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Abstract

Oscillation of torque and speed occurs in the electric powertrain based on permanent magnet synchronous motor under field-oriented control when we set an unreasonable proportional control parameter of proportional–integral regulator. Thus, it influences the stability and reliability of electric powertrain. The objectives of this article are to study nonlinear dynamics of electric powertrain under various complex operating conditions and settle a minimum stable range of proportional control parameter of proportional–integral regulator. To achieve these goals, nonlinear dynamic model of electric powertrain was established. Then, we solved equilibrium points and analyzed the stability of equilibrium points. Finally, we set different control parameters of proportional–integral regulator and various complex working conditions of electric powertrain to simulate nonlinear dynamics of electric powertrain. The simulation results show the electric powertrain operates stably when the control parameter is set in the area where there is only one stable equilibrium point. Chaos do exist in the electric powertrain with field-oriented control under different working conditions. Our analysis reveals the dynamics of electric powertrain are dependent on proportional control parameters of proportional–integral regulator and electric powertrain performs unstably most likely under the operating condition with no-load and zero reference rotational speed.

Keywords

Electric powertrain permanent magnet synchronous motor field-oriented control proportional–integral regulator nonlinear dynamics

Introduction

In recent years, electrification has become a trend due to the depletion of fossil energy and the pressure of environmental protection. In the field of power transmission system, more and more attention has been paid to electric powertrain. Permanent magnet synchronous motor (PMSM) has been applied increasingly in electric powertrain for its simple structure, high efficiency and large power density. In many cases, FOC strategy is applied to regulate PMSM and proportional–integral (PI) regulator is used to control the speed and current of PMSM. The output voltage of PI controller is proportional to the input error between speed of PMSM and reference speed. However, instable behaviors such as oscillation of torque and speed, irregular current noise and so on occur in electric powertrain when the proportional control parameter is set unreasonably.

Chaotic motion, such as strong oscillation of speed and torque, influences the stable operation of PMSM. Li et al.¹ studied steady-state and dynamic characteristics of PMSM when subject to constant input voltage and constant external torque. Conditions were derived under which the dynamic characteristics of the system are either of steady-state type, limit cycles, or chaotic. Huang and Wu² analyzed the basic dynamics of PMSM by applying nonlinear dynamic theory such as Lyapunov exponents, bifurcation diagram, and phase diagram, and the mathematical model of PMSM has been confirmed to exhibit prominent chaotic characteristics under certain parameters and working conditions.³ Jing et al.⁴ analyzed the dynamics of PMSM extended to the PMSM model with a non-smooth-air-gap and studied the stability, the number of equilibrium points, and the pitchfork and Hopf bifurcations. Analytical stability boundary were given by solving the local quadratic approximation of the two-dimensional (2D) stable manifold at an second-order saddle point.

Owing to chaos of PMSM under certain parameters or operating conditions, it is essential to study the method of suppressing chaos in electric powertrain based on PMSM. FOC is one of the widely used control methods. Considering that chaos of PMSM threatens the stability and reliability of electric powertrain, Huang and Xiao⁵ applied a proportionally differential method to eliminate chaos in electric powertrain. The control method only controls a part of the subsystem. The results show that the method is insensitive to system parameters and has fast control speed. Zhang et al.⁶ proposed a speed tracking control method for PMSM based on the nonlinear dynamics of PMSM. He designed a controller by adding an auxiliary state variable in the nonlinear dynamic model of PMSM and used the proportional integral control principle. The simulation results show that PI controller can realize the asymptotic tracking to the speed reference. Souhail et al.⁷ studied the coexistence and bifurcation of dynamic equilibrium points of PMSM. The synchronization of chaos was dealt with using a PI controller. Wahid et al.⁸ described certain dynamic responses of the PMSM under FOC with variation of a parameter. Equilibrium point coordinates were solved from the differential system equations and their stability can be showed from characteristic polynomial solutions. The multi-stability property was put into evidence by the identification of several attractors for the same motor. Control parameters were set differently for different initial conditions. Although we use PI controller to suppress instability of electric powertrain, electric powertrain exhibits chaotic characteristics when we set an unreasonable proportional control parameter of PI regulator. So it is essential to settle a stable range of control parameter for electric powertrain under various complex working conditions.

This article studied different nonlinear dynamics of electric powertrain with the variable proportional control parameter of PI regulator and determined a minimum stable range of proportional control parameter under various complex operating conditions. In this article, nonlinear dynamic model of electric powertrain based on PMSM under FOC was built. The model was transformed into a dimensionless one through coordinate and time-scale transformation. We applied nonlinear dynamics theory to study the bifurcation and chaos. The equilibrium points were solved and the stability of equilibrium points was analyzed. Hopf bifurcation under the variation of proportional control parameter leads to instable oscillation in electric powertrain. The stable regions under four typical operating conditions can be used for design and control purposes. The main contributions of this article are listed as follows:

The nonlinear dynamic model of electric powertrain based on PMSM under FOC was first established. We concentrate on the influence of control parameter on the nonlinear dynamic behaviors of electric powertrain, and the nonlinear dynamics theory was applied to determine the stability domain of electric powertrain under various operating conditions.

The nonlinear dynamics of electric powertrain with FOC under various operating conditions were analyzed to reveal the relationship between control parameter of PI regulator and stability of electric powertrain. A minimum stable range of proportional control parameter was settled for the stable operation of electric powertrain.

Nonlinear dynamic model of electric powertrain

PMSM

The d-axis inductance equals with q-axis inductance^9–11 for the surface-mounted PMSM. The nonlinear dynamic equations of PMSM are given using $i_{d}$ , $i_{q}$ , and $ω$ as variables and the model in the synchronously revolving $dq$ frame¹² is given as below

(\begin{array}{l} \frac{d i_{d}}{d t} = (u_{d} - R i_{d} + L i_{q} ω) / L \\ \frac{d i_{q}}{d t} = (u_{q} - R i_{q} - L i_{d} ω + ω ψ_{f d}) / L \\ \frac{d ω}{d t} = [n_{p} ψ_{f d} i_{q} - T_{L} - β ω] / J \end{array}

(1)

where $u_{d}$ and $u_{q}$ represent direct-axis and quadrature-axis voltages, respectively; direct-axis and quadrature-axis currents are represented by $i_{d}$ and $i_{q}$ ; $n_{p}$ represents pole pairs of PMSM; $L$ represents d-axis or q-axis inductance; $R$ is the stator resistance; permanent magnetic flux represented by $ψ_{fd}$ ; the load torque $T_{L}$ is another system input; and the mechanical parameters of the model are inertia $J$ and friction coefficient $β$ .

PMSM controller

The d-axis and q-axis stator current of PMSM can be controlled by $u_{d}$ and $u_{q}$ , respectively, for the usage of voltage source inverter.^13,14 The constant value of $u_{d}$ can be deduced by substituting $d i_{d} / dt = 0$ into equation (1), and $u_{q}$ is controlled by PI controller.^15,16 So, the control strategy can be set as follows

{\begin{matrix} u_{d} = R i_{d} + ω i_{q} L_{q} \\ u_{q} = (k_{p} + \frac{k_{i}}{s}) (ω_{r} - ω_{ref}) \end{matrix}

(2)

where $k_{p}$ represents proportional control parameter, $k_{i}$ represents integral control parameter, $ω_{r}$ is the rotational speed of rotor, and $ω_{ref}$ represents the reference speed.

Nonlinear dynamic model of electric powertrain

A schematic representation of electric powertrain is shown in Figure 1. PMSM and mechanical load are connected by shaft. Motor speed is compared with reference speed $ω_{ref}$ to generate a reference current $i_{q}^{*}$ by an outer loop. Three phase currents— $i_{a}$ , $i_{b}$ , and $i_{c}$ —are measured by current sensor. Clark transformation and park transformation are applied to transform $abc$ coordinate system to $dp$ rotating coordinates. The voltages— $u_{d}^{*}$ and $u_{q}^{*}$ —are adjusted by PI regulator. The output voltages are transformed into $α β$ coordinate system using the inverse park transformation, and the voltage $u_{α}^{*}$ and $u_{β}^{*}$ are applied to drive the motor by an inverter using space vector pulse width modulation (SVPWM) technique.

Figure 1.

Schematic structure of the electric powertrain.

The proportional control parameter of PI regulator reflects the deviation signal of the control system. The large proportional control parameter can speed up the adjustment. However, when it increases to a certain degree, the electric powertrain becomes unstable. So, we set the integral control parameter $k_{i} = 0$ to study the different nonlinear behaviors of electric powertrain under the variation of proportional control parameter. Moreover, the value $u_{d}$ is set to $u_{d} = 0$ for the sake of simplicity. The nonlinear dynamic model of electric powertrain under FOC is shown as follows

{\begin{matrix} \frac{d i_{d}}{dt} = (- R i_{d} + L i_{q} ω) / L \\ \frac{d i_{q}}{dt} = (k_{p} (ω - ω_{ref}) - R i_{q} + ω ψ_{fd}) / L \\ \frac{d ω}{dt} = [n_{p} ψ_{fd} i_{q} - T_{L} - β ω] / J \end{matrix}

(3)

The dynamic model of electric powertrain under FOC is similar to the famous Lorenz equations. So, we can use nonlinear affine transformation and time-scale transformation to rewrite the dynamic model. The nonlinear affine transformation equation^17,18 is given below

x = λ \tilde{x}

(4)

where $x = [i_{d} i_{q} ω]^{T}$ , $\tilde{x} = \tilde{i_{d}} \tilde{i_{q}} {\tilde{ω}}^{T}$ , $λ = {\begin{matrix} bk & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & \frac{1}{τ} \end{matrix}}$ , $b = L_{q} / L_{d} = 1$ , and $k = β / (n_{p} τ ψ_{fd})$ .

The time-scale transformation equation¹⁹ is shown

t = τ \tilde{t}

(5)

where $τ = L_{q} / R$ . So

\frac{dx}{dt} = \frac{λ}{τ} \frac{d \tilde{x}}{d \tilde{t}}

(6)

The nonlinear dynamic model of electric powertrain under FOC after nonlinear affine and time-scale transformation is expressed as follows

{\begin{matrix} \frac{d \tilde{i_{d}}}{d \tilde{t}} = - \tilde{i_{d}} + \tilde{ω} \tilde{i_{q}} \\ \frac{d \tilde{i_{q}}}{d \tilde{t}} = - \tilde{i_{q}} - \tilde{i_{d}} \tilde{ω} + γ \tilde{ω} + c (\tilde{ω} - {\tilde{ω}}_{ref}) \\ \frac{d \tilde{ω}}{d \tilde{t}} = σ (\tilde{i_{q}} - \tilde{ω}) - {\tilde{T}}_{L} \end{matrix}

(7)

where $γ = - ψ_{fd} / kL$ , $σ = β τ / J$ , $c = k_{p} / kL$ , and ${\tilde{T}}_{L} = (τ^{2} / J) T_{L}$ .

Determination methodology for stable control domain of electric powertrain

Stability analysis of equilibrium points

We note $x$ , $y$ , $z$ , and $T$ to represent $\tilde{i_{d}}$ , $\tilde{i_{q}}$ , $\tilde{ω}$ , and ${\tilde{T}}_{L}$ , respectively, for convenience. The equilibrium points^20,21 of electric powertrain under FOC²² follow the equations below

{\begin{matrix} - x + yz = 0 \\ - y - xz + γ z + c (z - z_{ref}) = 0 \\ σ (y - z) - T = 0 \end{matrix}

(8)

The simplified balance equation is given as follows

z^{3} + \frac{T}{σ} z^{2} + (1 - γ - c) z + \frac{T}{σ} + c z_{ref} = 0

(9)

The other two equations are $x = (z + (T / σ)) z$ and $y = z + (T / σ)$ .

We note $A$ to be the Jacobi matrix²³ at the equilibrium point²⁴ of equation (8). According to the research about the stability of the linear system, we give sufficient conditions for the stability of the nonlinear system. The eigenvalue of matrix $A$ is $λ_{r}$ , $r = 1, 2, 3$ , if $Re (λ_{r}) < 0, r = 1, 2, 3$ , the nonlinear system has asymptotic stability. It is important to know the real part of eigenvalues of matrix $A$ to judge whether the electric powertrain is asymptotic stable or not

A = [\begin{matrix} - 1 & z & y \\ - z & - 1 & - x + γ + c \\ 0 & σ & - σ \end{matrix}]

(10)

The eigenvalue equation of matrix $A$ is given below

\begin{matrix} λ^{3} + (σ + 2) λ^{2} + [1 + σ (2 + x - γ - c)] λ \\ + σ (1 + yz + z^{2} + x) = 0 \end{matrix}

(11)

The Routh–Hurwitz criterion²⁵ is used to judge the asymptotic stability of the electric powertrain under FOC at the equilibrium points. The stable conditions at the equilibrium points are expressed as shown

{\begin{matrix} Δ_{0} = a_{0} = 1 > 0 \\ Δ_{1} = a_{1} = σ + 2 > 0 \\ Δ_{2} = \det [\begin{matrix} a_{1} & a_{0} \\ a_{3} & a_{2} \end{matrix}] > 0 \\ Δ_{3} = \det [\begin{matrix} a_{1} & a_{0} & 0 \\ a_{3} & a_{2} & a_{1} \\ 0 & 0 & a_{3} \end{matrix}] > 0 \end{matrix}

(12)

where $a_{0} = 1$ , $a_{1} = σ + 2$ , $a_{2} = 1 + σ (2 + x - γ - c)$ , and $a_{3} = σ (1 + yz + z^{2} + x)$ .

Theoretical determination of stable control domain

The equilibrium point will get unstable when the real part of eigenvalues transforms from negative to positive, and this transformation may cause Hopf bifurcation. Conditions that are subject to any minor disturbances will get away from the equilibrium point, but the disturbed movement may eventually become a steady state due to the nonlinearity in the system. The conditions that system produces Hopf bifurcation²⁶ are given as follows

{\begin{matrix} Re (λ {) |}_{h = h_{0}} = 0 \\ Im (λ {) |}_{h = h_{0}} \neq 0 \\ \frac{d}{dh} Re (λ {) |}_{h = h_{0}} \neq 0 \end{matrix}

(13)

where $h_{0}$ represents the critical point when the Hopf bifurcation occurs.

If there is a non-zero pure imaginary eigenvalue $λ = nj, n \neq 0$ , we get equation (14) when the eigenvalue is substituted into the eigenvalue equation. The real part and imaginary part are separated

{\begin{matrix} n^{2} = 1 + σ (2 + z^{2} + \frac{T}{σ} z - γ - c) \\ n^{2} = \frac{σ (1 + 3 z^{2} + 2 Tz / σ)}{σ + 2} \end{matrix}

(14)

The conditions that system produces Hopf bifurcation are shown below

{\begin{matrix} 1 + 3 z^{2} + 2 Tz / σ > 0 \\ 1 + σ (2 + z^{2} + \frac{T}{σ} z - γ - c > 0 \\ b_{0} z^{2} + b_{1} z + b_{2} = 0 \end{matrix}

(15)

where $b_{0} = σ^{2} - σ$ , $b_{1} = T σ + 2 T - 1$ , and $b_{2} = (σ^{2} + 2 σ) (2 - γ - c)$ .

Some complex behaviors may happen when the Hopf bifurcations occur. Analysis of different dynamic behaviors of electric powertrain needs to recognize a particular set of bifurcation points. The transition from an equilibrium point to a limit cycle through a Hopf bifurcation leads to route to chaos. Lyapunov exponents are needed to calculate to confirm that chaos occurs. There is at least one index to be positive when chaos occurs. For the three-dimensional (3D) state space described by three first-order differential equations, the corresponding attractors (fixed point, limit cycle, quasi-periodic torus, and chaos) can be judged by the sign of Lyapunov exponent. The corresponding relationship is shown as follows: attractor type is fixed point if the symbol of the Lyapunov exponent is (–,–,–), attractor type is limit cycle if the symbol of the Lyapunov exponent is (0,–,–), attractor type is quasi-periodic torus if the symbol of the Lyapunov exponent is (0,0,–), and attractor type is chaos if the symbol of the Lyapunov exponent is (+,0,–).

Simulated results of stable control domain of electric powertrain

Under no-load operating condition

The reference speed ω_ref = 0

The parameters of PMSM are shown as follows: d-axis inductance is 14.25 mH, stator resistance is 0.9 Ω, pole-pairs is 1, permanent magnet flux is 0.031 Wb, polar moment of inertia is $4.7 \times 10^{- 5} kg m^{2}$ , and viscous damping coefficient is 0.0162 N/(rad s).

The equilibrium points are solved by work out the balance equation. The three equilibrium points are $E_{1} (0, 0, 0)$ , $E_{2} (c - 1.066, \sqrt{c - 1.066}, \sqrt{c - 1.066})$ , and $E_{3} (c - 1.066, - \sqrt{c - 1.066}, - \sqrt{c - 1.066})$ . When $0 < c < 1.066$ , there is only one equilibrium point $E_{1}$ . When $1.066 < c < 25$ , there are three equilibrium points. The eigenvalue equation of matrix $A$ is solved. The stable ranges of equilibrium points $E_{1}$ , $E_{2}$ , and $E_{3}$ are $0 < c < 1.98$ , $1.066 < c < 3.35$ , and $1.066 < c < 3.35$ , respectively. The stabilities of the equilibrium points are shown in Figure 2. $H_{1}$ , $H_{2}$ , and $H_{3}$ represent Hopf bifurcation points. Solid and dotted lines represent stable and unstable equilibrium points, respectively. According to Figure 2, there is only one stable equilibrium point $E_{1}$ when $c = 0.5$ . There are three equilibrium points when $c = 2.5$ . The equilibrium points $E_{2}$ and $E_{3}$ are stable whereas equilibrium point $E_{1}$ is unstable. There are three unstable equilibrium points when $c = 18$ .

Figure 2.

The stabilities of the equilibrium points underno-load operating condition with $ω_{ref} = 0$ .

Time history diagrams and phase portraits obtained by simulation when $c = 0.5$ are shown in Figure 3(a) and (b). The initial values are set to ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.83$ . When $\tilde{t} < 1$ , the speed of PMSM declines quickly from initial speed. It can be concluded from Figure 3(a) and (b) that electric powertrain operates stably at the equilibrium point $E_{1}$ , and the speed tracks reference speed fast and accurately. Time history diagrams and phase portraits at different initial values when $c = 2.5$ are shown in Figure 3. Figure 3(c) and (d) is drawn under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.83$ , and Figure 3(e) and (f) is drawn under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = - 0.83$ . The initial values influence the final running state of electric powertrain. The electric powertrain ends at equilibrium point $E_{2}$ or $E_{3}$ stably. Time history diagram and phase portraits under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.83$ when $c = 18$ are shown in Figure 3(g) and (h). The electric powertrain loses control due to chaos attractor.

Figure 3.

Time history and phase portraits for electric powertrain at different parameters $c$ : (a, c, e, g) time history and (b, d, f, h) phase portrait.

In order to study the global dynamic behavior of electric powertrain under FOC with the variation of parameter $c$ from 0 to 25, the initial values are set to ${\tilde{i}}_{d 0} = 0.1$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.83$ . The Poincaré bifurcation diagram of the rotational speed of PMSM with the variation of parameter $c$ is shown in Figure 4. When $0 < c < 1.98$ , the electric powertrain stabilizes at equilibrium point $E_{1}$ . When $1.98 < c < 7.35$ , the electric powertrain stabilizes at equilibrium point $E_{2}$ . When $7.35 < c < 14.25$ , the electric powertrain stabilizes at equilibrium point $E_{3}$ . When $14.25 < c < 25$ , the electric powertrain enters into chaotic state. The Lyapunov exponents at different typical parameter $c$ are given below. When $c = 0.5$ , Lyapunov exponents are −0.5460,−10.0000,−61.9173, the attractor type is fixed point. When $c = 2.5$ , Lyapunov exponents are −4.0616,−4.0656,−68.5680. When $c = 14.25$ , Lyapunov exponents are 3.805869,0,−80.3231, the attractor type is chaos. So, electric powertrain enters into chaotic state when $c = 14.25$ and loses control.

Figure 4.

The Poincaré bifurcation diagram under no-load operating condition with the reference speed $ω_{ref} = 0 rad / s$ .

The reference speed ω_ref = 6 rad/s

The stabilities of equilibrium points of electric powertrain under no-load operating condition with $ω_{ref} = 6 rad / s$ are shown in Figure 5. Solid and dotted lines represent stable and unstable equilibrium points, respectively. According to Figure 5, there is only one stable equilibrium point $E_{1}$ when $c = 2$ . There are three equilibrium points when $c = 5$ . The equilibrium points $E_{1}$ and $E_{3}$ are stable, whereas the equilibrium point $E_{2}$ is unstable. There are two stable equilibrium points when $c = 18$ . The equilibrium points $E_{1}$ and $E_{3}$ are stable, whereas the equilibrium point $E_{2}$ is unstable. All of the three equilibrium points are unstable when $c = 22$ .

Figure 5.

The stabilities of equilibrium points under no-load condition with $ω_{ref} = 6$ .

Time history diagrams and phase portraits obtained by simulation when $c = 2$ are shown in Figure 6(a) and (b). The initial values are set to ${\tilde{i}}_{d} = 0$ , ${\tilde{i}}_{q} = 0.1$ , and $\tilde{ω} = 0.5$ . It can be concluded from Figure 6(a) and (b) that electric powertrain operates stably at the equilibrium point $E_{1}$ . The speed tracks reference speed fast and accurately. The time history diagrams and phase portraits at different initial values when $c = 5$ are shown in Figure 6(c) and (d). Figure 6(c) and (d) is drawn under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = - 0.5$ . The final running state of electric powertrain is influenced by the initial values. The electric powertrain will end at equilibrium point $E_{1}$ or $E_{3}$ stably under different initial values, respectively. The time history diagrams and phase portraits at different initial values when $c = 18$ are shown in Figure 6(e)–(h). Figure 6(g) and (h) is drawn under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = - 0.5$ , and Figure 6(e) and (f) is drawn at ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ . It can be concluded from Figure 6(g) and (h) that chaos occurs in electric powertrain and the trajectory of PMSM is attracted by the chaotic attractor into chaotic state. The state of electric powertrain ends at equilibrium point $E_{1}$ , but local oscillation occurs and it is shown in Figure 6(e) and (f). The time history diagrams and phase portraits under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ when $c = 22$ are shown in Figure 6(i) and (j). Speed of the PMSM oscillates violently. Electric powertrain loses control due to chaos attractor.

Figure 6.

Time history and phase portraits for electric powertrain at different parameters $c$ : (a, c, e, g, i) time history and(b, d, f, h, j) phase portrait.

The Poincaré bifurcation diagram of the rotational speed of PMSM with variation of parameter $c$ is shown in Figure 7, and the initial values are set to ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ . When $0 < c < 2.17$ , the electric powertrain stabilizes at equilibrium point $E_{1}$ . When $2.17 < c < 6.15$ , electric powertrain stabilizes at equilibrium point $E_{3}$ . When $6.15 < c < 18.43$ , electric powertrain stabilizes at equilibrium point $E_{1}$ . When $18.43 < c < 25$ , the electric powertrain enters into chaotic state. When $c = 2$ , the Lyapunov exponents are −3.6334,−3.6340,−69.6388, attractor type is fixed point. When $c = 18.43$ , the Lyapunov exponents are 4.2254,0,−80.1333, the attractor type is chaos. The electric powertrain enters into chaotic state when $c = 18.43$ and loses control.

Figure 7.

The Poincaré bifurcation diagram under no-load operating condition with the reference speed $ω_{ref} = 6 rad / s$ .

Under non-zero load torque operating condition

The reference speed ω_ref = 0

The stabilities of equilibrium points of electric powertrain with $ω_{ref} = 0$ and ${\tilde{T}}_{L} = 5.46$ are shown in Figure 8. Solid and dotted lines represent stable and unstable equilibrium points, respectively. According to Figure 8, there is only one stable equilibrium point $E_{1}$ when $c = 2$ . There is only one stable equilibrium point when $c = 12$ . All of the three equilibrium points are unstable when $c = 20$ . Time history diagrams and phase portraits obtained by simulation when $c = 2$ are shown in Figure 9(a) and (b). The initial values are set to ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ . The rational speed of PMSM drops fast when $\tilde{t} < 2$ and then makes small amplitude fluctuation. Finally, the electric powertrain end at a stable state but cannot track the reference speed accurately. The time history diagrams and phase portraits under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ when $c = 12$ are shown in Figure 9(c) and (d). The electric powertrain ends at equilibrium point $E_{1}$ , but local oscillation occurs when $\tilde{t} = 2.3$ and lasts for some time until $\tilde{t} = 38$ . Electric powertrain cannot track the reference speed accurately. The time history diagrams and phase portraits under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ when $c = 20$ are shown in Figure 9(e) and (f). We can observe rapid fluctuations of speed and long-time oscillation from Figure 9(e). Chaos occurs in electric powertrain and the speed loses control.

Figure 8.

The stabilities of equilibrium points under non-zero load torque with $ω_{ref} = 0$ .

Figure 9.

Time history and phase portraits for electric powertrain at different variables $c$ : (a, c, e) time history and (b, d, f) phase portrait.

The Poincaré bifurcation diagram of the rotational speed of PMSM with variation of parameter $c$ is shown in Figure 10, and the initial values are that ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ . When $0 < c < 18$ , the electric powertrain stabilizes at equilibrium point $E_{1}$ . When $18 < c < 25$ , electric powertrain enters into chaotic state. The Lyapunov exponents at different typical parameter $c$ are given below. When $c = 2$ , the Lyapunov exponents are −6.2702,−6.2709,−63.5294, attractor type is fixed pointed. When $c = 12$ , the Lyapunov exponents are −0.7185,−0.722108,−73.6099, attractor type is fixed point. When $c = 20$ , the Lyapunov exponents are 5.0947,0,−81.0573, attractor type is chaos.

Figure 10.

The Poincaré bifurcation diagram under non-zero load torque operating condition with the reference speed $ω_{ref} = 0 rad / s$ .

The reference speed ω_ref = 6 rad/s

The stabilities of equilibrium points of electric powertrain with $ω_{ref} = 6 rad / s$ and ${\tilde{T}}_{L} = 5.46$ are shown in Figure 11. Solid and dotted lines represent stable and unstable equilibrium points, respectively. According to Figure 11, there is only one stable equilibrium point $E_{1}$ when $c = 4$ . There is only one stable equilibrium point when $c = 20$ . All of the three equilibrium points are unstable when $c = 24$ .

Figure 11.

The stabilities of equilibrium points under non-zero load torque operating condition with $ω_{ref} = 6$ .

Time history diagram and phase portrait obtained by simulation when $c = 4$ are shown in Figure 12(a) and (b). The initial values are set to ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ . It can be concluded from Figure 12 that electric powertrain operates stably at the equilibrium point $E_{1}$ . The speed tracks reference speed accurately but less fast. The time history diagram and phase portrait under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ when $c = 20$ are shown in Figure 12(c) and (d). The electric powertrain ends at equilibrium point $E_{1}$ , but local oscillation occurs and it influences the stability of electric powertrain. The time history diagram and phase portrait under initial values of ${\tilde{i}}_{d 0} = 0$ , ${\tilde{i}}_{q 0} = 0.1$ , and ${\tilde{ω}}_{0} = 0.5$ when $c = 24$ are shown in Figure 12(e) and (f).

Figure 12.

Time history and phase portraits for electric powertrain at different parameters $c$ : (a, c, e) time history and (b, d, f) phase portrait.

The Poincaré bifurcation diagram with variable $c$ is shown in Figure 13. When $0 < c < 23.6$ , the electric powertrain stabilizes at equilibrium point $E_{1}$ and occurs chaos when $23.6 < c < 25$ .

Figure 13.

The Poincaré bifurcation diagram under non-zero load torque operating condition with the reference speed $ω_{ref} = 6 rad / s$ .

Discussion

In this article, we analyzed the relationship between nonlinear dynamics of electric powertrain and variable control parameter of PI under four typical operating conditions. A minimum stable range of proportional control parameter under various complex working conditions is solved.

The simulation results demonstrate that the dynamic behaviors of electric powertrain are dependent on the proportional control parameter. Chaotic motion, such as strong oscillation of speed and torque, arises when the proportional control parameter of PI regulator increases to a specific value. In contrast to other research,⁵ which use proportional differential controller to eliminate chaos of PMSM under no-load and zero input voltage operating condition. A controller which uses the proportional integral control principle by adding an auxiliary state variable in the nonlinear dynamic model of PMSM was designed to suppression chaos and track speed.⁶ But they don’t reveal the relationship between control parameter of PI regulator and stability of electric powertrain under various complex operating conditions. It is found that the impact of proportional control parameters on dynamic motion of electric powertrain can’t be underestimated.

In summary, we have shown different nonlinear dynamics of electric powertrain with variable parameter PI and give a stable region. This can be used for control and design purposes. But more work can be considered in the future, and more in-depth research can be done when there is more than one PI regulator applied to control speed and current.

Conclusion

The nonlinear dynamic model of the electric powertrain under FOC was established. By applying nonlinear dynamic theory, we analyzed the influence of proportional control parameter of PI regulator on the stability of electric powertrain under four typical operating conditions. The dynamic behaviors of electric powertrain are dependent on the proportional control parameters through the simulation results. Important conclusions are summarized as follows:

The electric powertrain operates stably when the proportional control parameter is set in the area where there is only one stable equilibrium point. The final running state of electric powertrain is affected by initial values when the proportional control parameter is chosen in two stable equilibrium points area. Local oscillation occurs during the early operating period. The electric powertrain can not operate stably when proportional control parameter is selected in the region without any stable equilibrium point;

Chaos occurs with the increase of proportional control parameter. Stable regions of electric powertrain diverse under different operating conditions. Especially the stable range of the area where there is only one stable equilibrium point under no-load and zero reference rotational speed is smallest. That is to say, the electric powertrain operates unstably most likely under no-load and zero reference rotational speed operating condition;

We can determine a minimum stable range of proportional control parameter under four typical operating conditions, the smallest stable region is $c < 1.98$ for the electric powertrain, and also it can be used for control and design purposes.

Footnotes

Handling Editor: Sung-Cheon Han

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant no. 51705208) and China Postdoctoral Science Foundation (grant no. 2018M632240).

ORCID iD

Donghai Hu

Yanzhi Yan

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Determination methodology for stable control domain of electric powertrain based on permanent magnet synchronous motor

Abstract

Keywords

Introduction

Nonlinear dynamic model of electric powertrain

PMSM

PMSM controller

Nonlinear dynamic model of electric powertrain

Determination methodology for stable control domain of electric powertrain

Stability analysis of equilibrium points

Theoretical determination of stable control domain

Simulated results of stable control domain of electric powertrain

Under no-load operating condition

The reference speed ωref = 0

The reference speed ωref = 6 rad/s

Under non-zero load torque operating condition

The reference speed ωref = 0

The reference speed ωref = 6 rad/s

Discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

The reference speed ω_ref = 0

The reference speed ω_ref = 6 rad/s

The reference speed ω_ref = 0

The reference speed ω_ref = 6 rad/s