The PV module is tested taking into account the values of the different given parameters and according to the conditions specified in Table 1(T = 25 degrees, I = 1000 W/m2). The output voltage is illustrated in Figure 5 (between 0 and 8 s). The PV module is tested taking into account the values of the different given parameters and according to the conditions specified in Table 1(T = 25, degrees I = 1000 W/m2). The output voltage is illustrated in Figure 5 (between 0 and 8 s).
Figure 5.
Output voltage (Pv source).
In this part, the simulation is performed by varying only the speed and load torque. In the interval [0, 5 s], the speed is equal to 90 rad/s, and in the interval [5 s, 8 s], the speed is equal to 140 rad/s. At t = 2 s, the torque goes from 0 to 5 Nm, and at t = 6 s, the torque goes from 5 Nm to 0 Nm. Figure 6 shows the evolution of the three voltages obtained at the output of the nine-level converter, with a zoom on the interval [0.4 s, 0.5 s] illustrated in Figure 7. The stator currents are shown in Figure 8, and a zoom on the interval [0.4 s, 0.5 s] is provided in Figure 9. Figure 10 presents the evolution of the mechanical speed in both cases (ADRC, PI), clearly demonstrating the faster response time and better performance of the ADRC control compared to the PI control.
Figure 6.
Waveforms of line-to-line voltages of three phases (a,b,c).
Figure 7.
Waveforms of line-to-line voltages loop (0.4 sec–0.46 sec).
Figure 8.
Waveforms of stator currents.
Figure 9.
Waveforms of stator currents loop (0.4 sec–0.46 sec).
Figure 10.
Speed and reference speed.
Figure 11 shows the evolution of the electromagnetic torque in both scenarios (ADRC, PI). Additionally, the torque obtained with ADRC control exhibits sudden variations after a change in load torque, unlike the torque obtained with PI control.
Figure 11.
Torque, load torque, and reference torque.
To verify the validity of the state observer, rotor fluxes were also simulated. Figure 12 illustrates the evolution of the two components of the flux along the α and β axes, and a zoom on the interval [0.4 s, 0.5 s] is provided in Figure 13.
Figure 12.
Components of the flux along the α and β axes.
Figure 13.
Components of the flux along the α and β axes loop (0.4 sec – 0.46 sec).
9.2 Scenario 2: consideration of parameter variations (resistances and moment of inertia)
In the second part, the robustness of the proposed system is examined following variations in internal parameters (machine resistances) and external parameters (moment of inertia). The simulation tests were conducted with the following variations: For the machine, the rotor resistance was increased by 50% at t = 3.5 s, and the moment of inertia was also increased by 50% at t = 5 s. Figure 14 illustrates the evolution of the mechanical speed (actual and estimated) in both cases (ADRC, PI), clearly indicating the faster response time and superior performance of ADRC control compared to PI control. Similarly, the error between the actual speed and the estimated speed in the ADRC system is lower than in the PI system. Figure 15 shows the evolution of the electromagnetic torque in both cases (ADRC, PI). Additionally, the torque obtained with ADRC control exhibits lower interference than that obtained with PI control.
Figure 14.
Speed and reference speed.
Figure 15.
Torque, load torque, and reference torque.
The simulated results at the end of this work involve varying the parameters of the PV sources (at t = 0, we take T = 35 degrees and G = 1100 W/m2). We still retain the variations of the machine parameters mentioned above. Figure 16 depicts the profile of the voltage at the output of the PV module. A slight difference can be noticed compared to the one obtained in Figure 5. Figure 17 shows the evolution of the actual and estimated speed in the case of ADRC control. One can clearly observe the robustness of this control as it rejects all disturbances.
The simulated prototype allows the variation of the mechanical speed of an induction machine without the use of a speed sensor. This prototype can be implemented in an electric vehicle. The combination of the 9-level converter, PWM, ADRC, PI, and Luenberger observer has helped reduce signal harmonics and electromagnetic torque ripples. The system has also been designed to reject the effects of internal and external disturbances. Operational principles and mathematical models (PV source, induction motor, ADRC control, PI control, indirect field-oriented control, and state observers) were analyzed in this study to ensure a good stability of the machine speed and a response time adapted to the evolution of this mechanical quantity. The results show that the system with ADRC is superior and more robust than the system with PI in various scenarios and disturbance conditions.
Conflict of interest
The authors declare no conflict of interest.
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Written By
Abdellah Oukassi
Submitted: 23 May 2024Reviewed: 26 June 2024Published: 04 June 2025
Open access peer-reviewed chapter
