CN105634357A - Efficiency optimization control method for linear induction motor - Google Patents

Abstract

The invention discloses an efficiency optimization control method for a linear induction motor. The influence on the motor characteristics from a longitudinal end effect, a transverse end effect and a primary end part half-filling groove is corrected by adopting a correction coefficient; the influence on the output power and loss of the motor from primary leakage inductance and secondary leakage inductance is completely analyzed; and a dynamic loss function, including primary copper loss, secondary copper loss, iron loss caused by excitation inductance, and iron loss caused by primary leakage inductance and secondary leakage inductance, is established. By adoption of the control policy, the efficiency of the linear induction motor is greatly improved; and the efficiency can reach or can be close to the efficiency under a rated operation condition under different operation conditions.

Description

Efficiency optimization control method for linear induction motor

Technical Field

The invention belongs to the field of linear induction motors, and particularly relates to an efficiency optimization control method of a linear induction motor.

Background

The linear induction motor can generate direct thrust through the interaction of current between the primary and secondary stages, thereby saving an intermediate transmission link and being particularly suitable for direct-drive occasions. Meanwhile, the linear induction motor is widely applied to the industrial field such as urban rail transit, oil pumping units and the like due to the advantages of simple structure, large thrust, small mechanical loss, small maintenance amount and the like.

However, the efficiency of the linear induction motor is generally low because the mechanical air gap of the linear induction motor is larger than that of the common rotary induction motor due to the limitations of manufacturing and assembling processes, operation safety requirements and the like. In addition, the linear induction motor has transverse edge effect and longitudinal edge effect during operation due to the special structure of primary disconnection, unequal primary and secondary widths and semi-filled slots at the ends of the primary. These two effects will cause the increase of equivalent secondary resistance and the attenuation of equivalent excitation inductance, and under the same operating condition, it needs to input larger current, thus leading to the increase of motor loss and the decrease of efficiency.

Currently, there are two main types of methods for optimizing and controlling efficiency of a linear induction motor: model methods and search methods. The search method minimizes input power by monitoring motor or drive system input power and using a search algorithm to adjust a given flux linkage or other controlled quantity. Although the search method is not influenced by motor parameters, the convergence rate is low, the requirement on controller hardware is high, and the search method is easy to fall into local repeated optimization, so that the output fluctuation of a motor and the instability of a system are caused. The model method is based on a motor equivalent model, and motor efficiency optimization control is achieved by optimizing a motor loss function. Compared with a search method, the method has the advantages of fast calculation, small output fluctuation, simplicity, practicability and strong dependence on motor model parameters. The linear induction motor is influenced by a transverse edge effect and a longitudinal edge effect, the parameter change of the linear induction motor is complex, the coupling among all parameters is serious, and the ideal control effect can be obtained only by comprehensively considering the influence of all parameters. At present, no relatively complete loss model exists in the research on the efficiency optimization control of a linear induction motor model method.

Disclosure of Invention

Aiming at the defects or improvement requirements of the prior art, the invention provides the efficiency optimization control method of the linear induction motor, which can quickly calculate the minimum loss working point of the linear induction motor, obviously improve the motor efficiency under different running conditions and realize the efficiency optimization control in the global range.

In order to achieve the above object, the present invention provides a method for optimally controlling efficiency of a linear induction motor, which is characterized by comprising the following steps:

(1) collecting primary three-phase current i of linear induction motor_A、i_BAnd i_CAnd the motor running speed v₂According to the motor running speed v₂Calculating to obtain the secondary angular frequency omega_r；

(2) From primary three-phase currents i_A、i_BAnd i_CAnd secondary angular frequency ω_rCalculating to obtain the secondary flux linkage angle theta of the linear induction motor₁And an electromagnetic thrust F; from secondary flux linkage angle theta by coordinate transformation₁And primary three-phase current i_A、i_BAnd i_CObtain a primary d-axis current i_dsAnd primary q-axis current i_qs；

(3) According to secondary angular frequency ω_rAnd electromagnetic thrust F, and calculating to obtain the optimized secondary d-axis flux linkage psi_drFurther obtain the primary d-axis current control quantityReference value of secondary angular frequencyWith secondary angular frequency omega_rPerforming PI regulation on the difference value to obtain a primary q-axis current control quantity

Wherein the optimized secondary d-axis flux linkage ${\mathrm{\ψ}}_{dr}=\sqrt{\frac{1}{2}\sqrt{\frac{2a_2}{3a_1}+\mathrm{\Δ}}+\frac{1}{2}\sqrt{\frac{4a_2}{3a_1}-\mathrm{\Δ}-\frac{\frac{16a_3}{a_1}}{4\sqrt{\frac{2a_2}{3a_1}}+\mathrm{\Δ}}}},$ $\begin{array}{cc}\mathrm{\Δ}=\frac{\sqrt[3]{2}{\mathrm{\Δ}}_1}{3a_1\sqrt[3]{{\mathrm{\Δ}}_2+\sqrt{-4{\mathrm{\Δ}}_1^3+{\mathrm{\Δ}}_2^2}}}+\frac{\sqrt[3]{{\mathrm{\Δ}}_2+\sqrt{-4{\mathrm{\Δ}}_1^3+{\mathrm{\Δ}}_2^2}}}{3\sqrt[3]{2}{a}_1},& {\mathrm{\Δ}}_1=a_2^2-36a_1{a}_4\end{array},$ a₁、a₂、a₃And a₄To make the total loss of the linear induction motor be a loss factor $P_{loss}=a_0+a_1{\mathrm{\ψ}}_{dr}^2+a_2{\mathrm{\ψ}}_{dr}^{-2}+a_3{\mathrm{\ψ}}_{dr}^{-4}+a_4{\mathrm{\ψ}}_{dr}^{-6}$ Minimum, a₀Is the loss factor;

(4) controlling the primary d-axis currentWith primary d-axis current i_dsPerforming PI regulation on the difference value to obtain primary d-axis voltage control quantityControlling the primary q-axis currentWith primary q-axis current i_qsPerforming PI regulation on the difference value to obtain primary q-axis voltage control quantity

(5) Controlling the primary d-axis voltageAnd primary q-axis voltage control quantityCoordinate transformation is carried out to obtain primary α axis voltage control quantityAnd primary β axis voltage control quantityThe space vector pulse width modulation input is used as the input of space vector pulse width modulation, and the optimization control of the efficiency of the linear induction motor is realized.

Preferably, the loss factor a₁、a₂、a₃And a₄Respectively as follows:

$a_1=\frac{1}{L_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}{L}_{ls}^2{L}_{me}^4{\mathrm{\ω}}_r^2+L_{ls}^2{L}_{me}^4{R}_{Fe}+L_{ls}^2{L}_{me}^4{R}_s{\mathrm{\ω}}_r^2+L_{ls}^2{L}_{me}^4{R}_s+2L_{ls}{L}_{me}^5{R}_{Fe}{\mathrm{\ω}}_r^2\\ +2L_{ls}{L}_{me}^5{R}_{Fe}+2L_{ls}{L}_{me}^5{R}_s{\mathrm{\ω}}_r^2+2L_{ls}{L}_{me}^5{R}_s+2L_{ls}{L}_{me}^4{R}_s{R}_{Fe}\\ +L_{me}^6{R}_{Fe}{\mathrm{\ω}}_r^2+L_{me}^2+L_{me}^6{R}_{Fe}+L_{me}^6{R}_s{\mathrm{\ω}}_r^2+L_{me}^6{R}_s+L_{me}^5{R}_s{R}_{Fe}+L_{me}^4{R}_s{R}_{Fe}^2\end{array}\right\},$

$a_2=\frac{{\mathrm{\τ}}^2{F}^2}{{\mathrm{\π}}^2{R}_{Fe}^2{L}_{me}^6}\left\{\begin{array}{c}{L}_{ls}^2{L}_{lr}^4{L}_{me}^2(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}{L}_{lr}^4{L}_{me}^3(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}{L}_{lr}^4{L}_{me}^2{R}_s{R}_{Fe}\\ +L_{lr}^4{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{lr}^4{L}_{me}^3{R}_s{R}_{Fe}+L_{lr}^4{L}_{me}^2{R}_s{R}_{Fe}^2+4L_{ls}^2{L}_{lr}^3{L}_{me}^3(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +6L_{ls}{L}_{lr}^3{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+8L_{ls}{L}_{lr}^3{L}_{me}^3{R}_s{R}_{Fe}+2L_{lr}^3{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +6L_{lr}^3{L}_{me}^4{R}_s{R}_{Fe}+4L_{lr}^3{L}_{me}^3{R}_s{R}_{Fe}^2+6L_{ls}^2{L}_{lr}^2{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+L_{ls}^2{L}_{lr}^2{L}_{me}^2{R}_{re}^2(R_s+R_{Fe})\\ +6L_{ls}{L}_{lr}^2{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+12L_{ls}{L}_{lr}^2{L}_{me}^4{R}_s{R}_{Fe}+2L_{ls}{L}_{lr}^2{L}_{me}^3{R}_{re}^2(R_s+R_{Fe})+6L_{lr}^2{L}_{me}^5{R}_s{R}_{Fe}\\ +L_{lr}^2{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+L_{lr}^2{L}_{me}^4(R_{Fe}^2{R}_{re}+6R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)\\ +4L_{ls}^2{L}_{lr}{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}^2{L}_{lr}{L}_{me}^3{R}_{re}^2(R_s+R_{Fe})+2L_{ls}{L}_{lr}{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +8L_{ls}{L}_{lr}{L}_{me}^5{R}_s{R}_{Fe}+4L_{ls}{L}_{lr}{L}_{me}^4{R}_{re}^2(R_s+R_{Fe})+2L_{lr}{L}_{me}^6{R}_s{R}_{Fe}{R}_s{R}_{Fe}+L_{ls}^2{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +2L_{lr}{L}_{me}^5(R_{Fe}^2{R}_{re}+2R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)+L_{ls}^2{L}_{me}^4{R}_{re}^2(R_s+R_{Fe})+2L_{ls}{L}_{me}^6{R}_s{R}_{Fe}\\ +2L_{ls}{L}_{me}^5{R}_{re}^2(R_s+R_{Fe})+L_{me}^6(R_{Fe}^2{R}_{re}+R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)\end{array}\right\},$

$a_3=\frac{(R_s+R_{Fe})R_{re}{\mathrm{\ω}}_r{\mathrm{\τ}}^3{F}^3}{{\mathrm{\π}}^3{L}_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}2L_{ls}^2{L}_{Lr}^5{L}_{me}+4L_{ls}{L}_{lr}^5{L}_{me}^2+2L_{lr}^5{L}_{me}^3+10L_{ls}^2{L}_{Lr}^4{L}_{me}^2+16L_{ls}{L}_{lr}^4{L}_{me}^3\\ +6L_{lr}^4{L}_{me}^4+20L_{ls}^2{L}_{Lr}^3{L}_{me}^3+6L_{lr}^3{L}_{me}^5+24L_{ls}{L}_{lr}^3{L}_{me}^4+20L_{ls}^2{L}_{Lr}^2{L}_{me}^4\\ +16L_{ls}{L}_{lr}^2{L}_{me}^5+2L_{lr}^2{L}_{me}^6+10L_{ls}^2{L}_{lr}{L}_{me}^5+4L_{ls}{L}_{lr}{L}_{me}^6+2L_{ls}^2{L}_{me}^6\end{array}\right\}$ and

$a_4=\frac{(R_s+R_{Fe})R_{re}^2{\mathrm{\τ}}^4{F}^4}{{\mathrm{\π}}^{47}{L}_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}{L}_{ls}^2{L}_{lr}^6+2L_{ls}{L}_{lr}^6{L}_{me}+L_{lr}^6{L}_{me}^2+6L_{ls}^2{L}_{lr}^5{L}_{me}+10L_{ls}{L}_{lr}^5{L}_{me}^2+4L_{lr}^5{L}_{me}^3\\ +15L_{ls}^2{L}_{lr}^4{L}_{me}^2+20L_{ls}{L}_{lr}^4{L}_{me}^3+6L_{lr}^4{L}_{me}^4+20L_{ls}^2{L}_{lr}^3{L}_{me}^3+20L_{ls}{L}_{lr}^3{L}_{me}^4+4L_{lr}^3{L}_{me}^5\\ +15L_{ls}^2{L}_{lr}^2{L}_{me}^4+10L_{ls}{L}_{lr}^2{L}_{me}^5+L_{lr}^2{L}_{me}^6+6L_{ls}^2{L}_{lr}{L}_{me}^5+2L_{ls}{L}_{lr}{L}_{me}^6+L_{ls}^2{L}_{me}^6\end{array}\right\},$

wherein tau is polar distance, F is electromagnetic thrust, omega_rFor secondary angular frequency, R_sIs a primary resistance, R_FeIs an equivalent iron loss resistance, L_lsFor primary leakage inductance, L_lrFor secondary leakage inductance, L_meFor equivalent excitation inductance, R_reIs an equivalent secondary resistance.

Preferably, the equivalent excitation inductance L_meAnd an equivalent secondary resistance R_reRespectively as follows:

L_me＝K_xC_xL_mand

R_re＝K_rC_rR_r，

wherein, K_rCorrection factor, K, for the longitudinal side effect of the secondary resistor_xCorrection factor for longitudinal side effect of exciting inductance, C_rCorrection of coefficient for secondary resistance lateral edge effect, C_xFor correction of the coefficient of transverse edge effect of the excitation inductance, L_mFor exciting inductance, R_rIs the secondary resistance.

Preferably, the primary d-axis current control quantityComprises the following steps:

$i_{ds}^{*}=\frac{L_s+R_{Fe}}{L_{me}{R}_{Fe}}{\mathrm{\ψ}}_{dr}-\[{\mathrm{\ω}}_r+\frac{{\mathrm{\τR}}_{re}{L}_{r}F}{{\mathrm{\πL}}_{me}{{\mathrm{\ψ}}_{dr}}^2}\]\frac{{\mathrm{\τL}}_{r}F(L_r{L}_{ls}+L_{me}{L}_{lr})}{{\mathrm{\πL}}_{me}^2{R}_{Fe}{\mathrm{\ψ}}_{dr}},$

wherein L is_sIs equivalent to primary inductance, R_FeIs an equivalent iron loss resistance, L_meFor equivalent excitation inductance, τ is the pole pitch, R_reIs an equivalent secondary resistance, L_rEquivalent secondary inductance, F electromagnetic thrust, L_lsFor primary leakage inductance, L_lrIs the secondary leakage inductance.

Generally, compared with the prior art, the above technical solution conceived by the present invention has the following beneficial effects: the correction coefficient is adopted to correct the influence of the longitudinal side effect, the transverse edge effect and the primary end semi-filling groove of the linear induction motor on the motor characteristics, the influence of primary leakage inductance and secondary leakage inductance on the output power and the loss of the motor is completely analyzed, and a dynamic loss function containing primary and secondary copper loss, iron loss caused by excitation inductance and iron loss caused by the primary and secondary leakage inductances is established. By adopting the control strategy, the efficiency of the linear induction motor is obviously improved, and the efficiency under the rated operation condition can be reached or approached under different operation conditions.

Drawings

FIG. 1 is a schematic view of a one-dimensional structure of a linear induction motor;

FIG. 2 is a d-q axis dynamic equivalent circuit considering half-filled slots, lateral edge effects, longitudinal edge effects, and iron loss effects, where (a) is the d axis dynamic equivalent circuit and (b) is the q axis dynamic equivalent circuit;

FIG. 3 is a functional block diagram of an efficiency optimization control method of a linear induction motor according to an embodiment of the present invention;

fig. 4 is a graph comparing the efficiency of the control method of the present invention and the conventional control method.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

During the steady-state operation of the linear induction motor, the influence of the longitudinal edge effect and the transverse edge effect is reflected on the change of 4 correction coefficients. Based on a d-q axis dynamic equivalent circuit, the invention provides a novel linear induction motor dynamic loss model which comprehensively reflects the influence of a primary end part half-filled slot, a transverse edge effect, a longitudinal edge effect, primary leakage inductance and secondary leakage inductance on the output power and the loss of a motor, and simultaneously completely comprises primary and secondary copper loss, iron loss caused by excitation inductance and iron loss caused by the primary and secondary leakage inductances. Based on the loss model, the efficiency optimization control strategy of the linear induction motor is realized. First, the theoretical basis of the method for controlling efficiency optimization of a linear induction motor according to the present invention will be described in detail.

1. d-q axis dynamic equivalent circuit

Fig. 1 is a schematic view of a one-dimensional structure of a linear induction motor. Due to the special structure of primary breaking and unequal primary width, the linear induction motor has transverse edge effect and longitudinal edge effect during operation. These two effects cause an increase in the equivalent secondary resistance and a decay in the equivalent magnetizing inductance, resulting in a decrease in the motor operating performance. Meanwhile, the primary end part is a half-filled slot, so that the actual number of pole pairs of the motor is increased, and the equivalent number of pole pairs p is defined_eComprises the following steps:

$p_e=\frac{{(2n_p-1)}^2}{4n_p-3+\mathrm{\ϵ}/\left(m_{1}q\right)}---\left(1\right)$

wherein n is_pIs the number of pole pairs of the motor, is the short pitch, m₁Q is the number of primary phases and the number of slots per pole per phase.

FIG. 2 is a cross-sectional view of a half-filled trenchA d-q axis dynamic equivalent circuit of edge effect, longitudinal edge effect and iron loss effect, wherein fig. 2(a) is the d axis dynamic equivalent circuit, and fig. 2(b) is the q axis dynamic equivalent circuit. In the figure, K_rCorrection factor, K, for the longitudinal side effect of the secondary resistor_xCorrection factor for longitudinal side effect of exciting inductance, C_rCorrection of coefficient for secondary resistance lateral edge effect, C_xAnd correcting the coefficient for the transverse edge effect of the excitation inductance. The expressions for these four coefficients are:

$K_r=\frac{sG}{2p_e\mathrm{\τ}\sqrt{1+{\left(sG\right)}^2}}\frac{C_1^2+C_2^2}{C_1}---\left(2\right)$

$K_x=\frac{1}{2p_e\mathrm{\τ}\sqrt{1+{\left(sG\right)}^2}}\frac{C_1^2+C_2^2}{C_2}---\left(3\right)$

$C_r=\frac{sG\[{\mathrm{Re}}^2\left(T\right)+{\mathrm{Im}}^2\left(T\right)\]}{\mathrm{Re}\left(T\right)}---\left(4\right)$

$C_x=\frac{{\mathrm{Re}}^2\left(T\right)+{\mathrm{Im}}^2\left(T\right)}{\mathrm{Im}\left(T\right)}---\left(5\right)$

wherein s is the slip of the linear induction motor, G is the quality factor, tau is the polar distance, T, C₁And C₂Re () represents the real part and Im () the imaginary part as a function of the slip and the figure of merit.

In FIG. 2, L_ls、L_lr、L_mRespectively primary leakage inductance, secondary leakage inductance, excitation inductance, R_s、R_r、R_FeRespectively a primary resistance, a secondary resistance and an equivalent iron loss resistance. The equivalent iron loss resistor is connected with the excitation inductor in parallel and is positioned on the left side of the primary leakage inductor, and can simultaneously and completely reflect the iron loss of the excitation inductor branch, the iron loss caused by the primary leakage inductor and the iron loss caused by the secondary leakage inductor.

2. Mathematical model of linear induction motor

According to the equivalent circuit of fig. 2, a d-q axis mathematical model of the linear induction motor can be obtained, wherein a voltage equation is as follows:

$\left\{\begin{array}{c}{u}_{ds}=R_s{i}_{ds}+{\mathrm{p\ψ}}_{ds}-{\mathrm{\ω}}_s{\mathrm{\ψ}}_{qs}\\ u_{qs}=R_s{i}_{qs}+{\mathrm{p\ψ}}_{qs}+{\mathrm{\ω}}_s{\mathrm{\ψ}}_{ds}\\ 0=K_r{C}_r{R}_r{i}_{dr}+{\mathrm{p\ψ}}_{dr}-({\mathrm{\ω}}_s-{\mathrm{\ω}}_r){\mathrm{\ψ}}_{qr}\\ 0=K_r{C}_r{R}_r{i}_{qr}+{\mathrm{p\ψ}}_{qr}+({\mathrm{\ω}}_s-{\mathrm{\ω}}_r){\mathrm{\ψ}}_{dr}\end{array}\right.---\left(6\right)$

in the formula u_ds、u_qsPrimary d-axis voltage, primary q-axis voltage, i_ds、i_qs、i_drAnd i_qrPrimary d-axis current, primary q-axis current, secondary d-axis current and secondary q-axis current, psi_ds、ψ_qs、ψ_dr、ψ_qrAre respectively primary d-axis flux linkage, primary q-axis flux linkage, secondary d-axis flux linkage and secondary q-axis flux linkage, omega_s、ω_rPrimary angular frequency, secondary angular frequency, respectively, and p is a differential operator.

The flux linkage equation is:

$\left\{\begin{array}{c}{\mathrm{\ψ}}_{ds}=L_{ls}{i}_{dsm}+K_x{C}_x{L}_m{i}_{dm}\\ {\mathrm{\ψ}}_{qs}=L_{ls}{i}_{qsm}+K_x{C}_x{L}_m{i}_{qm}\\ {\mathrm{\ψ}}_{dr}=L_{ls}{i}_{dr}+K_x{C}_x{L}_m{i}_{dm}\\ {\mathrm{\ψ}}_{qr}=L_{ls}{i}_{qr}+K_x{C}_x{L}_m{i}_{qm}\end{array}\right.---\left(7\right)$

in the formula i_dm、i_qmThe d-axis current and the q-axis current of the excitation branch are respectively.

The voltage equation of the iron loss resistance branch circuit is as follows:

$\left\{\begin{array}{c}0={\mathrm{p\ψ}}_{ds}-{\mathrm{\ω}}_s{\mathrm{\ψ}}_{qs}-R_{Fe}{i}_{dFe}\\ 0={\mathrm{p\ψ}}_{qs}+{\mathrm{\ω}}_s{\mathrm{\ψ}}_{ds}-R_{Fe}{i}_{qFe}\end{array}\right.---\left(8\right)$

in the formula i_dFeBy means of an equivalent iron loss resistor branch d-axis current i_qFeThe q-axis current of the equivalent iron loss resistance branch circuit.

The node KCL equation is:

$\left\{\begin{array}{c}{i}_{dsm}=i_{ds}-i_{dFe}\\ i_{qsm}=i_{qs}-i_{qFe}\\ i_{dm}+i_{dFe}=i_{ds}+i_{dr}\\ i_{qm}+i_{qFe}=i_{qs}+i_{qr}\end{array}\right.---\left(9\right)$

3. linear induction motor loss model

The controllable loss of the linear induction motor comprises three parts: primary copper, secondary copper and iron losses. According to the equivalent circuit of fig. 2, the total loss of the linear induction motor can be expressed as:

$P_{loss}=R_s(i_{ds}^2+i_{qs}^2)+K_r{C}_r{R}_r(i_{dr}^2+i_{qr}^2)+R_{Fe}(i_{dFe}^2+i_{qFe}^2)---\left(10\right)$

considering the electromagnetic thrust F and the motor speed (i.e. the secondary angular frequency ω) when the motor is in load operation and in steady state_r) Are all constants. When the secondary magnetic field orientation control is used and the motor is in a steady state, the secondary d-axis current of the motor is zero, and the secondary q-axis flux linkage is zero, namely:

i_dr＝0(11)

ψ_qr＝0(12)

according to the d-axis circuit of FIG. 2, the voltage balance equation is calculated

R_Fei_dFe＝-ω_sψ_qs+(i_ds-i_dFe)(L_ls+K_xC_xL_m)(13)

Can obtain the product

$i_{ds}-i_{dFe}=\frac{R_{Fe}{i}_{dFe}+{\mathrm{\ω}}_s{\mathrm{\ψ}}_{qs}}{L_{ls}+K_x{C}_x{L}_m}---\left(14\right)$

The secondary d-axis flux linkage equation in (7) shows

ψ_dr＝K_xC_xL_m(i_ds-i_dFe)(15)

Then, the two components (13) and (14) are simultaneously dissolved to obtain

$i_{dFe}=\frac{(L_{ls}+K_x{C}_x{L}_m){\mathrm{\ψ}}_{dr}-{\mathrm{\ω}}_s{K}_x{C}_x{L}_m{\mathrm{\ψ}}_{qs}}{K_x{C}_x{L}_m{R}_{Fe}}---\left(16\right)$

$i_{ds}=\frac{{\mathrm{\ψ}}_{dr}}{K_x{C}_x{L}_m}+\frac{(K_x{C}_x{L}_m+L_{ls}){\mathrm{\ψ}}_{dr}}{K_x{C}_x{L}_m{R}_{Fe}}-\frac{{\mathrm{\ω}}_s{\mathrm{\ψ}}_{qs}}{R_{Fe}}---\left(17\right)$

Similarly, the q-axis circuit voltage balance equation is written in the column, and can be known from the secondary q-axis flux linkage equation in (7)

R_Fei_qFe＝ω_sψ_ds+(i_qs-i_qFe)L_ls+(i_qs-i_qFe+i_qr)K_xC_xL_m＝ω_sψ_ds+ψ_qs(18)

ψ_qr＝L_lri_qr+K_xC_xL_m(i_qs-i_qFe+i_qr)＝0(19)

Is obtained by simultaneous decomposition of (17) and (18)

$i_{qFe}=\frac{{\mathrm{\ω}}_s{\mathrm{\ψ}}_{ds}+{\mathrm{\ψ}}_{qs}}{R_{Fe}}---\left(20\right)$

$i_{qs}=\frac{{\mathrm{\ω}}_s{\mathrm{\ψ}}_{ds}+{\mathrm{\ψ}}_{qs}}{R_{Fe}}-\frac{L_{lr}+K_x{C}_x{L}_m}{K_x{C}_x{L}_m}{i}_{qr}---\left(21\right)$

Primary q-axis flux linkage equations in column write alone (7), i.e.

ψ_qs＝L_ls(i_qs-i_qFe)+K_xC_xL_m(i_qs-i_qFe+i_qr)(22)

Is obtained by simultaneous decomposition of (19) and (22)

${\mathrm{\ψ}}_{qs}=-\frac{L_{ls}{L}_{lr}+(L_{ls}+L_{lr})K_x{C}_x{L}_m}{K_x{C}_x{L}_m}{i}_{qr}---\left(23\right)$

From the d-axis flux linkage relationship, it can be known

${\mathrm{\ψ}}_{ds}=(L_{ls}+K_x{C}_x{L}_m)(i_{ds}-i_{dFe})=(L_{ls}+K_x{C}_x{L}_m)\frac{{\mathrm{\ψ}}_{dr}}{K_x{C}_x{L}_m}---\left(24\right)$

The electromagnetic thrust of the motor is expressed as

$F=-\frac{\mathrm{\π}}{\mathrm{\τ}}\frac{K_x{C}_x{L}_m}{K_x{C}_x{L}_m+L_{lr}}{\mathrm{\ψ}}_{dr}{i}_{qr}---\left(25\right)$

Can be derived from (25)

$i_{qr}=-\frac{\mathrm{\τ}}{\mathrm{\π}}\frac{K_x{C}_x{L}_m+L_{lr}}{K_x{C}_x{L}_m}\frac{F}{{\mathrm{\ψ}}_{dr}}---\left(26\right)$

At steady state the slip angular frequency of the motor is

${\mathrm{\ω}}_{sl}=-\frac{K_r{C}_r{R}_r{i}_{qr}}{{\mathrm{\ψ}}_{dr}}---\left(27\right)$

Primary angular frequency of the motor is

${\mathrm{\ω}}_s={\mathrm{\ω}}_r+{\mathrm{\ω}}_{sl}={\mathrm{\ω}}_r-\frac{K_r{C}_r{R}_r{i}_{qr}}{{\mathrm{\ψ}}_{dr}}---\left(28\right)$

The current expressions described by equations (11), (16), (17), (20), (21), and (26) are now arranged as follows:

$\left\{\begin{array}{c}{i}_{ds}=\frac{{\mathrm{\ψ}}_{dr}}{K_x{C}_x{L}_m}+\frac{(K_x{C}_x{L}_m+L_{ls}){\mathrm{\ψ}}_{dr}}{K_x{C}_x{L}_m{R}_{Fe}}-\frac{{\mathrm{\ω}}_s{\mathrm{\ψ}}_{qs}}{R_{Fe}}\\ i_{dFe}=\frac{(L_{ls}+K_x{C}_x{L}_m){\mathrm{\ψ}}_{dr}-{\mathrm{\ω}}_s{K}_x{C}_x{L}_m{\mathrm{\ψ}}_{qs}}{K_x{C}_x{L}_m{R}_{Fe}}\\ i_{dr}=0\\ i_{qs}=\frac{{\mathrm{\ω}}_s{\mathrm{\ψ}}_{ds}+{\mathrm{\ψ}}_{qs}}{R_{Fe}}-\frac{L_{lr}+K_x{C}_x{l}_m}{K_x{C}_x{L}_m}{i}_{qr}\\ i_{qFe}=\frac{{\mathrm{\ω}}_s{\mathrm{\ψ}}_{ds}+{\mathrm{\ψ}}_{qs}}{R_{Fe}}\\ i_{qr}=-\frac{\mathrm{\τ}}{\mathrm{\π}}\frac{K_x{C}_x{L}_m+l_{lr}}{K_x{C}_x{L}_m}\frac{F}{{\mathrm{\ψ}}_{dr}}\end{array}\right.---\left(29\right)$

wherein,

$\left\{\begin{array}{c}{\mathrm{\ψ}}_{ds}=\frac{L_{ls}+K_x{C}_x{L}_m}{K_x{C}_x{L}_m}{\mathrm{\ψ}}_{dr}\\ {\mathrm{\ψ}}_{qs}=-\frac{L_{ls}{L}_{lr}+(L_{ls}+L_{lr})K_x{C}_x{L}_m}{K_x{C}_x{L}_m}{i}_{qr}\\ {\mathrm{\ω}}_s={\mathrm{\ω}}_r-\frac{K_r{C}_r{R}_r{i}_{qr}}{{\mathrm{\ψ}}_{dr}}\end{array}\right.---\left(30\right)$

substituting (30) into (29) and simplifying to obtain:

$\left\{\begin{array}{c}{i}_{ds}=\frac{L_s+R_{Fe}}{L_{me}{R}_{Fe}}{\mathrm{\ψ}}_{dr}-({\mathrm{\ω}}_r+\frac{{\mathrm{\τR}}_{re}{L}_{r}F}{{\mathrm{\πL}}_{me}{\mathrm{\ψ}}_{dr}^2})\frac{{\mathrm{\τL}}_{r}F(L_r{L}_{ls}+L_{me}{L}_{lr})}{{\mathrm{\πL}}_{me}^2{R}_{Fe}{\mathrm{\ψ}}_{dr}}\\ i_{dFe}=\frac{L_s}{L_{me}{R}_{Fe}}{\mathrm{\ψ}}_{dr}-({\mathrm{\ω}}_r+\frac{{\mathrm{\τR}}_{re}{L}_{r}F}{{\mathrm{\πL}}_{me}{\mathrm{\ψ}}_{dr}^2})\frac{{\mathrm{\τL}}_{r}F(L_r{L}_{ls}+L_{me}{l}_{lr})}{{\mathrm{\πL}}_{me}^2{R}_{Fe}{\mathrm{\ψ}}_{dr}}\\ i_{qs}=({\mathrm{\ω}}_r+\frac{{\mathrm{\τR}}_{re}{L}_{r}F}{{\mathrm{\πL}}_{me}{\mathrm{\ψ}}_{dr}^2})\frac{L_s}{L_{me}{R}_{Fe}}{\mathrm{\ψ}}_{dr}+\frac{{\mathrm{\τL}}_{r}F(L_r{R}_{Fe}+L_r{L}_{ls}+L_{me}{L}_{lr})}{{\mathrm{\πL}}_{me}^2{R}_{Fe}{\mathrm{\ψ}}_{dr}}\\ i_{qFe}=({\mathrm{\ω}}_r+\frac{{\mathrm{\τR}}_{re}{L}_{r}F}{{\mathrm{\πL}}_{me}{\mathrm{\ψ}}_{dr}^2})\frac{L_s}{L_{me}{R}_{Fe}}{\mathrm{\ψ}}_{dr}+\frac{{\mathrm{\τL}}_{r}F(L_r{L}_{ls}+L_{me}{l}_{lr})}{{\mathrm{\πL}}_{me}^2{R}_{Fe}{\mathrm{\ψ}}_{dr}}\\ i_{qr}=-\frac{\mathrm{\τ}}{\mathrm{\π}}\frac{L_r}{L_{me}}\frac{F}{{\mathrm{\ψ}}_{dr}}\\ i_{dr}=0\end{array}\right.---\left(31\right)$

wherein L is_me、R_re、L_s、L_rRespectively equivalent excitation inductance, equivalent secondary resistance, equivalent primary inductance, equivalent secondary inductance, its definition is respectively:

$\{\begin{array}{c}{L}_{me}=K_x{C}_x{L}_m\\ R_{re}=K_r{C}_r{R}_r\\ L_s=L_{ls}+K_x{C}_x{L}_m\\ L_r=L_{ls}+K_x{C}_x{L}_m\end{array}---\left(32\right)$

substituting (31) into (10), simplifying and sorting to obtain a general expression of the loss function:

$P_{loss}=a_0+a_1{\mathrm{\ψ}}_{dr}^2+a_2{\mathrm{\ψ}}_{dr}^{-2}+a_3{\mathrm{\ψ}}_{dr}^{-4}+a_4{\mathrm{\ψ}}_{dr}^{-6}---\left(33\right)$

wherein the loss factor a₀、a₁、a₂、a₃And a₄Is defined as follows:

$a_0=\frac{{\mathrm{\ω}}_r\mathrm{\τ}F}{{\mathrm{\πL}}_{me}^6{R}_{Fe}}\left\{\begin{array}{c}2L_{ls}^2{L}_{lr}{L}_{me}^3{R}_{Fe}{R}_{re}+2L_{ls}^2{L}_{lr}{L}_{me}^3{R}_s{R}_{re}+4L_{ls}{L}_{me}^4{R}_{Fe}{R}_{re}+4L_{ls}{L}_{lr}{L}_{me}^4{R}_s{R}_{re}\\ +2L_{lr}{L}_{me}^5{R}_{Fe}{R}_{re}+2L_{lr}{L}_{me}^5{R}_s{R}_{re}+2L_{ls}^2{L}_{me}^4{R}_{Fe}{R}_{re}+2L_{ls}^2{L}_{me}^4{R}_s{R}_{re}\\ +4L_{ls}{L}_{me}^5{R}_{Fe}{R}_{re}+4L_{ls}{L}_{me}^5{R}_s{R}_{re}+2L_{me}^6{R}_{Fe}{R}_{re}+2L_{me}^6{R}_s{R}_{Fe}+2L_{me}^6{R}_s{R}_{re}\end{array}\right\}---\left(34\right)$

$a_1=\frac{1}{L_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}{L}_{ls}^2{L}_{me}^4{R}_{Fe}{\mathrm{\ω}}_r^2+L_{ls}^2{L}_{me}^4{R}_{Fe}+L_{ls}^2{L}_{me}^4{R}_s{\mathrm{\ω}}_r^2+L_{ls}^2{L}_{me}^4{R}_s+2L_{ls}{L}_{me}^5{R}_{Fe}{\mathrm{\ω}}_r^2\\ +2L_{ls}{L}_{me}^5{R}_{Fe}+2L_{ls}{L}_{me}^5{R}_s{\mathrm{\ω}}_r^2+2L_{ls}{L}_{me}^5{R}_s+2L_{ls}{L}_{me}^4{R}_s{R}_{Fe}\\ +L_{me}^6{R}_{Fe}{\mathrm{\ω}}_r^2+L_{me}^2+L_{me}^6{R}_{Fe}+L_{me}^6{R}_s{\mathrm{\ω}}_r^2+L_{me}^6{R}_s+L_{me}^5{R}_s{R}_{Fe}+L_{me}^4{R}_s{R}_{Fe}^2\end{array}\right\}---\left(35\right)$

$a_2=\frac{{\mathrm{\τ}}^2{F}^2}{{\mathrm{\π}}^2{R}_{Fe}^2{L}_{me}^6}\left\{\begin{array}{c}{L}_{ls}^2{L}_{lr}^4{L}_{me}^2(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}{L}_{lr}^4{L}_{me}^3(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}{L}_{lr}^4{L}_{me}^2{R}_s{R}_{Fe}\\ +L_{lr}^4{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{lr}^4{L}_{me}^3{R}_s{R}_{Fe}+L_{lr}^4{L}_{me}^2{R}_s{R}_{Fe}^2+4L_{ls}^2{L}_{lr}^3{L}_{me}^3(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +6L_{ls}{L}_{lr}^3{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+8L_{ls}{L}_{lr}^3{L}_{me}^3{R}_s{R}_{Fe}+2L_{lr}^3{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +6L_{lr}^3{L}_{me}^4{R}_s{R}_{Fe}+4L_{lr}^3{L}_{me}^3{R}_s{R}_{Fe}^2+6L_{ls}^2{L}_{lr}^2{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+L_{ls}^2{L}_{lr}^2{L}_{me}^2{R}_{re}^2(R_s+R_{Fe})\\ +6L_{ls}{L}_{lr}^2{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+12L_{ls}{L}_{lr}^2{L}_{me}^4{R}_s{R}_{Fe}+2L_{ls}{L}_{lr}^2{L}_{me}^3{R}_{re}^2(R_s+R_{Fe})+6L_{lr}^2{L}_{me}^5{R}_s{R}_{Fe}\\ +L_{lr}^2{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+L_{lr}^2{L}_{me}^4(R_{Fe}^2{R}_{re}+6R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)\\ +4L_{ls}^2{L}_{lr}{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}^2{L}_{lr}{L}_{me}^3{R}_{re}^2(R_s+R_{Fe})+2L_{ls}{L}_{lr}{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +8L_{ls}{L}_{lr}{L}_{me}^5{R}_s{R}_{Fe}+4L_{ls}{L}_{lr}{L}_{me}^4{R}_{re}^2(R_s+R_{Fe})+2L_{lr}{L}_{me}^6{R}_s{R}_{Fe}{R}_s{R}_{Fe}+L_{ls}^2{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +2L_{lr}{L}_{me}^5(R_{Fe}^2{R}_{re}+2R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)+L_{ls}^2{L}_{me}^4{R}_{re}^2(R_s+R_{Fe})+2L_{ls}{L}_{me}^6{R}_s{R}_{Fe}\\ +2L_{ls}{L}_{me}^5{R}_{re}^2(R_s+R_{Fe})+L_{me}^6(R_{Fe}^2{R}_{re}+R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)\end{array}\right\}---\left(36\right)$

$a_3=\frac{(R_s+R_{Fe})R_{re}{\mathrm{\ω}}_r{\mathrm{\τ}}^3{F}^3}{{\mathrm{\π}}^3{L}_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}2L_{ls}^2{L}_{Lr}^5{L}_{me}+4L_{ls}{L}_{lr}^5{L}_{me}^2+2L_{lr}^5{L}_{me}^3+10L_{ls}^2{L}_{Lr}^4{L}_{me}^2+16L_{ls}{L}_{lr}^4{L}_{me}^3\\ +6L_{lr}^4{L}_{me}^4+20L_{ls}^2{L}_{Lr}^3{L}_{me}^3+6L_{lr}^3{L}_{me}^5+24L_{ls}{L}_{lr}^3{L}_{me}^4+20L_{ls}^2{L}_{Lr}^2{L}_{me}^4\\ +16L_{ls}{L}_{lr}^2{L}_{me}^5+2L_{lr}^2{L}_{me}^6+10L_{ls}^2{L}_{lr}{L}_{me}^5+4L_{ls}{L}_{lr}{L}_{me}^6+2L_{ls}^2{L}_{me}^6\end{array}\right\}---\left(37\right)$

$a_4=\frac{(R_s+R_{Fe})R_{re}^2{\mathrm{\τ}}^4{F}^4}{{\mathrm{\π}}^{47}{L}_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}{L}_{ls}^2{L}_{lr}^6+2L_{ls}{L}_{lr}^6{L}_{me}+L_{lr}^6{L}_{me}^2+6L_{ls}^2{L}_{lr}^5{L}_{me}+10L_{ls}{L}_{lr}^5{L}_{me}^2+4L_{lr}^5{L}_{me}^3\\ +15L_{ls}^2{L}_{lr}^4{L}_{me}^2+20L_{ls}{L}_{lr}^4{L}_{me}^3+6L_{lr}^4{L}_{me}^4+20L_{ls}^2{L}_{lr}^3{L}_{me}^3+20L_{ls}{L}_{lr}^3{L}_{me}^4+4L_{lr}^3{L}_{me}^5\\ +15L_{ls}^2{L}_{lr}^2{L}_{me}^4+10L_{ls}{L}_{lr}^2{L}_{me}^5+L_{lr}^2{L}_{me}^6+6L_{ls}^2{L}_{lr}{L}_{me}^5+2L_{ls}{L}_{lr}{L}_{me}^6+L_{ls}^2{L}_{me}^6\end{array}\right\}---\left(38\right)$

4. efficiency optimization control strategy for linear induction motor

First and second derivatives are taken for (33), respectively:

$\frac{{\mathrm{dP}}_{loss}}{{\mathrm{d\ψ}}_{dr}}=2a_1{\mathrm{\ψ}}_{dr}-2a_2{\mathrm{\ψ}}_{dr}^{-3}-4a_3{\mathrm{\ψ}}_{dr}^{-5}-6a_4{\mathrm{\ψ}}_{dr}^{-7}---\left(39\right)$

$\frac{d^2{P}_{loss}}{{{\mathrm{d\ψ}}_{dr}}^2}=2a_1+6a_2{\mathrm{\ψ}}_{dr}^{-4}+20a_3{\mathrm{\ψ}}_{dr}^{-6}+42a_4{\mathrm{\ψ}}_{dr}^{-8}---\left(40\right)$

due to the fact that $\begin{array}{cc}\∀F>0,& \∀{\mathrm{\ω}}_r>0\end{array},$ Is invariably provided with

a_i＞0(i＝0,1,2,3,4)(41)

Therefore toThe formula (40) is constantly positive, i.e.

$\frac{d^2{P}_{loss}}{{{\mathrm{d\ψ}}_{dr}}^2}>0---\left(42\right)$

Thus, if the first derivative of (33) has a zero point in the real domain, the zero point must be the (33) extreme point, and the minimum point can be determined according to (42), which is the optimal flux linkage value corresponding to the minimum loss. Let (39) equal zero, i.e.

$\frac{{\mathrm{dP}}_{loss}}{{\mathrm{d\ψ}}_{dr}}=0---\left(43\right)$

The following proves toWhen psi_drAt > 0, a positive real solution always exists for equation (43), and this solution is unique.

Due to the fact thatThus, it is possible to provideIs a monotone increasing functionAnd (4) counting. At the same time due to

$\underset{{\mathrm{\ψ}}_{dr}\→0^{+}}{\lim }\frac{{\mathrm{dP}}_{loss}}{{\mathrm{d\ψ}}_{dr}}=\underset{{\mathrm{\ψ}}_{dr}\→0^{+}}{\lim }(-2a_2{\mathrm{\ψ}}_{dr}^{-3}-4a_3{\mathrm{\ψ}}_{dr}^{-5}-6a_4{\mathrm{\ψ}}_{dr}^{-7})=-\mathrm{\∞}---\left(44\right)$

$\underset{{\mathrm{\ψ}}_{dr}\→\mathrm{\∞}}{\lim }\frac{{\mathrm{dP}}_{loss}}{{\mathrm{d\ψ}}_{dr}}=\underset{{\mathrm{\ψ}}_{dr}\→\mathrm{\∞}}{\lim }\left(2a_1{\mathrm{\ψ}}_{dr}\right)=+\mathrm{\∞}---\left(45\right)$

It can be known thatThere is a unique zero at (0, + ∞), i.e., there is a unique positive real solution for equation (43). This positive real solution can be solved as:

${\mathrm{\ψ}}_{dr}=\sqrt{\frac{1}{2}\sqrt{\frac{2a_2}{3a_1}+\mathrm{\Δ}}+\frac{1}{2}\sqrt{\frac{4a_2}{3a_1}-\mathrm{\Δ}-\frac{\frac{16a_3}{a_1}}{4\sqrt{\frac{2a_2}{3a_1}}+\mathrm{\Δ}}}}---\left(46\right)$

wherein,

$\mathrm{\Δ}=\frac{\sqrt[3]{2}{\mathrm{\Δ}}_1}{3a_1\sqrt[3]{{\mathrm{\Δ}}_2+\sqrt{-4{\mathrm{\Δ}}_1^3+{\mathrm{\Δ}}_2^2}}}+\frac{\sqrt[3]{{\mathrm{\Δ}}_2+\sqrt{-4{\mathrm{\Δ}}_1^3+{\mathrm{\Δ}}_2^2}}}{3\sqrt[3]{2}{a}_1}---\left(47\right)$

$\{\begin{array}{c}{\mathrm{\Δ}}_1=a_2^2-36a_1{a}_4\\ {\mathrm{\Δ}}_2=-2a_2^3+108a_1{a}_3^2-216a_1{a}_2{a}_4\end{array}---\left(48\right)$

from equation (31), the primary d-axis current control amount for generating the optimal flux linkage value is:

$i_{ds}^{*}=\frac{L_s+R_{Fe}}{L_{me}{R}_{Fe}}{\mathrm{\ψ}}_{dr}-\[{\mathrm{\ω}}_r+\frac{{\mathrm{\τR}}_{re}{L}_{r}F}{{\mathrm{\πL}}_{me}{{\mathrm{\ψ}}_{dr}}^2}\]\frac{{\mathrm{\τL}}_{r}F(L_r{L}_{ls}+L_{me}{L}_{lr})}{{\mathrm{\πL}}_{me}^2{R}_{Fe}{\mathrm{\ψ}}_{dr}}---\left(49\right)$

as shown in fig. 3, the method for optimizing and controlling efficiency of a linear induction motor according to an embodiment of the present invention includes the following steps:

(1) collecting primary three-phase current i of linear induction motor_A、i_BAnd i_CAnd the motor running speed v₂According to the motor running speed v₂Calculating to obtain the secondary angular frequency omega_r。

(2) Based on direct magnetic field orientation method, using primary three-phase current i_A、i_BAnd i_CAnd secondary angular frequency ω_rCalculating to obtain the secondary flux linkage angle theta of the linear induction motor₁And an electromagnetic thrust F; from secondary flux linkage angle theta by coordinate transformation₁And primary three-phase current i_A、i_BAnd i_CObtain a primary d-axis current i_dsAnd primary q-axis current i_qs。

(3) According to secondary angular frequency ω_rAnd electromagnetic thrust F, and the primary d-axis current control amount is calculated by using the formula (49)Reference value of secondary angular frequencyWith secondary angular frequency omega_rPerforming PI regulation on the difference value to obtain a primary q-axis current control quantity

(4) Controlling the primary d-axis currentWith primary d-axis current i_dsPerforming PI regulation on the difference value to obtain primary d-axis voltage control quantityControlling the primary q-axis currentWith primary q-axis current i_qsPerforming PI regulation on the difference value to obtain primary q-axis voltage control quantity

(5) Controlling the primary d-axis voltageAnd primary q-axis voltage control quantityCoordinate transformation is carried out to obtain primary α axis voltage control quantityAnd primary β axis voltage control quantityThe Space Vector Pulse Width Modulation (SVPWM) is used as the input of Space Vector Pulse Width Modulation (SVPWM), and the output signal controls the inverter to drive the linear induction motor to operate, so that the efficiency of the linear induction motor is optimally controlled.

FIG. 4 is a graph showing the comparison between efficiency optimization control and the efficiency of conventional vector control of a linear induction motor under different loads and different operating speeds by using the method of the present invention. It can be seen that under the traditional vector control, the motor efficiency is increased along with the increase of the load, the efficiency is low under light load, and the efficiency is only 37% at the speed of 5m/s and the thrust of 20N. By adopting the efficiency optimization control method, the efficiency of the motor can be maintained at a relatively high level, the efficiency can be close to the efficiency under the rated working condition under different loads and different speeds, the efficiency is 52% at the speed of 5m/s and the thrust of 20N, and the efficiency is improved by 15% compared with the traditional control mode. Therefore, the efficiency optimization control method can obviously improve the motor efficiency in the global range, especially under the condition of light load.

It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (4)

1. The efficiency optimization control method of the linear induction motor is characterized by comprising the following steps:

(1) collecting primary three-phase current i of linear induction motor_A、i_BAnd i_CAnd the motor running speed v₂According to the motor running speed v₂Calculating to obtain the secondary angular frequency omega_r；

(2) From primary three-phase currents i_A、i_BAnd i_CAnd secondary angular frequency ω_rCalculating to obtain the secondary flux linkage angle theta of the linear induction motor₁And an electromagnetic thrust F; from secondary flux linkage angle theta by coordinate transformation₁And primary three-phase current i_A、i_BAnd i_CObtain a primary d-axis current i_dsAnd primary q-axis current i_qs；

(3) According to secondary angular frequency ω_rAnd electromagnetic thrust F, and calculating to obtain the optimized secondary d-axis flux linkage psi_drFurther obtain the primary d-axis current control quantityReference value of secondary angular frequencyWith secondary angular frequency omega_rPerforming PI regulation on the difference value to obtain a primary q-axis current control quantity

Wherein the optimized secondary d-axis flux linkage ${\mathrm{\ψ}}_{dr}=\sqrt{\frac{1}{2}\sqrt{\frac{2a_2}{3a_1}+\mathrm{\Δ}}+\frac{1}{2}\sqrt{\frac{4a_2}{3a_1}-\mathrm{\Δ}-\frac{\frac{16a_3}{a_1}}{4\sqrt{\frac{2a_2}{3a_1}}+\mathrm{\Δ}}}},$ $\mathrm{\Δ}=\frac{\sqrt[3]{2}{\mathrm{\Δ}}_1}{3a_1\sqrt[3]{{\mathrm{\Δ}}_2+\sqrt{-4{\mathrm{\Δ}}_1^3+{\mathrm{\Δ}}_2^2}}}+\frac{\sqrt[3]{{\mathrm{\Δ}}_2+\sqrt{-4{\mathrm{\Δ}}_1^3+{\mathrm{\Δ}}_2^2}}}{3\sqrt[3]{2}{a}_1},$ ${\mathrm{\Δ}}_1=a_2^2-36a_1{a}_4,$ a₁、a₂、a₃And a₄To make the total loss of the linear induction motor be a loss factor $P_{loss}=a_0+a_1{\mathrm{\ψ}}_{dr}^2+a_2{\mathrm{\ψ}}_{dr}^{-2}+a_3{\mathrm{\ψ}}_{dr}^{-4}+a_4{\mathrm{\ψ}}_{dr}^{-6}$ Minimum, a₀Is the loss factor;

(4) controlling the primary d-axis currentWith primary d-axis current i_dsPerforming PI regulation on the difference value to obtain primary d-axis voltage control quantityControlling the primary q-axis currentWith primary q-axis current i_qsPerforming PI regulation on the difference value ofObtain primary q-axis voltage control quantity

(5) Controlling the primary d-axis voltageAnd primary q-axis voltage control quantityCoordinate transformation is carried out to obtain primary α axis voltage control quantityAnd primary β axis voltage control quantityThe space vector pulse width modulation input is used as the input of space vector pulse width modulation, and the optimization control of the efficiency of the linear induction motor is realized.

2. The method of claim 1, wherein the loss factor a is₁、a₂、a₃And a₄Respectively as follows:

$a_1=\frac{1}{L_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}{L}_{ls}^2{L}_{me}^4{R}_{Fe}{\mathrm{\ω}}_r^2+L_{ls}^2{L}_{me}^4{R}_{Fe}+L_{ls}^2{L}_{me}^4{R}_s{\mathrm{\ω}}_r^2+L_{ls}^2{L}_{me}^4{R}_s+2L_{ls}{L}_{me}^5{R}_{Fe}{\mathrm{\ω}}_r^2\\ +2L_{ls}{L}_{me}^5{R}_{Fe}+2L_{ls}{L}_{me}^5{R}_s{\mathrm{\ω}}_r^2+2L_{ls}{L}_{me}^5{R}_s+2L_{ls}{L}_{me}^4{R}_s{R}_{Fe}\\ +L_{me}^6{R}_{Fe}{\mathrm{\ω}}_r^2+L_{me}^6{R}_{Fe}+L_{me}^6{R}_s{\mathrm{\ω}}_r^2+L_{me}^6{R}_s+L_{me}^5{R}_s{R}_{Fe}+L_{me}^4{R}_s{R}_{Fe}^2\end{array}\right\},$

$a_2=\frac{{\mathrm{\τ}}^2{F}^2}{{\mathrm{\π}}^2{R}_{Fe}^2{L}_{me}^6}\left\{\begin{array}{c}{L}_{ls}^2{L}_{lr}^4{L}_{me}^2(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}{L}_{lr}^4{L}_{me}^3(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}{L}_{lr}^4{L}_{me}^2{R}_s{R}_{Fe}\\ +L_{lr}^4{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{lr}^4{L}_{me}^3{R}_s{R}_{Fe}+L_{lr}^4{L}_{me}^2{R}_s{R}_{Fe}^2+4L_{ls}^2{L}_{lr}^3{L}_{me}^3(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +6L_{ls}{L}_{lr}^3{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+8L_{ls}{L}_{lr}^3{L}_{me}^3{R}_s{R}_{Fe}+2L_{lr}^3{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +6L_{lr}^3{L}_{me}^4{R}_s{R}_{Fe}+4L_{lr}^3{L}_{me}^3{R}_s{R}_{Fe}^2+6L_{ls}^2{L}_{lr}^2{L}_{me}^4(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+L_{ls}^2{L}_{lr}^2{L}_{me}^2{R}_{re}^2(R_s+R_{Fe})\\ +6L_{ls}{L}_{lr}^2{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+12L_{ls}{L}_{lr}^2{L}_{me}^4{R}_s{R}_{Fe}+2L_{ls}{L}_{lr}^2{L}_{me}^3{R}_{re}^2(R_s+R_{Fe})+6L_{lr}^2{L}_{me}^5{R}_s{R}_{Fe}\\ +L_{lr}^2{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+L_{lr}^2{L}_{me}^4(R_{Fe}^2{R}_{re}+6R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)\\ +4L_{ls}^2{L}_{lr}{L}_{me}^5(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)+2L_{ls}^2{L}_{lr}{L}_{me}^3{R}_{re}^2(R_s+R_{Fe})+2L_{ls}{L}_{lr}{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +8L_{ls}{L}_{lr}{L}_{me}^5{R}_s{R}_{Fe}+4L_{ls}{L}_{lr}{L}_{me}^4{R}_{re}^2(R_s+R_{Fe})+2L_{lr}{L}_{me}^6{R}_s{R}_{Fe}+L_{ls}^2{L}_{me}^6(R_s+R_{Fe})({\mathrm{\ω}}_r^2+1)\\ +2L_{lr}{L}_{me}^5(R_{Fe}^2{R}_{re}+2R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)+L_{ls}^2{L}_{me}^4{R}_{re}^2(R_s+R_{Fe})+2L_{ls}{L}_{me}^6{R}_s{R}_{Fe}\\ +2L_{ls}{L}_{me}^5{R}_{re}^2(R_s+R_{Fe})+L_{me}^6(R_{Fe}^2{R}_{re}+R_s{R}_{Fe}^2+R_{Fe}{R}_{re}^2+2R_s{R}_{Fe}{R}_{re}+R_s{R}_{re}^2)\end{array}\right\},$

$a_3=\frac{(R_s+R_{Fe})R_{re}{\mathrm{\ω}}_r{\mathrm{\τ}}^3{F}^3}{{\mathrm{\π}}^3{L}_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}2L_{ls}^2{L}_{Lr}^5{L}_{me}+4L_{ls}{L}_{lr}^5{L}_{me}^2+2L_{lr}^5{L}_{me}^3+10L_{ls}^2{L}_{Lr}^4{L}_{me}^2+16L_{ls}{L}_{lr}^4{L}_{me}^3\\ +6L_{lr}^4{L}_{me}^4+20L_{ls}^2{L}_{me}^3+6L_{lr}^3{L}_{me}^5+24L_{ls}{L}_{lr}^3{L}_{me}^4+20L_{ls}^2{L}_{Lr}^2{L}_{me}^4\\ +16L_{ls}{L}_{lr}^2{L}_{me}^5+2L_{lr}^2{L}_{me}^6+10L_{ls}^2{L}_{lr}{L}_{me}^5+4L_{ls}{L}_{lr}{L}_{me}^6+2L_{ls}^2{L}_{me}^6\end{array}\right\}$ and

$a_4=\frac{(R_s+R_{Fe})R_{re}^2{\mathrm{\τ}}^4{F}^4}{{\mathrm{\π}}^4{l}_{me}^6{R}_{Fe}^2}\left\{\begin{array}{c}{L}_{ls}^2{L}_{lr}^6+2L_{ls}{L}_{lr}^6{L}_{me}+L_{lr}^6{L}_{me}^2+6L_{ls}^2{L}_{lr}^5{l}_{me}+10L_{ls}{L}_{lr}^5{L}_{me}^2+4L_{lr}^5{L}_{me}^3\\ +15L_{ls}^2{L}_{lr}^4{L}_{me}^2+20L_{ls}{L}_{lr}^4{L}_{me}^3+6L_{lr}^4{L}_{me}^4+20L_{ls}^2{L}_{lr}^3{L}_{me}^3+20L_{ls}{L}_{lr}^3{L}_{me}^4+4L_{lr}^3{L}_{me}^5\\ +15L_{ls}^2{L}_{lr}^2{L}_{me}^4+10L_{ls}{L}_{lr}^2{L}_{me}^5+L_{lr}^2{L}_{me}^6+6L_{ls}^2{L}_{lr}{L}_{me}^5+2L_{ls}{L}_{lr}{L}_{me}^6+L_{ls}^2{L}_{me}^6\end{array}\right\},$

wherein tau is polar distance, F is electromagnetic thrust, omega_rFor secondary angular frequency, R_sIs a primary resistance, R_FeIs an equivalent iron loss resistance, L_lsFor primary leakage inductance, L_lrFor secondary leakage inductance, L_meFor equivalent excitation inductance, R_reIs an equivalent secondary resistance.

3. The method of claim 2, wherein the equivalent exciting inductance L is_meAnd an equivalent secondary resistance R_reRespectively as follows:

L_me＝K_xC_xL_mand

R_re＝K_rC_rR_r，

wherein, K_rCorrection factor, K, for the longitudinal side effect of the secondary resistor_xCorrection factor for longitudinal side effect of exciting inductance, C_rCorrection of coefficient for secondary resistance lateral edge effect, C_xFor correction of the coefficient of transverse edge effect of the excitation inductance, L_mFor exciting inductance, R_rIs the secondary resistance.

4. The method of optimally controlling efficiency of a linear induction motor according to any one of claims 1 to 3, wherein the primary d-axis current control amountComprises the following steps:

$i_{ds}^{*}=\frac{L_s+R_{Fe}}{L_{me}{R}_{Fe}}{\mathrm{\ψ}}_{dr}-\[{\mathrm{\ω}}_r+\frac{{\mathrm{\τR}}_{re}{L}_{r}F}{{\mathrm{\πL}}_{me}{{\mathrm{\ψ}}_{dr}}^2}\]\frac{{\mathrm{\τL}}_{r}F(L_r{L}_{ls}+L_{me}{L}_{lr})}{{\mathrm{\πL}}_{me}^2{R}_{Fe}{\mathrm{\ψ}}_{dr}},$

wherein L is_sIs equivalent to primary inductance, R_FeIs an equivalent iron loss resistance, L_meFor equivalent excitation inductance, τ is the pole pitch, R_reIs an equivalent secondary resistance, L_rEquivalent secondary inductance, F electromagnetic thrust, L_lsFor primary leakage inductance, L_lrIs the secondary leakage inductance.