Considering the variability of motor parameters during operation, there is a need to propose a parametric motor loss estimation method. In current research, estimating techniques relying on loss data are limited to specific parameter configurations, hindering the ability to directly connect loss with motor parameters. Furthermore, current mechanistic estimating methods do not adequately account for the complexities of the torque and current relationships across various motor control techniques. To overcome these limitations and provide accurate and flexible economic optimization goals for online control allocation strategies, this section attempts to create a decoupled, demand torque-variable-based parametric online estimating method for PMSM loss.
Based on the above discussion, this section proposes a two-stage motor loss estimation method that relies on a mechanistic model and incorporates real-time system sampling data. This method decouples motor loss estimation into two independent steps: estimating the dq-axis currents from the total demand torque, and then converting these currents into motor losses. The relationship between current and motor loss, as detailed in the previously mentioned drive system mechanistic model, maintains the direct influence of motor parameters on loss. Regarding current estimation, due to the variety of motor control methods, it is difficult to immediately derive explicit expressions from the mechanistic model. Therefore, a data-based approach to identify the dynamic torque–current mapping curves is adopted in this paper. Given the motor’s low rotational inertia and high response speed, this paper has decided to disregard the impact of system inertia. Instead, real-time sampling data are used directly to construct dynamic torque–current mapping curves, accurately depicting the relationship between torque and current under current operating conditions [
24].
3.1. Initial Mapping Identification Based on Conv-Fusion Curve
The identification of dynamic torque–current mapping unfolds in two stages: initial mapping identification and the continuous update of the mapping results obtained from the initial identification. Dynamic torque–current mapping can be decomposed into two independent curves, namely and . By utilizing multiple sets of torque–current data pairs collected from the system, which include noise, the process of mapping acquisition is transformed into the problem of fitting two curves within a two-dimensional coordinate system.
As an integral part of the online control allocation strategy, ensuring the real-time aspect of curve fitting is critical. Additionally, to reduce the impact of noise and enhance the accuracy of system energy loss estimation, improving curve smoothness is essential. Therefore, it is necessary to adopt a fitting method that is computationally convenient, offers high fitting accuracy, and ensures curve smoothness.
Given that the mapping curves lack typical functional form characteristics, it is difficult to fit them with specific functions. Neural network-based approaches [
25] may fulfill accuracy and smoothness criteria but are not practical for real-time applications because of their high computing complexity. Furthermore, methods that rely on full torque range data not only have long sampling times but are also prone to overfitting. However, the mapping curves have distinct linear features within certain intervals. Therefore, piecewise linear fitting effectively circumvents the aforesaid problems, enhancing both precision and computing efficiency. As a result, this part proposes a curve convolution fusion method based on piecewise linear fitting to achieve a smooth transition between two straight line segments.
To provide a more precise depiction of the curve, a series of torque and currents data is obtained at uniform intervals
spanning from 0 to the motor maximum output torque. The torque corresponding
or
value is denoted as
.
Figure 5 is a schematic diagram of curve convolution fusion, where curve
L represents the originally fitted curve, and
represents the newly fitted curve in the current period. The elucidation of the particular approach for implementation is provided as follows.
The core technique of the proposed curve fusion method lies in allocating gradual transition weights to the two linear segments of the area to be fused. This method smooths and diffuses by conducting vector convolution operations on the difference between the two linear segments. Subsequently, the error curve after convolution is integrated and normalized to determine the curve weight at certain places. The specific implementation strategy is as follows.
During the process of train traction, the output torque of each motor gradually diminishes following the traction curve, generating sample data of torque, d-axis current and q-axis current that encompasses a significant portion of the torque range. The system sampling data are acquired at a fixed time interval
, and the duration taken for each m number of samples is recorded as a time interval. In the
k-th time interval, the sampled data set
is obtained, where
m is the number of data in the data set
. The range of torque values represented in the sampling data is denoted as
. A least squares linear fitting is applied to the data set
, resulting in the determination of linear fitting parameters as
As a result, the fitted curve can be described as
:
, where
represents the ordinate of the curve
, meaning d-axis or q-axis current,
T represents the abscissa of the curve
, meaning torque. The torque range represented by
is extended from the original curve
L to the newly adjusted curve
, which is the area where the two curves overlap. Assume that there is an intersection point between two curves, and the torque corresponding to the intersection point is recorded as
, then calculate the curve difference in the torque range
, as shown in
Figure 5a. Otherwise, it can be considered that
, and the torque area range for calculating the curve difference is
, as depicted in
Figure 5b.
The error
is calculated by making a difference between the data points of the curves
and
L in the overlapping area. Then, Gaussian window function
is applied to operate vector convolution on the error so that the error propagates to the surrounding area and becomes smooth. The above process is described as follows.
where
represents the result of the originally fitted curve at time
k,
represents the result of error
after Gaussian smoothing,
u in Equation (
14) is the input of the Gaussian window function, and
is its standard deviation, which is used to control the width of the window.
The data closing to one curve at both ends of the fusion region should possess greater confidence and carry a higher weighting compared to the data associated with the curve at the other end. The difference curve, which has been smoothed using convolution, is subjected to integration and normalization. The resulting value is then employed as the weight for the curve, which is expressed as
where
is the weight curve, and
T means torque, which is the independent variable of the weight curve. Therefore, the description of the mapping curve
L is changed as
where
represents the result of the updated curve
L,
represents the ordinate of the torque–current mapping curve.
3.2. Mapping Iterative Update Based on Optimal Estimation
Considering the time-varying characteristics of the torque–current mapping curve affected by the motor control method, this paper locally updates the torque–current mapping curve using the latest sampling data to enhance the accuracy of loss estimation. The newly sampled data contains a large amount of noise, and its deviation may not only be from changes in the motor’s operating trajectory but also from other random factors, leading to insufficient data reliability. Hence, it is essential to devise an effective method to fuse the new sampling data with the existing mapping curve, ensuring a seamless alignment between the updated mapping curve and the new sample point.
Directly incorporating new sampling points for refitting does not facilitate maintaining computational simplicity, nor does it adequately consider the credibility of new data during the updating process. This paper introduces the concept of data fusion from the Kalman filter theory, implementing an optimal estimation based on the minimum standard deviation within the original mapping interval and new sampling data.
Kalman filter is a classical mathematical method [
26] for state estimation, capable of obtaining optimal estimates of unknown quantities from noisy data. Although curve updating is not its traditional application field, the core concepts of optimal estimation and data fusion are still worth adopting in curve updating. Consider the state variables
, which represents a series of dq-axis currents in the mapping curve
L results,
is the state variables at the
k-th time, where
n represents dimension of state vector, also the number of coordinate points used to describe the mapping curve. The state equation and observation equation of the system are formulated below.
where
represents the identity matrix,
represents the observation results in the
k-th time interval,
and
represent the corresponding system noise and measurement noise, respectively, and their covariances are recorded as
and
, respectively.
Assumes that both system noise and measurement noise follow Gaussian distributions. Observation noise and system prediction equation noise can be considered to be related to the sampling point density and the error between the sampling points and the original points, respectively. Based on this, the covariance matrix of each noise can be constructed. The specific implementation is as follows.
Once the new sampling data set
has been acquired, the density estimation of the data is performed using the Gaussian kernel function. This allows us to obtain the density estimate
for each sampling point, as shown below. Next, the sampling point and its corresponding original mapping curve are subjected to linear interpolation. The resulting interpolated curve is then compared to the original curve to calculate the error
.
where
m is the number of sampled data set
;
h is a constant parameter in the Gaussian kernel function.
The sampling point density and error mean are calculated within the range of
, as the density and error at
. The aforementioned procedure is depicted as follows.
where
represents the total number of data in the range near
,
represents the corresponding average density, and
represents the corresponding average error,
,
when · is true and
when · is false. Then, convolution is employed to effectively smooth and propagate the density and error, in the same way as Equation (
14), and
and
are obtained, respectively. As a result, the covariance matrix of the system noise and measurement noise at
k-th time intervals can be expressed as follows.
The optimal estimate of the mapping curve in differential form can be expressed as
where
is a priori estimate at
k-th time interval, calculated by
,
is called gain at
k-th time interval. Therefore, the error of the estimate is expressed as
where
, is called a priori state error. The covariance matrix of the estimation error is expressed as
where
,
. The covariance matrix of a priori state error
is calculated by
In the optimal estimate representation shown in Equation (
24), the gain
needs to be determined. To minimize the covariance matrix of the estimation error
, the gain
should be calculated as
Therefore, the optimal estimate of the state variable can be obtained according to Equations (
24)–(
28).
In the implementation process, the mapping update and the initial mapping identification use the same update interval to perform a local update operation. As depicted in
Figure 6, assuming that the data points sampled within a certain time interval deviate from the mapping curve, the mapping curve needs to approach the new sampling point smoothly and gradually. If the sampling points of multiple consecutive time intervals are of this type of distribution, the mapping curve will pass through the area of the sampled data, as illustrated in
Figure 6. Conversely, if the sampling points that deviate from the mapping curve are due to sporadic factors within a certain time interval, the mapping curve will not exhibit significant deviations.
3.3. Loss Estimation of PMSM
After the initial identification and iterative updating of the torque–current mapping, a dynamic mapping relationship that can automatically adjust based on the operating condition is obtained, denoted as
. The current to motor loss calculation formula described by Equations (
5) and (
6) is denoted as
, thus, the two-stage loss estimation method is described as
In the implementation process, the initial identification or update of the dynamic torque–current mapping is performed at fixed time intervals , as detailed in Algorithm 1. The initial identification operation runs as the drive system transitions from startup to stable operation, and then it enters the update phase. This identification and update process is executed only once at each time point without setting a loop termination condition, ensuring the controllability of the running time, with time complexity maintained at O(n). Furthermore, whether it is the initial identification or subsequent updates, they are only aimed at the local range of the mapping, effectively shortening the execution time.
Figure 7 presents the implementation framework of the self-updating parametric PMSM loss estimation method. The blue section of
Figure 7 indicates that the initial recognition mapping result is marked as
, then the updated mapping result which marked as
is obtained. Finally, by integrating with the PMSM loss model, the mapping relationship from torque to loss can be established. The purple section demonstrates the steps to be executed when performing loss estimation. When motor loss needs to be estimated, the control allocation unit takes torque and angular speed as inputs. First, the dq-axis current value is estimated by mapping
, and then the motor loss is calculated by the current value using mapping
. It is worth noting that the loss estimation and the recognition of the torque–current mapping do not follow the same execution cycle, loss estimation is only activated when performing control allocation calculations; whereas mapping updates are carried out at a lower priority and frequency.
Algorithm 1 Mapping : Acquisition and Update |
Input: Sampled data of drive system, old mapping curve L Output: new mapping curve - 1:
Obtain samples and Sample torque space - 2:
if then - 3:
Reset initial identification flag, - 4:
end if - 5:
Compute density through the kernel function method based on Equation ( 20) - 6:
if then - 7:
Compute linear fit parameters based on Equation ( 12) - 8:
Compute errors based on Equation ( 13) - 9:
Compute the error after convolution based on Equations ( 14) and ( 15) - 10:
Compute the weight of the curve based on Equation ( 16) - 11:
Update based on Equation ( 17) - 12:
else if then - 13:
Compute density through the kernel function method based on Equation ( 20) - 14:
Compute the density and error at based on Equation ( 21) - 15:
Compute the density and error after convolution and , respectively - 16:
Update covariance matrix and based on Equations ( 22) and ( 23) - 17:
Update covariance matrix and based on Equations ( 22) and ( 23) - 18:
Update , , , of Kalman filter equation based on Equations ( 24)–( 28) - 19:
Update - 20:
end if - 21:
return
|