Enhanced Raccoon Optimization Algorithm for PMSM Electrical Parameter Identification

1. Introduction

Permanent magnet synchronous motors (PMSMs) are widely used in industry due to their high efficiency and small size [1,2,3]. However, realizing the high-performance control for PMSMs requires accurate motor parameters. For example, model predictive control [4,5] and speed sensorless control [6,7] require accurate motor parameters as a basis. The PMSM parameter identification is categorized into online identification [8,9,10] and offline identification [11,12,13]. Online identification is performed when the motor is integrated into the control system. This method relies on the real-time response of the motor’s input and output, combining the motor control model with the identification algorithm to estimate key parameters. Online identification is often coupled with motor control techniques [14,15] to continuously update the motor parameters. In contrast, offline identification is typically performed when the motor is disconnected from the operating system. This method uses feedback data to estimate the motor’s electrical and mechanical parameters by applying a predetermined signal to the motor. Offline identification is commonly used to initialize parameters before motor startup [16], and it is widely applied in industry.

The main online identification methods are recursive least squares (RLS), model reference adaptive (MRAS), extended Kalman filtering, and high-frequency signal injection. RLS has good mathematical-statistical properties and a wide range of adaptability. In article [17], inductance, flux linkage, and resistance parameters are recognized by a recursive total least-square excitatory and inhibitory learning to update the model parameters of the velocity-free sensor. Reference [18] addresses the rank-deficiency problem in motor parameter identification using the current derivative method in combination with RLS to obtain parameter results. However, RLS requires careful consideration of the convergence speed and solution accuracy of the algorithm. In [19], inductance, resistance, and flux linkages are identified using MRAS. Ref. [20] improves the tracking accuracy of inductance parameters and enhances the precision of speed sensorless control by combining the improved Virtual Rotating Axis High-Frequency Injection (VHFSI) method with MRAS. However, the adaptive rate design process of MRAS is more complex.

In [21], system states and two key motor parameters are estimated using a dual extended Kalman filter. However, this method operates the motor under steady-state conditions, and there is a rank-deficiency issue in the parameter identification process, making it difficult to identify all electrical parameters, particularly for Interior PMSMs (IPMSMs). In [22], motor parameters are obtained by injecting high-frequency sinusoidal voltages into the stationary coordinate system (

α β

-axis) to update the model parameters in the position observer. However, high-frequency signals can interfere with motor control, and motor feedback signals are prone to distortion. Reference [23] proposes using a triangle wave injection method to address the rank-deficiency problem in parameter identification, employing RLS to extract multiple motor parameters after receiving the motor feedback signal.

Offline identification methods include finite element analysis (FEA) [24], signal injection [25], and artificial intelligence algorithms [26]. FEA can provide high-precision parameters that accurately reflect the nonlinear characteristics of the motor, but FEA is commonly used in the pre-design stage of the motor. In article [27], the effectiveness of a parameter identification scheme is verified by FEA. However, finite element analysis requires a powerful computing system and is often used when designing engines. In article [16], the amplitude-auto-adjusting signal injection (ASI) method is proposed to recognize the inductance and stator resistance in different states offline. Selecting an appropriate signal injection method is crucial for motor parameter identification, as it helps decouple the parameters in the identification model and compensates for rank deficiencies in the identification process. In this study, the issue of rank deficiency in the identification equation is addressed by injecting different currents into the d-axis. Intelligent algorithms have rapidly developed, and many types of intelligent algorithms have emerged [28]. Depending on the source of inspiration, they can be categorized as swarm-based, evolution-based, physics-based, game-rule-based, and human-social-relationship-based [29]. Artificial intelligence algorithms have outstanding abilities to solve high-dimensional functions, and scholars have applied them to the solution of motor parameters [30]. Commonly, an improved particle swarm algorithm (PSO) performs parameter identification. In article [31], a modified PSO is used to solve the motor parameter identification objective function. In article [32], the particle motion states are improved to form an interactive and dynamically learned PSO to solve for motor parameters.

In recent years, the number of swarm intelligence algorithms has steadily increased, expanding the range of methods available for parameter identification. Ref. [33] employs PSO to identify key motor parameters, enhancing the effectiveness of model predictive control. Ref. [34] uses PSO to optimize the peak cogging torque and no-load back electromotive force (EMF) amplitude, thereby improving motor control performance. Ref. [35] introduces the improved CGCRAO algorithm to enhance parameter identification accuracy, offering new insights in the field. Reference [36] takes into account the negative effects of the drive circuit, such as the dead time of the switching devices, the time delay of the low-pass filter, and angle deviation, to improve the accuracy of parameter identification results.

ROA [37] has a better solution at lower population sizes. In article [38], ROA reduces the scheduling time compared to other algorithms. In article [39], an algorithm based on ROA is implemented for precise localization to improve the accuracy of range measurements. In article [40], ROA is used to significantly improve job scheduling. By improving ROA, motor parameters can be more accurately identified, yielding new insights. To ensure the algorithm meets the requirements of parameter identification, this paper introduces three improvements aimed at enhancing the accuracy and stability of the process.

In this article, the signal injection method combined with intelligent algorithms is used for PMSM parameter identification. Since the direct application of ROA does not solve the problem effectively, this article improves ROA using the adaptive exploration radius, raccoon-washing-food-inspired, and escaping-predator strategies. To validate the effectiveness of EROA, the more classical algorithms ROA, PSO [41], E_WOA [42], and GA [43] were selected for comparison using some of the IEEE CEC 2015 test sets. When compared with recent algorithms such as the Snow ablation optimizer [44] and the Walrus Optimization Algorithm [29], the newly proposed EROA reaches an advanced level of performance. EROA has a large advantage in convergence speed and solution accuracy. Parameter identification objective functions are established in simulation and motor experiments for comparison. EROA can find the motor parameters more quickly.

2. Parameter Identification of PMSM

The parameter identification method in this article is based on the voltage equations in the steady state of the PMSM. To facilitate the analysis, the motor voltage Equation (1), which ignores the effect of core saturation and losses of PMSM, is considered in the

d q

rotating coordinate system.

$\{\begin{array}{l} u_{d} = R_{s} i_{d} + L_{d} \frac{d i_{d}}{d t} - ω_{e} L_{q} i_{q} \\ u_{q} = R_{s} i_{q} + L_{q} \frac{d i_{q}}{d t} + ω_{e} L_{d} i_{d} + ψ_{f} ω_{e} \end{array}$

(1)

where $u_{d}$ and $u_{q}$ are the $d q$ -axis voltages; $i_{d}$ and $i_{q}$ are the $d q$ -axis currents; $ω_{e}$ are the electromechanical angular velocities; $R_{s}$ is the stator resistance; $L_{d}$ and $L_{q}$ are the $d q$ -axis inductances; and $ψ_{f}$ is the permanent flux linkage. Through PMSM vector control, the motor steadily operates at a certain speed. At this point, the $d q$ -axis currents of the PMSM remain stable in a very small range. It may be assumed that $d i_{d} / d t = d i_{q} / d t = 0$ . From Equation (1), we obtain Equation (2).

$\{\begin{array}{l} u_{d} = R_{s} i_{d} - ω_{e} L_{q} i_{q} \\ u_{q} = R_{s} i_{q} + ω_{e} L_{d} i_{d} + ψ_{f} ω_{e} \end{array}$

(2)

Injecting different currents into the d-axis increases the number of equations. The system of full-rank voltage equations is shown in (3), which can be used as a reference model for the motor.

$\{\begin{array}{l} u_{d 0} = - ω_{e 0} L_{q} i_{q 0} \\ u_{q 0} = R_{s} i_{q 0} + ψ_{f} ω_{e 0} \\ u_{d 1} = R_{s} i_{d 1} - ω_{e 1} L_{q} i_{q 1} \\ u_{q 1} = R_{s} i_{q 1} + ω_{e 1} L_{d} i_{d 1} + ψ_{f} ω_{e 1} \end{array}$

(3)

where $i_{d}$ = 0 A defines the formula subscript as 0, and $i_{d}$ = −1 A defines the formula subscript as 1. The adjustable model (4) is formed using the PMSM run data and the reference model.

$\{\begin{array}{l} {\hat{u}}_{d 0} = - ω_{e 0} {\hat{L}}_{q} i_{q 0} \\ {\hat{u}}_{q 0} = {\hat{R}}_{s} i_{q 0} + {\hat{ψ}}_{f} ω_{e 0} \\ {\hat{u}}_{d 1} = {\hat{R}}_{s} i_{d 1} - ω_{e 1} {\hat{L}}_{q} i_{q 1} \\ {\hat{u}}_{q 1} = {\hat{R}}_{s} i_{q 1} + ω_{e 1} {\hat{L}}_{d} i_{d 1} + {\hat{ψ}}_{f} ω_{e 1} \end{array}$

(4)

Using the sum of squares of the differences between the reference and adjustable models, we obtain (5) as a fitness function for parameter identification.

$\{\begin{cases} f_{1} ({\hat{L}}_{q}) = [u_{d 0} (k) - {\hat{u}}_{d 0} (k)]^{2} \\ f_{2} ({\hat{R}}_{s}, {\hat{ψ}}_{f}) = [u_{q 0} (k) - {\hat{u}}_{q 0} (k)]^{2} \\ f_{3} ({\hat{R}}_{s}, {\hat{L}}_{q}) = [u_{d 1} (k) - {\hat{u}}_{d 1} (k)]^{2} \\ f_{4} ({\hat{R}}_{s}, {\hat{L}}_{d}, {\hat{ψ}}_{f}) = [u_{q 1} (k) - {\hat{u}}_{q 1} (k)]^{2} \end{cases}$

(5)

When suitable motor parameters are encountered such that the sum of the four functions is 0, the PMSM parameters in (6) are found.

$F u n (p) = \sum_{i = 1}^{4} f_{i} (p), p = [{\hat{R}}_{s}, {\hat{L}}_{d}, {\hat{L}}_{q}, {\hat{ψ}}_{f}]$

(6)

$F u n (p)$ is the fitness of the parameter vector p. In this paper, this function is solved by improving the raccoon optimization algorithm, and the final parameter vector p can be obtained when the function converges to a certain range.

3. Enhanced ROA

3.1. Characteristics of ROA

ROA is mainly characterized by generating two sets of scenarios,

R_{b e s t 0}

and

V_{b e s t 0}

, in two ranges around the initial location

l o c_{0}

, namely the reachable zone radius (RZR) and the visible zone radius (VZR). We compare the fitness values of

R_{b e s t 0}

V_{b e s t 0}

, and

l o c_{0}

to obtain their best solution (7).

$F u n (l o c_{1}) = m i n [F u n (l o c_{0}), F u n (R_{b e s t}), F u n (V_{b e s t})]$

(7)

By comparing the fitness value of the new generation $l o c_{1}$ with $G_{o p t}$ , the algorithm determines whether to update the globally optimal solution $G_{o p t}$ . If no good solution is selected around $l o c_{0}$ , we infer that $l o c_{1}$ = $l o c_{0}$ , which represents no movement in the raccoon location and requires a cumulative migration factor (MF). When MF reaches a certain number of times, the $l o c_{1}$ position is forced to update. If a certain number of times is not reached, we generate a different population. We perform the iteration NI times and finally output $G_{o p t}$ . To quickly find the optimal solution, it is necessary to rationalize the migration scheme and the new position after the migration.

To improve the algorithm’s accuracy, ROA is improved using the adaptive exploration radius, raccoon-washing-food-inspired, and escaping-predator strategies to form EROA. Figure 1 shows the overall flow of EROA. In Figure 1, i is the current number of iterations;

n_{t}

is the current number of times

l o c_{i}

l o c_{0}

;

L o c_{0}

is the raccoon natural community; and

F l o c_{0}

F l o c_{1}

F R_{b e s t}

, and

F V_{b e s t}

are the values of

l o c_{0}

l o c_{1}

R_{b e s t}

, and

V_{b e s t}

for the fitness values, respectively.

3.2. Adaptive Exploration Radius Strategy

The algorithm uses the adaptive exploration radius strategy to facilitate the solution, as the initial exploration radius is a human-set initial value, but each dimension is almost unpredictable for the problem that requires solving. To improve the convergence accuracy of the algorithm, one approach is to slowly converge the exploration radius during the iteration of the algorithm. The second approach is to explore different positions so that the algorithm can be automatically adjusted to the appropriate exploration radius.

The strategy to decrease the exploration radius with the number of iterations is expressed as (8).

$\{\begin{array}{l} R Z R = (X_{m a x} - X_{m i n}) \times k & i f (i < N_{i}) \\ R Z R = (X_{m a x} - X_{m i n}) \times k \div i & i f (i \geq N_{i}) \end{array}$

(8)

where $X_{\min}$ and $X_{\max}$ constitute the exploration range, and k controls the range of RZR initial values. To adapt the radius of exploration so that the algorithm can automatically learn, we define (9) as follows:

$R_{r} = G_{o p t} - L o c (r o u n d (N o p), 1)$

(9)

where $L o c_{0}$ is the raccoon community, Nop is the set population density, $R_{r}$ is the adaptive radius updated with iterations, and round is defined to generate random positive integers. To ensure that $R_{r}$ has self-learning properties, we set (10).

$F L o c_{r} = F u n (G_{o p t} + r a n d \times R_{r})$

(10)

where $F L o c_{r}$ is the evaluated value of another raccoon that arrives at the new location under the influence of $G_{o p t}$ , and rand denotes a 0–1 uniform distribution. The adaptive updating of RZR is performed during the iteration of the algorithm, as shown in (11).

$R Z R = R_{r}, i f (F l o c_{r} < F G_{o p t})$

(11)

If $G_{o p t}$ is updated, the adaptive radii RZR and VZR are updated with $G_{o p t}$ . The adaptive exploration radius strategy allows the algorithms to exchange optimal target information during the optimization process, improving the accuracy of the algorithm solution.

3.3. Raccoon-Washing-Food-Inspired Strategy

The raccoons must find a nearby water source to wash their food. From this perspective, the behavior of raccoons searching for wetlands after capturing food is modeled. Assume that the current optimal raccoon is the earliest to find food and arrive at the water source to wash the food, and the other raccoons come relatively late. At this point, the cluster of raccoons by the water edge can be represented as a partial cluster near the optimal raccoon, which produces a better solution by combining the optimal raccoon with the later ones in conjunction with one another. A mathematical model of the process in which the optimal raccoon and other raccoons head out to a nearby water source is established as follows. The small cluster gRZR (12) represents the concentration of raccoons near a given water source.

$g R Z P = {[\begin{matrix} g R Z P_{1} \\ ⋮ \\ g R Z P_{i} \\ ⋮ \\ g R Z P_{m} \end{matrix}]}_{m \times 1}$

(12)

where the new location of each raccoon is updated according to (13).

$g R Z P_{i} = G_{o p t} + (r a n d - 0.5) \times (G_{o p t} - L o c (s, :))$

(13)

Loc(s,:) is the location of another raccoon with the same objective. The smallest

F g R Z P_{m i n}

(14) is selected in gRZR (12).

$F g R Z P_{m i n} = m i n (F u n (g R Z P_{i}))$

(14)

The fitness of

g R Z P_{m i n}

relative to

G_{o p t}

determines whether

G_{o p t}

is updated, as shown in Equation (15).

$G_{o p t} = g R Z P_{(m i n)}, i f (F u n (g R Z P_{(m i n)}) < F G_{o p t})$

(15)

This produces better results and improves the algorithm solution accuracy. Using the raccoon-washing-food-inspired strategy, a new generation of raccoons can be provided with a new advantageous position, guiding the algorithm to quickly pursue the optimal goal.

3.4. Escaping-Predator Strategy

A raccoon ducking under attack often creates new possibilities. The raccoon escape strategy is modeled by assuming that a raccoon looks for a direction to quickly escape after being attacked. During this behavior, the ability of the raccoon to arrive at a new safe location implies that the algorithm finds a new target. Failure to do so means that the raccoon fails to escape, and a better target is not found.

M r i

is the new location of the raccoon after it has escaped, and its update process can be tabulated as shown in (16).

$\begin{array}{l} M r i = L o c_{0} (x i, :) + (M X m i n + (r a n d + 0.5) \times (M X m a x - M X m i n)) \\ M X m a x = \frac{X_{m a x}}{i}, M X m i n = \frac{X_{m i n}}{i} \end{array}$

(16)

M X_{m i n}

and

M X_{m a x}

are the current minimum and maximum ranges of escape of the raccoon, which continuously change as the algorithm iterates; xi is a random integer, and 0 < xi ≤ Nop. The resulting superior solution is used to update the raccoon colony (17).

$\{\begin{array}{l} L o c_{0} (x i, :) = M r i \\ F L o c_{x i} = F u n (M r i) \end{array}, i f (F u n (M r i) < F u n (L o c_{0} (x i, :)))$

(17)

which can be compared with $G_{o p t}$ (18).

$\{\begin{array}{l} G_{o p t} = M r i \\ F G_{o p t} = F u n (M r i) \end{array}, i f (F u n (M r i) < F G_{o p t})$

(18)

Here, the algorithm can greatly jump out of the local optimum and find better results. Under certain conditions, the escaping-predator strategy offers an effective solution for the raccoon’s new location, aiding in the optimization of the algorithm and potentially improving its solution speed.

In summary, this article employs the adaptive exploration radius, raccoon-washing-food-inspired, and predator-escape strategies to enhance ROA, resulting in the generation of EROA.

5. Conclusions

In this article, $R_{s}$ , $L_{s}$ , and $ψ_{f}$ of PMSMs were identified based on an enhanced raccoon optimization algorithm. ROA was improved using the adaptive exploration radius, raccoon-washing-food-inspired, and escaping-predator strategies to form EROA with better solution efficiency. The simulation results show that EROA can achieve faster converge speed and higher robustness than ROA. From the experimental results, EROA can quickly find the motor parameters, which are consistent with the simulation results, and the error of the identification results does not exceed 2%.

In conclusion, this study offers significant implications for the field of motor control. The flexibility of intelligent algorithms has expanded the application scope of parameter identification techniques. In future work, the research will focus on reducing the computational complexity of intelligent algorithms, enhancing the speed and accuracy of parameter identification, and expanding its applications to various contexts.

Source link

Zhihong Hu www.mdpi.com

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Enhanced Raccoon Optimization Algorithm for PMSM Electrical Parameter Identification

1. Introduction

2. Parameter Identification of PMSM

3. Enhanced ROA

3.1. Characteristics of ROA

3.2. Adaptive Exploration Radius Strategy

3.3. Raccoon-Washing-Food-Inspired Strategy

3.4. Escaping-Predator Strategy

5. Conclusions

Greenberg

1. Introduction

2. Parameter Identification of PMSM

3. Enhanced ROA

3.1. Characteristics of ROA

3.2. Adaptive Exploration Radius Strategy

3.3. Raccoon-Washing-Food-Inspired Strategy

3.4. Escaping-Predator Strategy

5. Conclusions

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