The previous research (Deng. et al., 2019), as shown in Table 1, a π model for long cables was established for the first time, which compensates for voltage and current deviations through coordinate transformation, thereby improving the starting torque. The simulation and experiment of 50 Hz no-load and sudden load start-up are carried out. There is no quantitative analysis of the THD metric and no stability proof. Specifically, it didn’t add the function to the system’s transfer function, didn’t prove its stability, didn’t analyze the parameters of the long cable that introduces time sensitivity, and didn't perform any field test and verification under actual deep-sea conditions.

Reference (Tao et al., 2024) proposed a novel super-twisting algorithm-based sliding mode observer for the multi-vector model-free predictive control (STASMO-MVMPC) method, applied to permanent magnet synchronous motors, which improved control performance. Reference (Liu et al., 2025) by reducing the number of voltage vectors and calculating the operating time of each level based on volt-second balance, the voltage balance issue of the inverter is resolved, offering the advantages of simplicity and low loss. Reference (Zhu et al., 2024) proposes an improved decoupling control method combining the grey wolf optimization algorithm and LSSVM, with mathematical model derivation conducted. The proposed algorithm demonstrates superior performance. Reference (Qiao et al., 2025) proposes a multi-vector direct model predictive torque control algorithm, which optimizes the issue of difficulty in guaranteeing the optimal characteristics of voltage vectors. Comparisons with recent control strategies demonstrate its excellent stability and dynamic response. Reference (Dursun, 2023) proposed a Runge-Kutta (RK) algorithm, featuring a robust mathematical structure, that addresses the high-dimensional issues of asynchronous motors by employing control algorithms such as FDB-RK, RK, etc. Reference (Adamczyk and Orlowska-Kowalska, 2024) established an observer-based vector control algorithm for induction motor control. Reference (Boukhili et al., 2022) proposes a current sensing disturbance control algorithm to address the critical importance of current in vector control, requiring a method to detect current loss. The proposed control algorithm demonstrates effective detection performance and improves motor and generator performance. Reference (Feng et al., 2024) to address the issue of reduced accuracy in LIM speed identification caused by system disturbances, a speed observation method based on the Coleman filter is proposed, which enhances the effectiveness of the LIM algorithm. Reference (Qin et al., 2015) introduced an improved PID controller aimed at enhancing acceleration performance, and its effectiveness was verified through comparisons with other controllers. As shown in Table 1, recent studies (Tao et al., 2024) have improved the dynamic response of multi-vector control. The simulation results show that the proposed method has better performance than the general algorithm in dynamic speed, THD, and torque fluctuation. The THD metric is analyzed quantitatively, but it is an open-loop stability hypothesis (Dursun 2023). proposed a Runge-Kutta (RK) control algorithm, which can induce the high-dimensional problem of motor tracking through mathematical methods. The developed RK algorithm was tested and verified on the CEC17 benchmark problems in a 30-dimensional search space. The reference was statistically analyzed, and it puts forward the problem of the unquantified “robustness under parameter perturbation”.

The principle of the routine vector control algorithm involves detecting the three-phase output voltage and current of the VFD, which are then transformed into two-phase voltage and current in the d-q coordinate system via a Park transformation. The speed and flux estimation unit estimates the speed and flux of the induction motor. The two-phase current is fed back through the rotating park transformation to the stator voltage generation unit. Finally, these signals generate PWM waves to control the VFD, achieving closed-loop control of the induction motor.

As shown in Table 1, this paper constructed a mathematical model of the entire system by developing models for long cables and filters, by calculating the voltage and current on the induction motor side from the voltage and current on the VFD’s side, to achieve its adaptability. Specifically, the transfer function of the entire system under the long-cable vector control algorithm is derived, proposing a stability proof based on Lyapunov functions, and the error transfer function is formulated and subjected to stability verification. The results demonstrate that the stability constant is greater than 0, indicating the system is asymptotically stable. Then, the Lyapunov functions and functions of the long-cable model with time-sensitivity introduced are proposed and analyzed. To further validate the correctness of algorithmic stability and quantify performance improvement percentages, simulations were conducted on rotor current, torque, and rotor speed under various starting conditions: no-load startup, loaded startup, and instantaneous loading at different frequencies, no-load startup with the introduction of time-sensitivity parameters for long cable, and then, experimental verifications on no-load startup, sudden load, and sudden unload were carried out. By comparing the original algorithm with the improved version, it is demonstrated that the improved vector control algorithm, under long cable conditions, simulating parasitic inductance, parasitic capacitance, parasitic resistance, improved torque, speed, and current waveforms, enabling smoother operation, and has certain adaptability to time-sensitivity parameters. The theoretical value of the method proposed in this paper is that it goes from “Experience Optimization” to “Strict Stability.” The value of experiment and simulation is that the dynamic performance is improved and the stability is increased. The value of the THD metric is to reduce harmonic distortion and improve dynamic response performance. On the proof of stability, a new solution is provided.

Additional FFT simulations were performed for 10 Hz, 30 Hz, and 50 Hz no-load operating. The THD measurements reveal:

1. For 10 Hz: Pre-improvement THD was 8.28%, post-improvement 5.41%.

THD frequencies 10 Hz, 30 Hz, and 50 Hz dropped by 2.87%, 3.35%, and 0.04%, which are crucial for induction motors. Cable capacitive reactance increases at low frequencies (The capacitance value is C = 0.29 μF/km), easily triggering harmonic oscillations. However, the improved algorithm prevents shutdowns caused by inadequate heat dissipation; if not resolved, induction motor repairs require production stoppages exceeding 30 days.

The improved control algorithm significantly reduces THD in operating current.

Chapter 2 of this paper presents the models of long cables and filters, Chapter 3 details the improved vector control algorithm, Chapter 4 covers simulations and experiments, and Chapter 5 provides the conclusions.

The basic idea of the vector control algorithm oriented by the rotor flux is to obtain an equivalent DC motor model in the synchronous rotating orthogonal reference coordinate system through an appropriate coordinate transformation. By analogy with DC motor control methods, the electromagnetic torque T _e and the magnitude of the rotor flux ψ _r are controlled independently. Then, the control quantities in the rotating orthogonal coordinate system are inverse-transformed to obtain the corresponding control quantities in the three-phase coordinate system for implementation of control. The motor model after rotor flux orientation is as follows:

The speed sensorless control technology adopted in this paper is based on the Model Reference Adaptive System (MRAS) of rotor flux linkage. This method uses the current model as the adjustable model and the voltage model as the reference model. The deviation between the outputs of the two models forms an adaptive system to correct the speed in the adjustable model, where the output value of the adaptive regulator becomes the observed speed value. The system equations are as follows:

The conventional speed sensorless vector control algorithm for power supply systems is a speed closed-loop control system. Through coordinate transformation, speed and flux observers, it achieves speed estimation, decoupling of electromagnetic torque and rotor flux, as well as closed-loop control of electromagnetic torque, rotor flux, speed, and current.

However, in this control algorithm, the estimation of speed ω, the magnitude of rotor flux ψ _r , rotor phase angle θ, and the calculation of electromagnetic torque T _e are all based on the measured three-phase voltage and current signals output by the inverter. The measured three-phase voltage and current signals from the inverter side are directly treated as the motor’s input three-phase voltage and current signals and applied to the feedforward decoupling and motor stator voltage generation unit shown in Section 3.1.1, as well as the MRAS-based speed estimation and flux calculation unit shown in Section 3.1.2. The influence of long cable distributed parameters on the motor-side three-phase voltage and current signals is completely disregarded.

To address the influence of sinusoidal wave filters and distributed parameters of long cables, this paper proposes a speed sensorless vector control algorithm for the power supply system based on Lyapunov stability theory, utilizing a comprehensive mathematical model of sinusoidal wave filters and the two-port characteristics of long cables. This algorithm is primarily designed for motor loads connected via long cables, and its control system block diagram is shown in Figure 8. The proposed control algorithm employs the three-phase voltage and current signals output by the inverter to calculate the precise motor terminal voltage and current signals using the comprehensive mathematical model of sinusoidal wave filters and long cables in the α-β stationary reference frame. These calculated motor terminal signals are then used to estimate the speed, rotor flux amplitude, and rotor flux phase angle of the induction motor, rather than directly using the measured inverter output signals. This approach enables the accurate construction of a rotor flux linkage model, achieving precise observation of rotor flux linkage and speed identification, as well as closed-loop control of torque, flux, and current.

As shown in Figure 8, the improved speed sensorless vector control algorithm independently controls the electromagnetic torque Te and the rotor flux amplitude ψ _r . To obtain an accurate mathematical model for the improved algorithm, the motor terminal stator voltage components u _{s_d} and u _{s_q} , the motor terminal stator current components i _{s_d} and i _{s_q} , and the rotor flux amplitude ψ _{r_r} , calculated in the d-q rotating reference frame based on the comprehensive mathematical model of sinusoidal wave filters and long cables, must be substituted into the rotor flux-oriented motor model (Equations 33, 34). This yields an accurate dynamic mathematical model of the rotor flux-oriented induction motor based on the comprehensive mathematical model of sinusoidal wave filters and long cables. i s _ d · = β i s _ d + ω 1 i s _ q + L m σ L s L r T r ψ r _ r + 1 σ L s u s _ d (43) i s _ q · = − ω 1 i s _ d + β i s _ q − L m σ L s L r ω ψ r _ r + 1 σ L s u s _ q (44)

In the equation β = − R s L r 2 + R r L m 2 σ L s L r 2 ,T _r is motor rotor time.

The rotor flux linkage function can be expressed as: ψ r r · = − 1 T r ψ r r + L m T r i s d (45)

The electromagnetic torque equation can be expressed as: T e = 3 n p L m 2 L r ψ r r i s q (46)

The dynamic function of the motor is: x · = J x + g · u s d q (47)

The state variables in the transfer function are: x = i s d i s q ψ r r T (48)

The input transfer function is: u s d q = u s d u s q T (49)

There is g = 1 σ L s 0 0 0 1 σ L s 0 T (50) J x = β ω 1 L m σ L s L r T r − ω 1 β L m ω σ L s T r L m T r 0 − 1 T r i s d i s q ψ r r (51)

By analyzing the stability of this transfer function, an improved equation for long cable conditions can be derived. Substituting the voltages u _{s_α} and u _{s_β} and the currents i _{s_α} and i _{s_β} into Equation 37 yields: ψ r α = L r L m ∫ u s α − R s i s α d t − σ L s i s α ψ r β = L r L m ∫ u s β − R s i s β d t − σ L s i s β (52)

In the equation, ψ _{r_α} and ψ _{r_β} are the two-phase magnetic fluxes calculated from the output voltage and current of the VFD. ψ r r = ψ r α 2 + ψ r β 2 (53)

In the equation, ψ _{r_r} is the calculated amplitude of the magnetic flux linkage. From this, the transfer function after the Park transformation can be derived. u s d i s d = cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ 1 + Z L Z C − Z L − Z L − 2 Z C Z C 2 1 + Z L Z C l c 1 − Z F − 1 Z R 1 + Z F Z R u V F D α i V F D α (54) u s q i s q = cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ 1 + Z L Z C − Z L − Z L − 2 Z C Z C 2 1 + Z L Z C l c 1 − Z F − 1 Z R 1 + Z F Z R u V F D β i V F D β (55)

Equations 54, 55 are the output voltage and current of the filter model and the long cable model in the d-q coordinate system. By incorporating them into the vector control algorithm, the response characteristics of the vector control algorithm under long cable conditions can be improved. From this, the motor stator voltages u _{s_d} and u _{s_q}, the motor stator currents i _{s_d} and i _{s_q}, and the motor rotor flux linkage ψ _{r_r} can be obtained. Thus, the parameter model is given as: K 1 K 2 K 3 K 4 = cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ 1 + Z L Z C − Z L − Z L − 2 Z C Z C 2 1 + Z L Z C l c 1 − Z F − 1 Z R 1 + Z F Z R (56)

Then Equations 54, 55 are transformed into: u s d i s d = K 1 K 2 K 3 K 4 u V F D α i V F D α (57) u s q i s q = K 1 K 2 K 3 K 4 u V F D β i V F D β (58)

As shown in Figure 8, the output of the control algorithm is torque and rotor flux linkage: y 1 = T e y 2 = ψ r r (59)

Define the deviation between torque and rotor flux linkage as: e 1 = T * e − T e e 2 = ψ r r * − ψ r r (60)

In Equation 60, e ₁ and e ₂ represent the errors of torque and rotor flux linkage, respectively, where T e * , ψ r r * denote the desired torque and desired rotor flux linkage. Thus, we have: T e * · = 0 (61) ψ r r * · = 0 (62)

The input-output relationship of the control system shown in Figure 8 is: e · = e · 1 e · 2 = J 1 x J 2 x + g 1 g 2 u s d u s q (63)

Differentiating the torque and rotor flux deviation shown in Equation 60 yields: e · 1 = T e * · − T e · (64) e · 2 = ψ r r * · − ψ r r · (65)

Substituting Equations 46 and 61 and Equations 45 and 62 into Equations 64, 65 respectively, we can calculate: e · 1 = − 3 n p L m 2 L r ψ r r i s q · (66) e · 2 = 1 T r ψ r r − L m T r i s d (67)

Substituting Equation 44 into Equation 66, the torque deviation differential is calculated as: e · 1 = 3 n p L m 2 L r ψ r r ω 1 i s d − β i s q + L m σ L s L r ω ψ r r − 1 σ L s u s q (68)

Substituting the torque and rotor flux deviation differentials from Equations 67, 68 into Equation 69 yields the input-output relationship as: e · = e · 1 e · 2 = 3 n p L m ψ r r 2 L r ω 1 i s d − β i s q + ω L m ψ r r σ L s L r 1 T r ψ r r − L m T r i s d + 0 − 3 n p L m ψ r r 2 σ L s L r 0 0 u s d u s q (69)

In Equation 69, the voltages in the d-q coordinate system are u _{s_d} , u _{s_q,} and the motor stator currents are i _{s_d} , i _{s_q} . The model calculated through Equations 57, 58 is the final model, while the motor rotor flux linkage ψ _{r_r} is calculated by Equation 53. By substituting the motor voltages, currents, and rotor flux linkage into Equations 67, 68, we obtain: e · 1 = 3 n p L m ψ r r 2 L r K 4 ω 1 i V F D α − β K 4 i V F D β − K 2 σ L s i V F D β + ω L m ψ r r σ L s L r + 3 n p L m ψ r r 2 L r K 3 ω 1 − 3 n p L m ψ r r 2 L r β K 3 + K 1 σ L s u V F D α u V F D β (70) e · 2 = − K 4 L m T r i V F D α + 1 T r ψ r r − K 3 L m T r u V F D α (71)

Substituting Equations 70, 71 into the input-output function mentioned earlier, namely, Equation 69, we obtain J x = J 1 x J 2 x = 3 n p L m ψ r r 2 L r K 4 ω 1 i V F D α − β K 4 i V F D β − K 2 σ L s i V F D β + ω L m ψ r r σ L s L r − K 4 L m T r i V F D α + 1 T r ψ r r (72) g x = g 1 g 2 = 3 n p L m ψ r r 2 L r K 3 ω 1 − 3 n p L m ψ r r 2 L r β K 3 + K 1 σ L s − K 3 L m T r 0 (73)

Then the input-output relationship shown in Figure 8 becomes: e · = e · 1 e · 2 = J 1 x J 2 x + g x u V F D α u V F D β (74)

The motor stator voltages u _{s_d}, u _{s_q} are converted into the motor stator voltages u _{VFD_α,} u _{VFD_β} . From Equation 74, it can be concluded that the effects of long cables and filters can be eliminated.

The matrix g(x) in Equation 75 is derived from the product of the motor flux linkage and the rotor flux linkage. Therefore, g(x) is a non-singular matrix and is not equal to zero, leading to the input as: u = u V F D α u V F D β = g − 1 x − J 1 x − J 2 x + u 1 u 2 (75)

Where u ₁ and u ₂ are adjustment input values, and to ensure favorable characteristics, there exists: u 1 u 2 = − K 5 0 0 − K 6 ξ 1 ξ 2 (76)

Substituting Equation 75 and Equation 76 into Equation 74 yields e · = e · 1 e · 2 = − K 5 0 0 − K 6 ξ 1 ξ 2 (77)

K ₅ and K ₆ both are select appropriate constants, choose the Lyapunov function as: V ξ = 1 2 ξ T ξ > 0 (78)

In the equation ξ T = ξ 1 ξ 2 T (79)

The derivative of the selected Lyapunov function V(ξ) with respect to time is: V · ξ = − K 5 ξ 1 2 − K 6 ξ 2 2 (80)

Since K ₅ and K ₆ are positive numbers, V · ξ <0, which proves that the improved vector control algorithm proposed in this paper is asymptotically stable.

Suppose P(t) describes the change of R, L, C, over time t. This model is affected by multiple deep-sea environmental factors x 1 t , x 2 t , ⋯ , x n t , can be expressed as: P t = f x 1 t , x 2 t , ⋯ , x n t , t (81)

To analyze the sensitivity of the proposed method to the time-varying cable parameters, we introduce the sensitivity coefficient S P , x i , which represents the sensitivity of parameter P to the environmental factor x i . The sensitivity coefficient can be defined by partial derivatives: S P , x i = ∂ P ∂ x i · x i P (82)

By calculating the sensitivity coefficients, we can determine which environmental factors have the most significant impact on cable parameters.

To consider the influence of time factors, we can further calculate the rate of change of the sensitivity coefficient concerning time d S P , x i : d S P , x i d t = d d t ∂ P ∂ x i · x i P (83)

Specifically, let the time-varying parameters R, C, and L of the long cable in the deep-sea environment be R(t), C(t), and L(t), respectively. In the deep-sea environment, factors such as temperature, pressure, and humidity can affect the resistivity of the long cable conductor, thereby influencing the resistance. Assuming that the change in resistance over time is related to the temperature T(t), pressure P(t), and humidity H(t), Equations 84–92 can be derived. R t = R 0 + S R , T T t + S R , P P t + S R , H H t (84)

Where R 0 is the initial resistance, S R , T , S R , P , S R , H are the coefficients of the influence of temperature, pressure, and humidity on time, respectively.

The capacitance of a long cable is related to the dielectric constant of the insulating material, and the dielectric constant is affected by temperature and humidity. Equation 85 can be derived. C t = C 0 + S C , T T t + S C , H H t (85)

Where C 0 is the initial capacitance, S C , T , S C , H are the coefficients of the influence of temperature and humidity on capacitance respectively.

Inductance is related to the geometric structure and magnetic permeability of the cable. In the deep-sea environment, pressure may cause slight changes in the cable’s geometric structure, thus affecting the inductance. L t = L 0 + S L , P P t (86)

Where L 0 is the initial inductance, S L , P is the coefficient of the influence of pressure on inductance.

Sensitivity refers to the ratio of the rate of change of a parameter to the rate of change of an influencing factor. The sensitivity of resistance coefficient S R , T of resistance to temperature is defined as: S R , T = d R t R t d T t T t = T t R t d R t d T t (87)

The sensitivity coefficient of resistance to pressure S R , P is defined as: S R , P = P t R t d R t d P t (88)

The sensitivity coefficient of resistance to humidity S R , H is defined as: S R , H = H t R t d R t d H t (89)

Similarly, the sensitivity coefficient of capacitance to temperature S C , T is defined as: S C , T = T t C t d C t d T t (90)

The sensitivity coefficient of capacitance to humidity S C , H is defined as: S C , H = H t C t d C t d H t (91)

The sensitivity coefficient of inductance to pressure S L , P is defined as: S L , P = P t L t d L t d P t (92)

By calculating these parameters, the magnitude and severity of the impacts of various conditions on the R, C, and L of the long cable can be determined.

Substituting the changing parameters R(t), C(t), and L(t) into the long cable Equations 22, 23, we can obtain: Z L t = R t + j ω L t (93) Z C t = 1 j ω C t (94)

Substitute it into Equations 27–29, It can be concluded that the long cable model varying with time is as follows: u m _ u i m _ u = 1 + Z L t Z C t − Z L t − Z L t − 2 Z C t Z C 2 t 1 + Z L t Z C t l c u f _ u i f _ u (95) u m _ v i m _ v = 1 + Z L t Z C t − Z L t − Z L t − 2 Z C t Z C 2 t 1 + Z L t Z C t l c u f _ v i f _ v (96) u m _ w i m _ w = 1 + Z L t Z C t − Z L t − Z L t − 2 Z C t Z C 2 t 1 + Z L t Z C t l c u f _ w i f _ w (97)

Then, Equations 54, 55 become: u s d i s d = cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ 1 + Z L t Z C t − Z L t − Z L t − 2 Z C t Z C 2 t 1 + Z L t Z C t l c 1 − Z F − 1 Z R 1 + Z F Z R u V F D α i V F D α (98) u s q i s q = cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ 1 + Z L t Z C t − Z L t − Z L t − 2 Z C t Z C 2 t 1 + Z L t Z C t l c 1 − Z F − 1 Z R 1 + Z F Z R u V F D β i V F D β (99)