Keywords

1 Introduction

In the era of new infrastructure, the cornerstone industry of engineering construction confronts novel demands and challenges [1, 2]. As a vital component of engineering construction, engineering machinery must also align with the imperative for green and low-carbon solutions by transitioning toward electrification [3]. Within the journey toward the comprehensive electrification of construction machinery, the permanent magnet synchronous motor (PMSM) emerges as a pivotal subject of research, with its related investigations progressively delving deeper. As part of the electrification of construction machinery, the reliability of PMSM puts forward new requirements, and the researchers [4] use the fast digital twin system constructed in Simulink to construct a PMSM monitoring system, and use it as a basis for determining the remaining structural life of the permanent magnet. PMSM can suffer from undervoltage during sudden changes in torque, which reduces the dynamic performance of the motor. Researchers [5] proposed a predictive trajectory control strategy with improved dynamic performance, combining trajectory optimisation with a dynamic overmodulation algorithm, verifying that the proposed control scheme has good performance and low computational cost.

For PMSM, their control schemes such as field oriented control (FOC), sliding mode control (SMC) and model predictive control (MPC) are hotspots in current research [6,7,8]. Among them, the MPC algorithm has attracted much attention because of its good dynamic performance, high adaptability and multi-objective optimization. In the recent past, researchers [9] proposed a novel control strategy based on MPC algorithm and used it for distributed drive electric vehicles, which will be possibly applied to construction machinery in the next step.

Within the context of the MPC algorithm for PMSM, a pivotal element is the cost function, serving to designate the algorithm’s controlled entity and its respective weightage. To advance the exploration of the MPC algorithm, this study centers its attention on investigating the cost function linked to dq-axis current and its associated weight factor. Experimental results encompassing dq-axis current, motor output torque and speed, and three-phase current are also provided. Facilitating a quantitative analysis, this study constructs a novel evaluation function employing current ripple and current harmonics. This evaluation function assesses the performance of the MPC algorithm across diverse weight factor combinations, culminating in the synthesis of multiple outcomes to identify an optimal weight factor composition. The efforts in this study establish a foundational framework for the prospective widespread adoption of electric drive construction machinery.

2 Simulation Model

For PMSM, the following assumptions are made when establishing its mathematical model when it is located in a rotating coordinate system:

  1. (1)

    The eddy currents as well as hysteresis losses in the motor are neglected;

  2. (2)

    The three-phase windings are strictly symmetrical, and the waveform of the induced electromotive force present in the windings is recognized as a sinusoidal waveform;

  3. (3)

    There is no damping winding in the rotor, and the influence of external factors such as temperature on the motor is ignored.

Under the above assumptions, the current differential equation of PMSM can be obtained after finishing as:

$$ \left\{ \begin{gathered} i_{d} (k + 1) = i_{d} (k) + \frac{{T_{s} }}{{L_{d} }}\left[ {u_{d} (k) + w_{r} (k)L_{d} i_{q} (k) - R_{S} i_{d} (k)} \right] \hfill \\ i_{q} (k + 1) = i_{q} (k) + \frac{{T_{s} }}{{L_{q} }}\left[ {u_{q} (k) - w_{r} (k)L_{q} i_{d} (k) - R_{S} i_{q} (k) - w_{r} (k)\Psi_{f} } \right] \hfill \\ \end{gathered} \right. $$
(1)

In the Eq. (1), Ld and Lq are the corresponding inductances of the d and q axes; RS is the stator resistance of the motor; Ts is the sampling period; uq(k) is the torque voltage at the moment of k, which is the q-axis component of the stator terminal voltage; ud(k) is the magnetic chain voltage at the moment of k, which is the d-axis component of the stator terminal voltage; id(k) and iq(k) are the dq-axis currents at the moment of k; Ψf is the permanent magnet chain. The Eq. (1) is used to build the mathematical model of PMSM. For MPC algorithm, the cost function also needs to be constructed to complete the control closed loop. In the algorithm, the cost function is utilized to constrain the output variables to obtain the switching state of the inverter at the next moment. The traditional cost function is generally the sum of squares of the dq-axis current error, which can be written as:

$$ \left\{ \begin{gathered} e_{{i_{d} }} (k + 1) = i_{d} (k + 1) - i_{d}^{ref} \hfill \\ e_{{i_{q} }} (k + 1) = i_{q} (k + 1) - i_{q}^{ref} \hfill \\ f = e_{{i_{d} }}^{2} (k + 1) + e_{{i_{q} }}^{2} (k + 1) \hfill \\ \end{gathered} \right. $$
(2)

In Eq. (2), \({\text{e}}_{{{\text{i}}_{{\text{d}}} }} ({\text{k}}\, + \,{1})\) and \({\text{e}}_{{{\text{i}}_{{\text{q}}} }} ({\text{k}}\, + \,{1})\) are the errors between the predicted current and the set value of the dq-axis at the k + 1 moment in the discrete domain. In order to further improve the performance of MPC algorithm, this paper constructs the cost function containing the dq-axis weight factor, which is expressed as:

$$ f = \lambda_{d} e_{{i_{d} }}^{2} (k + 1) + \lambda_{q} e_{{i_{q} }}^{2} (k + 1) $$
(3)

Differing from the conventional cost function, this paper introduces specific weight factors prior to squaring the dq-axis error. In particular, λd signifies the weight factor pertaining to the squared summation of chain current error, while λq denotes the weight factor linked to the squared summation of torque current error. Varied combinations of weight factors are employed to conduct simulation experiments, ultimately identifying the optimal combination of weight factors for PMSM’s MPC. In the course of the study, the paper explores seven diverse weight factor combinations within an expansive scope, facilitating experimental investigations. To streamline presentation, the designations of corresponding working conditions in Table 1 are employed to signify distinct weight factor combinations.

Table 1 Working conditions designations of weight factor combinations
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Equations (1) and (3) are employed to form a MPC system for PMSM, and the simulation system shown in Fig. 1 is built within Simulink.

Fig. 1
A simulation model of the P M S M system. The system is composed of a signal source, P I D, current disturbance calculation module, M P C, p w m, frequency counting module, rotor speed, d q axis current, and three phase current.

Simulation system

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To provide a more comprehensive demonstration of the various combinations of weight factors, a current ripple measurement module is incorporated into the simulation system. This module can be represented as follows:

$$ \vartriangle i_{RMS} = \left| {\left\{ {\frac{1}{N}\sum\limits_{k = 0}^{N - 1} {\left\{ {\left[ {i_{d} \left( k \right) - \overline{{i_{d} }} } \right]^{2} + \left[ {i_{q} \left( k \right) - \overline{{i_{q} }} } \right]^{2} } \right\}} } \right\}} \right| $$
(4)

3 Results

The MPC algorithm model for PMSM is built in Simulink, and the specific simulation parameters are shown in Table 2.

Table 2 Simulation parameters
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Simulation experiments are carried out for the working conditions corresponding to different weighting factors, and the results obtained are shown below. In the simulation system, PMSM is accelerated to 1200 rpm and decelerated to −1200 rpm at 0.2 s. In order to further verify the performance of the algorithms, the system applies an external load of 10 n*m at 0.35 s. Figure 2 shows the operation of the dq-axis current under different working conditions. It can be seen that the q-axis current fluctuates relatively less under the working conditions represented by C, D and E, while the fluctuation of the d-axis current is more obvious, which is also related to the fact that the weighting factor correspondingly enlarges the proportion of the torque current in the cost function; and the opposite is true for the working conditions corresponding to B, F and G. Taken together, only the A case can suppress all the dq-axis current fluctuations well, which proves the excellent performance of the algorithm to the A case.

Fig. 2
2 simulation graphs plot i q and i d versus time. a. It plots 7 step functions, A to G, with some fluctuations between negative 40 and 80. b. It plots 7 distortion signals, A to G, between negative 8 and 16.

Simulation results: a q-axis current b d-axis current

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For a deeper investigation into the MPC algorithm’s performance across varying weight factors, Fig. 3 displays both motor speed and torque. Analysis of the Fig. 3 results reveals that when the weighting factor for torque current is increased (cases C, D, and E), motor torque exhibits enhanced stability with reduced fluctuations arising from torque oscillations. This consistency corresponds to the performance of torque currents shown in Fig. 2. Nevertheless, the augmentation of the torque current weighting factor concurrently impacts the chain currents, resulting in minor motor speed fluctuations. Conversely, the A case exerts superior control over PMSM speed and torque fluctuations, facilitating a more balanced and consistent torque and speed output.

Fig. 3
2 simulation graphs plot n and T e versus time. a. It plots 7 trapezoidal functions, A to G, between negative 1000 and 1000. b. It plots 7 step functions, A to G, between negative 8 and 16 with respect to the time period from 0 to 0.5.

Simulation results: a motor speed b motor torque

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Building upon the preceding analysis outcomes, Fig. 4 illustrates the behavior of motor three-phase currents across distinct weight factor scenarios. Observing the figure, it becomes apparent that the A case yields a three-phase current output with reduced irregularities and a narrower fluctuation span compared to others. Particularly, at 0.35 s when an external load is imposed, all conditions exhibit a substantial upsurge in A-phase current. However, the A case generates a comparatively minor increase, implying enhanced system robustness when confronted with external perturbations. This attribute proves invaluable when constructing machinery under intricate operational settings or challenging environments.

Fig. 4
A simulation graph plots i A B C versus time. It plots 7 fluctuating curves, A to G, between negative 60 and 60. 3 functions begin along the y axis, fluctuating pattern up to 0.35 seconds, and follow the sinusoidal pattern up to 0.5 seconds.

Three-phase current operation

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Many of the aforementioned findings are derived from qualitative analyses and may lack robust persuasiveness. To provide deeper insight into the algorithm’s performance, we compute and present in Table 3 the current ripple, current harmonics, and average switching frequency across the entire simulation timeframe, encompassing a range of operational scenarios.

Table 3 Comparison of algorithm performance under different working conditions
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Table 3 demonstrates a clear trend where the A case exhibits the smallest current ripple and harmonics. This observation substantiates the algorithm’s efficacy under the A case and aligns with findings regarding three-phase currents. Regarding the switching frequency, the A case remains viable; however, a notable surge in current ripple and harmonics is evident when contrasted with the lower switching frequency linked to the A case. This escalation compromises the MPC algorithm’s efficacy and contributes to heightened losses during PMSM system operation. To provide further insight into the Table 3 outcomes, the article introduces an evaluation function M for validation, computed as follows:

$$ M = {}_{\Delta }i_{RMS} + I_{THD} $$
(5)

It’s important to emphasize that the numerical value embodied by the evaluation function M entails a level of intricacy. For now, we consider it solely as a dimensionless parameter, initially utilized for comparing the performance of the MPC algorithm across diverse working conditions. We refrain from delving into its current physical interpretation. Refer to Table 4 for its specific values.

Table 4 Comparison of evaluation functions under different working conditions
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The data presented in Table 4 reveals that the A case yields the lowest evaluation function value. While the C case results in a slightly higher evaluation function value compared to the A case, an analysis considering preceding parameters like dq-axis current fluctuation, three-phase current operation, and average switching frequency further supports the conclusion that the A case stands as the optimal weight factor combination for application in the MPC algorithm for PMSM.

4 Conclusion

This paper delves into the cost function of the MPC algorithm for PMSM. The model is built within a simulation system, exploring and contrasting various combinations of weight factors for the dq-axis current within the cost function. Beyond examining parameters such as dq-axis current, motor speed, and three-phase current, this study introduces current ripple and current harmonics to formulate a novel evaluation function. Leveraging this approach alongside the average switching frequency during algorithm execution, the study quantitatively assesses the working conditions depicted by distinct weight factors, ultimately deriving optimal performance from the amalgamation of weight factor combinations. In a comprehensive sense, this paper’s research content holds specific implications for the advancement of electric power-driven construction machinery.