Introduction

The electric submersible pump (ESP) is an important production device for oil exploitation in deep-sea oil fields, particularly applicated for the fluids that contain minimal amounts of gas [1]. In the actual production of deep-sea oil fields, this device is installed at the bottom of subsea wells up to 7000 m deep for oil extraction, and is driven by the inverter via long transmission cable of ranging from several kilometers to tens of kilometers [2]. The inverter can control the ESP to a speed well that better producing the well and keeping the ESP within its best efficiency range, it also offers soft motor starting for ESP systems and reduces impact to power grid during the ESP systems start-up [3]. Hence, the inverter is widely used in oil field facilities, some of latest facilities could have up to 80% loads supplied power by Inverter.

The connection between inverter and the ESP through the long submarine cable incurs some problems [4, 5], mainly long cable distribution parameters [2] which caused changes in the inverter output voltage, current and phase, introduce harmonics into the DC-link current. Moreover, the long cable impedance voltage drop which decreased the starting electromagnetic torque, increases the rise time and causes starting problems, especially in case with high-inertia motor loads [6]. The voltage drop across the long cable impedance increases speed dip and speed recovery time, when sudden load was applied. The voltage regulation needs to be increased and the inverter modulation index is apt to be saturated with the increase of the cable length [7]. These problems limit the application of the inverter, which making the various control algorithms of the inverter, especially the vector control algorithm, unable to operate effectively. For speed sensor-less vector control algorithm, the flux and velocity estimation technique are prerequisite [8, 9]. The observers which are used for IM flux and velocity estimation require the exact motor terminal voltage/current [10]. That direct measuring motor terminal voltage/current signals may be too costly or too difficult to achieve for long motor subsea cable drives due to subsea cable length or safety/environmental precautions. Hence, the motor terminal voltage/current must be calculated from the inverter side.

There are some control algorithms used for remote operation of inverter via a long motor subsea cable, such as the stability controller for open-loop operation which can be used for motor derived with an output filter and transformer in oil pump applications [11], the open loop current regulated sensor-less control scheme for an induction and permanent magnetic motor derived with output sinewave filter and transformer which provides sufficient starting torque with controlled drive/motor current [12], the robust voltage vector control strategy for a PMSM driving a high-power ESP [13]. These methods are to improve the conventional scalar control algorithm to control the motor drives. However, there is no literature on the research of the speed sensor-less vector control algorithm for IM based on long cable.

In this paper, the ISSVCA, needed for long motor cable, was proposed. The presented ISSVCA was based on the exact long cable model, which was the motor stator voltage and current could be calculated from the inverter output voltage/current using α − β model for the long cable in the stationary reference frame, and the correct command voltage could be given in current control cycle which was based on the state equation of the IM in the synchronous rotating reference frame. An experimental setup was implemented with a 1-km cable model. Practical aspects, such as cable selection and inverter limitations, were considered in the experimental setup to match a scaled-down actual inverter. Simulation in addition to experimental results verified the correctness and effectiveness of the proposed ISSVCA.

This paper is organized into six sections. Following the introduction in Sect. 1, speed sensor-less vector control technique is discussed in Sect. 2. In Sect. 3, the long motor subsea cable model and remote motor voltage/current calculator are illustrated. The proposed ISSVCA for IM is presented in Sect. 4. Simulation and experimental results are illustrated in Sect. 5. Finally, the conclusion forms Sect. 6.

Speed Sensorless Vector Control Technique

IM Model

The fundamental operation principle of the rotor field-oriented vector control (FOC) [14] is to obtain equivalent DC motor model in a synchronous rotating reference frame through an appropriate coordinate transformation. The torque Te and rotor flux ψr are controlled independently of each other following the control method of the DC motor, and then the control amounts in the synchronous rotating reference frame are inversely transformed to obtain the corresponding amounts of the three-phase coordinate system to implement the control. The motor model after rotor field-oriented [15] is as follows:

(1) Stator voltage equation

$$\left\{ {\begin{array}{l} {u_{sd} = R_{s} i_{sd} + p\sigma L_{s} i_{sd} - \sigma L_{s} \omega_{1} i_{sq} + p\frac{{L_{m} }}{{L_{r} }}\psi_{r} } \\ {u_{sq} = R_{s} i_{sq} + p\sigma L_{s} i_{sq} + \sigma L_{s} \omega_{1} i_{sd} + \frac{{L_{m} }}{{L_{r} }}\omega_{1} \psi_{r} } \\ \end{array} } \right.$$
(1)

where usd, \(u_{sq}\) are motor stator voltage components in the d − q rotating reference frame (V). isd, \(i_{sq}\) are motor stator current components in the d − q rotating reference frame (A). Rs, p, σ, Ls, ω1, Lr, Lm, ψr are motor stator resistance (Ω), differential operator, leakage coefficient, motor stator inductance (H), synchronous angular velocity (r/min), motor rotor inductance (H), motor mutual inductance (H), motor rotor flux linkage amplitude, respectively.

(2) Torque equation

$$T_{e} = \frac{{3n_{p} L_{m} }}{{2L_{r} }}\psi_{r} i_{sq}$$
(2)

where np, \(T_{e}\) are motor number of pole pairs, motor developed torque (N m), respectively.

(3) Decoupled voltage equation

$$\left\{\begin{array}{l}{u_{sd}^{**} = - \omega_{1} \sigma L_{s} i_{sq}^{*} }\\ {u_{sq}^{**} = \omega_{1} \sigma L_{s} i_{sd}^{*} + \omega_{1} \frac{{L_{m} }}{{L_{r} }}\psi_{r}^{*} } \\ \end{array} \right.$$
(3)
$$\omega_{1} = p\theta$$
(4)

where \(u_{sd}^{**}\), \(u_{sq}^{**}\) are coupling voltages (V). \(i_{sd}^{*}\), \(i_{sq}^{*}\) are motor stator reference current components in the \(d - q\) rotating reference frame (A). \(\psi_{r}^{*}\), \(\theta\) are motor rotor reference flux linkage amplitude, rotor flux phase angle, respectively.

The Model Reference Adaptive System Based on Rotor Flux

The model reference adaptive system (MRAS) [16] based on rotor flux, which is using the voltage model as a reference model and using the current model as an adjustable model. The error from cross multiplication of the voltage and current model outputs is used to generate the estimated speed using a PI controller. The system equations are as follows:

(1) Stator model (reference model)

$$\left\{ {\begin{array}{*{20}c} {\psi_{{{\text{r}}\upalpha}} = \frac{{L_{r} }}{{L_{m} }}\left[ {\int {(u_{s\alpha } - R_{s} i_{s\alpha } )dt - \sigma L_{s} i_{s\alpha } } } \right]} \\ {\psi_{{{\text{r}}\upbeta}} = \frac{{L_{r} }}{{L_{m} }}\left[ {\int {(u_{s\beta } - R_{s} i_{s\beta } )dt - \sigma L_{s} i_{s\beta } } } \right]} \\ \end{array} } \right.$$
(5)
$$\psi_{\text{r}} = \sqrt {\psi_{r\alpha }^{2} + \psi_{r\beta }^{2} }$$
(6)
$${\uptheta } = \sin^{ - 1} \frac{{\psi_{{{\text{r}}\upbeta}} }}{{\psi_{{{\text{r}}\upalpha }} }}$$
(7)

where \(\psi_{r\alpha }\), \(\psi_{r\beta }\) are motor rotor flux linkage components in the \(\alpha - \beta\) stationary reference frame (Wb). \(u_{s\alpha }\), \(u_{s\beta }\) are motor stator voltage components in the \(\alpha - \beta\) stationary reference frame (V). \(i_{s\alpha }\), \(i_{s\beta }\) are motor stator current components in the \(\alpha - \beta\) stationary reference frame (A).

(2) Rotor model (adjusted model)

$$\left\{ {\begin{array}{*{20}c} {\psi_{\text{ra}}^{r} = \int {\left( { - \frac{{R_{r} }}{{L_{r} }}\psi_{\text{ra}}^{r} - \omega \psi_{{{\text{r}}\upbeta}}^{r} + L_{m} \frac{{R_{r} }}{{L_{r} }}i_{s\alpha } } \right)dt} } \\ {\psi_{{{\text{r}}\upbeta }}^{r} = \int {\left( { - \frac{{R_{r} }}{{L_{r} }}\psi_{{{\text{r}}\upbeta}}^{r} + \omega \psi_{\text{ra}}^{r} + L_{m} \frac{{R_{r} }}{{L_{r} }}i_{s\beta } } \right)dt} } \\ \end{array} } \right.$$
(8)

where \(\psi_{r\alpha }^{r}\), \(\psi_{r\beta }^{r}\) are motor rotor flux linkage components in the \(\alpha - \beta\) stationary reference frame (generated from a rotor model based on the estimated speed)(Wb). \(\omega\), \(R_{r}\) are estimate motor angular speed (r/min), motor rotor resistance (Ω), respectively.

(3) Error equation

$${\text{e}}_{\text{speed}} = \psi_{{{\text{r}}\upbeta }} \psi_{\text{ra}}^{\text{r}} - \psi_{{{\text{r}}\upalpha }} \psi_{{{\text{r}}\upbeta }}^{\text{r}} .$$
(9)

(4) Estimated speed calculation

$$\omega = \left( {K_{p\_est} + \frac{{K_{i\_est} }}{s}} \right)e_{speed}$$
(10)

where \({\text{K}}_{{{\text{p}}\_{\text{est}}}}\), \({\text{K}}_{{{\text{i}}\_{\text{est}}}}\) are motor speed PI controller constants, respectively.

Figure 1 shows the control block diagram of feed-forward decoupling and motor stator voltages generated unit [15], where the coupling voltages \(u_{sd}^{**}\) and \(u_{sq}^{**}\) are generated according to the given motor stator d-axis current component \(i_{sd}^{*}\), q-axis current component \(i_{sq}^{*}\), and rotor flux linkage \(\psi_{r}^{*}\). Figure 2 shows the block diagram for MRAS-based speed estimation and flux calculation, where the estimated rotor angular speed \(\upomega\), rotor flux linkage amplitude \(\psi_{r}\) and the rotor phase angle \(\uptheta\) are calculated from the measured the motor voltage/current signals. The estimated motor speed, flux linkage amplitude and rotor phase angle are used in the ISSVCA as shown in the following sections.

Fig. 1
figure 1

Feed-forward decoupling and motor stator voltages generated unit

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Fig. 2
figure 2

MRAS-based speed estimation and flux calculation unit

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Figure 3 shows the conventional speed sensor-less vector control algorithm (CSSVCA) [17, 18] of IM based on long cable. This algorithm is a speed closed-loop control system. The aims are to achieve speed estimation, torque and flux decoupling through coordinate transformation, flux and speed observers. These give independent control of the flux and torque during both the steady state and dynamic condition.

Fig. 3
figure 3

Conventional speed sensorless vector control algorithm of IM based on long cable

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Long Cable Modeling and Calculation of Motor Stator Voltage and Phase Current

The analysis of the problems associated with long motor cable drives needs reliable modeling of the complete drive system, especially the long subsea cable. Where frequency-domain analysis is a powerful tool in long cable modeling [19]. Lumped R, L and C elements can represent the long cable effectively, even for transient analysis. Applying this modeling technique is simple and does not reduce the accuracy [20,21,22], this model is sufficient to give accurate results on long motor cable terminations [5]. Therefore, in this paper, a π-network with lumped elements R, L and C was used to model the long motor cable and output filter. Figure 4 shows the long cable and filter π-network model of inverter IM drive system. The simulated long motor cable was a subsea umbilical, which was also a PWM special insulation. Table 1 lists the parameters of this long cable.

Fig. 4
figure 4

Long cable and filter \(\pi\) network model of inverter induction motor drive system

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Table 1 Cable parameters
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The two-port \(\pi\) network model can be used to represent each phase model of the sinewave filter and long cable. Figure 5a shows the sinewave filter two-port impedance \(\pi\) network model. Figure 6a shows the long cable two-port impedance π-network model. As a balanced three-phase network [23, 24], the each phase parameters of the sinewave filter model and long cable model shown in Fig. 4 are equal, and can be transformed to the \(\alpha - \beta\) stationary reference frame as shown in Figs. 5b, c and 6b, c. There are:

$$L_{f\_\alpha } = L_{f\_\beta } = L_{f\_u} = L_{f\_v} = L_{f\_w} = L_{f}$$
(11)
$$C_{f\_\alpha } = C_{f\_\beta } = C_{f\_u} = C_{f\_v} = C_{f\_w} = C_{f}$$
(12)
$$R_{c\_\alpha } = R_{c\_\beta } = R_{c\_u} = R_{c\_v} = R_{c\_w} = R_{c}$$
(13)
$$L_{c\_\alpha } = L_{c\_\beta } = L_{c\_u} = L_{c\_v} = L_{c\_w} = L_{c}$$
(14)
$$C_{c\_\alpha } = C_{c\_\beta } = C_{c\_u} = C_{c\_v} = C_{c\_w} = C_{c}$$
(15)

where \(L_{f\_\alpha } , L_{f\_\beta }\) are filter inductances per phase in the \(\alpha - \beta\) stationary reference frame (H). \(L_{f\_u} , L_{f\_v} ,L_{f\_w}\) are filter inductances per phase (H). \(C_{f\_\alpha } , C_{f\_\beta }\) are filter capacitances per phase in the \(\alpha - \beta\) stationary reference frame (F). \(C_{f\_u} , C_{f\_v} ,C_{f\_w}\) are filter capacitances per phase (F). \(R_{c\_\alpha } , R_{c\_\beta }\) are cable resistances per phase in the \(\alpha - \beta\) stationary reference frame (Ω). \(R_{c\_u} ,R_{c\_v} ,R_{c\_w}\) are cable resistances per phase (Ω). \(L_{c\_\alpha } , L_{c\_\beta }\) are cable inductances per phase in the \(\alpha - \beta\) stationary reference frame (H). \(L_{c\_u} , L_{c\_v} ,L_{c\_w}\) are cable inductances per phase (H). \(C_{c\_\alpha } ,C_{c\_\beta }\) are cable capacitances per phase in the \(\alpha - \beta\) stationary reference frame (F).

Fig. 5
figure 5

Sinewave filter π-network model

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Fig. 6
figure 6

Long cable π-network model

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\(C_{c\_u} , C_{c\_v} ,C_{c\_w}\) are cable capacitances per phase (F).

Thanks to the inverter with filter is nearly sinusoidal output, the voltage/current are as well nearly sinusoidal. Figures 5 and 6 show the functional relationship between the voltage/current of the two-port π-network model and the model distribution parameters. It is not necessary to perform derivative calculation at the fundamental operating frequency, and by replacing the Laplace transform variable s with \(j\omega_{1}\), it is solved as the linear equations, without the derivative process, which simplified the calculation. Calculation equation can be expressed by

(1) Sinewave filter parameters function equations at both ends

$$\left[ {\begin{array}{*{20}c} {u_{f\_uvw} } \\ {i_{f\_uvw} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & { - Z_{F} } \\ { - \frac{1}{{Z_{R} }}} & {1 + \frac{{Z_{F} }}{{Z_{R} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{VFD\_uvw} } \\ {i_{VFD\_uvw} } \\ \end{array} } \right]$$
(16)

where

$$Z_{F} = j\omega_{1} L_{f}$$
(17)
$$Z_{R} = \frac{1}{{j\omega_{1} C_{f} }}$$
(18)

where \(u_{f\_uvw}\), \(i_{f\_uvw}\), \(u_{VFD\_uvw}\), \(i_{VFD\_uvw}\) are filter output three-phase voltage, filter output three-phase current, inverter output three-phase voltage, and inverter output three-phase current, respectively.

(2) Long cable parameters function equations at both ends

$$\left[ {\begin{array}{*{20}c} {u_{m\_uvw} } \\ {i_{m\_uvw} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {1 + \frac{{Z_{L} }}{{Z_{C} }}} & { - Z_{L} } \\ {\frac{{ - Z_{L} - 2Z_{C} }}{{Z_{C}^{2} }}} & {1 + \frac{{Z_{L} }}{{Z_{C} }}} \\ \end{array} } \right]^{{l_{c} }} \left[ {\begin{array}{*{20}c} {u_{f\_uvw} } \\ {i_{f\_uvw} } \\ \end{array} } \right]$$
(19)

where

$$Z_{L} = R_{C} + j\omega_{1} L_{C}$$
(20)
$$Z_{C} = \frac{1}{{j\omega_{1} C_{C} }}$$
(21)

where \(u_{m\_uvw}\), \(i_{m\_uvw}\), \(l_{c}\) are motor three-phase voltage, motor three-phase current, and cable length (km), respectively.

The inductive reactance and capacitive reactance of the unit length (km) of the cable are given by (20) and (21). For different cable lengths (\(l_{c}\)), by putting \(l_{c}\) into (19), the functional relationship values between the parameters at both ends of this cable length can be calculate.

By putting (16) into (19), the functional relationship between voltage/current of the motor terminal and output voltage/current of the inverter can be derived as

$$\left[ {\begin{array}{*{20}c} {u_{m\_uvw} } \\ {i_{m\_uvw} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {1 + \frac{{Z_{L} }}{{Z_{C} }}} & { - Z_{L} } \\ {\frac{{ - Z_{L} - 2Z_{C} }}{{Z_{C}^{2} }}} & {1 + \frac{{Z_{L} }}{{Z_{C} }}} \\ \end{array} } \right]^{{l_{c} }} \left[ {\begin{array}{*{20}c} 1 & { - Z_{F} } \\ { - \frac{1}{{Z_{R} }}} & {1 + \frac{{Z_{F} }}{{Z_{R} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{VFD\_uvw} } \\ {i_{VFD\_uvw} } \\ \end{array} } \right].$$
(22)

The equations in the \(\alpha - \beta\) stationary reference frame are expressed as

$$\left[ {\begin{array}{*{20}c} {u_{s\_\alpha } } \\ {i_{s\_\alpha } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {1 + \frac{{Z_{L} }}{{Z_{C} }}} & { - Z_{L} } \\ {\frac{{ - Z_{L} - 2Z_{C} }}{{Z_{C}^{2} }}} & {1 + \frac{{Z_{L} }}{{Z_{C} }}} \\ \end{array} } \right]^{{l_{c} }} \left[ {\begin{array}{*{20}c} 1 & { - Z_{F} } \\ { - \frac{1}{{Z_{R} }}} & {1 + \frac{{Z_{F} }}{{Z_{R} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{VFD\_\alpha } } \\ {i_{VFD\_\alpha } } \\ \end{array} } \right]$$
(23)
$$\left[ {\begin{array}{*{20}c} {u_{s\_\beta } } \\ {i_{s\_\beta } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {1 + \frac{{Z_{L} }}{{Z_{C} }}} & { - Z_{L} } \\ {\frac{{ - Z_{L} - 2Z_{C} }}{{Z_{C}^{2} }}} & {1 + \frac{{Z_{L} }}{{Z_{C} }}} \\ \end{array} } \right]^{{l_{c} }} \left[ {\begin{array}{*{20}c} 1 & { - Z_{F} } \\ { - \frac{1}{{Z_{R} }}} & {1 + \frac{{Z_{F} }}{{Z_{R} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{VFD\_\beta } } \\ {i_{VFD\_\beta } } \\ \end{array} } \right]$$
(24)

where \(u_{s\_\alpha } , u_{s\_\beta }\) are calculated motor voltage components from the inverter output voltage/current using a \(\alpha - \beta\) model for the long cable in the stationary reference frame (V). \(i_{s\_\alpha } , i_{s\_\beta }\) are calculated motor current components from the inverter output voltage/current using a \(\alpha - \beta\) model for the long cable in the stationary reference frame (A). \(u_{VFD\_\alpha } , u_{VFD\_\beta }\) are inverter voltage components in the \(\alpha - \beta\) stationary reference frame (V). \(i_{VFD\_\alpha } , i_{VFD\_\beta }\) are inverter current components in the \(\alpha - \beta\) stationary reference frame (A).

The actual motor terminal voltage and current can be calculated from the inverter side by (23) and (24).

ISSVCA of IM Based on Long Cable

The proposed ISSVCA was designed for a motor drive with long cable connection, which was commonly utilized for ESP drives, and the block diagram of algorithm is shown in Fig. 7. The proposed technique was capable of calculating the motor terminal voltage and current from the inverter output voltage/current signals using a \(\alpha - \beta\) model for the long cable in the stationary reference frame. The exact motor terminal voltage/current were calculated to estimate rotor angular speed, rotor flux amplitude and rotor flux phase angle, instead of directly using the measured inverter output voltage/current signals. So that the accurate rotor flux linkage model was constructed to realize the observation of the rotor flux, accurate velocity identification, and closed-loop control of the torque, flux and current.

Fig. 7
figure 7

ISSVCA of IM based on long cable

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Simulation and Experimental Results

Computer simulation and experiments were performed with the proposed ISSVCA for IM based on long cable. The experimental setup shown in Fig. 8 was used to emulate an long cable drive system, which typically has an inverter, motor unit, sinewave filer and cable model. The parameters of the three-phase IM and sinewave filter used in simulation and experiments are shown in Tables 2 and 3, and the cable model length is 1 km.

Fig. 8
figure 8

Experimental setup

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Table 2 IM parameters
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Table 3 Sinewave filter parameters
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Figure 9 shows simulation and experimental no-load response results for motor speed and current with a 1-km cable. Figure 9a, c show simulation and experimental no-load start waveforms for motor speed and current with the CSSVCA. Figure 9b, d show simulation and experimental no-load start waveforms for motor speed and current with the ISSVCA. The motor reaches the rated speed from zero speed in 3 s. The simulation and experimental results in Fig. 9 show that the ISSVCA improves the starting electromagnetic torque and increases the dynamic response.

Fig. 9
figure 9

Simulation and experimental no-load response results for motor speed and current with a 1 km cable

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Figure 10 shows experimental sudden load response results for motor speed and current with a 1-km cable. Figure 10b shows that the ISSVCA enables the motor to restore the speed to 1460 rpm in less than 500 ms, when a sudden rated load is applied. Figure 10a shows that the CSSVCA increases the time to restore the reference speed and causes the speed dip, when the rated load is applied.

Fig. 10
figure 10

Experimental sudden load response results for motor speed and current with a 1 km cable

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Figure 11 shows that the long cable introduces harmonics into the DC-link current as the first element experienced by the inverter is the stray capacitance of the long cable model. This capacitance acts as a short circuit to each inverter voltage pulse. Figure 11 shows that use of the ISSVCA can decrease the harmonics in DC-link current, avoid harmonics in DC-link current which caused problems such as radiated EMI, inaccurate current measurement and false protection system operation, and increase the DC-link capacitor lifetime.

Fig. 11
figure 11

DC-link current experimental results for motor operation with a 1 km cable

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Conclusion

In this paper, the proposed ISSVCA of IM based on long cable that could be effectively applied to long cable-connected motor loads. This algorithm was fast and accurate in flux and velocity identification, simple in algorithm, less in computation, easy to implement online.

Simulation and experimental results showed that the ISSVCA could control the reliable and stable operation of the motor, provided sufficient starting electromagnetic torque, and starting current of the motor, made the motor start smoothly, reduced the harmonics in DC-link current, increased the DC-link capacitor lifetime. And it has good dynamic and static characteristics, strong engineering practicality and application promotion value.