4.2. Remaining Useful Life Prediction
According to the method in the previous section, we treat a #1 battery as a single component m and predict its RUL, choose the starting point of prediction as
k = 125 (36 cycles before failure), initialize the model parameters, set the number of particles to 200, and predict the battery capacity for future cycle = 80 (cycles); after several adjustments of parameter optimization, we obtain the battery degradation capacity prediction curve. The prediction results are shown in
Figure 7,
Figure 8 and
Figure 9.
The blue curve in
Figure 9 indicates the failure probability of the battery, and the red line indicates its actual failure time.
From the prediction results, it can be seen that the actual RUL falls within the predicted RUL probability distribution and is located in the interval with higher probability density, which can be initially determined that the prediction effect of this method is good. In order to further evaluate the advantages of this method, we compare it with the traditional model-based method.
We know from the previous section that the battery degradation model can be represented by a double exponential model and the model parameters have been fitted by using the MATLAB 2020B Data Fitting Toolbox. We then obtain the degradation model expression for the battery as
By substituting the data calculations, we obtain the prediction results of the traditional model-based approach as shown in
Figure 10.
The green vertical line in
Figure 10 represents the point in time when the prediction was started. We conduct a comparative analysis between the prediction outcomes of the particle-filter-based method utilized in this study and the prediction results derived from the traditional model-based approach mentioned earlier. The comparison between these two curves is depicted in
Figure 11. By examining the curves displayed in the figure, it becomes evident that the predicted data obtained through the method employed in this research exhibits closer proximity to the actual degradation data of the battery. For the sake of comparative analysis, we define the absolute error of prediction as:
where
r is the actual battery service life, i.e., the actual number of usable cycles.
denotes the predicted battery service life value obtained at the prediction starting point
k.
Using Equation (24), we calculate that
Ck = 1.303 > 1.2995 (failure threshold) for
k = 143, and
Ck = 1.294 < 1.2995 for
k = 144, i.e., the predicted service life of the component using the traditional model-based method is 143. We compare the analysis with the particle-filter-based method used in this paper, and the results are shown in
Table 3.
As can be seen from the data in
Table 3, the relative error of prediction is lower for the method used in this study compared to the traditional model-based method when the starting point of prediction is the same.
After that, we choose the prediction starting point of
k = 120 (41 cycles before failure) and
k = 130 (31 cycles before failure) for prediction and calculate the prediction error of the particle-filter-based method used in this study three times, and the results are shown in
Table 4.
From the data in
Table 4, it can be seen that the relative error of prediction of the methods used in this study is below 8% for three different starting points, with an average value of about 6%.
After the above analysis, we can conclude that the prediction of the remaining service life of the components based on the particle filtering algorithm used in this research is more accurate and has a higher prediction stability.
4.3. Analysis of Maintenance Strategies
In order to verify the feasibility and effectiveness of the proposed maintenance strategy, we choose a multi-component system (l = 3) of three #1 batteries in series as the experimental example, and the degradation data of each #1 battery is known. After that, a random number is sampled from a normal distribution with a mean 0 and standard deviation of the mean of the degradation difference, and the random number is used as a perturbation and added to the original data points to obtain eight new sets of data
We first perform the maintenance analysis for part m. Its optimal maintenance moment and maintenance cost rate can be calculated based on its RUL prediction results, as in Equations (15) and (16):
, MCR = 5.56. As shown in
Table 5, the optimal maintenance moment of part m is influenced by the cost parameters
and
. When
is kept constant, the optimal maintenance moment of the part lags with the increase of
, and the part maintenance cost rate increases because the maintenance cost growth rate of the preventive replacement is greater than the optimal maintenance moment growth rate. From an economic point of view, when the ratio of
to
gradually increases, and when
gradually approaches
, the optimal maintenance moment is closer to the failure moment of the component, the cost rates of performing preventive replacement and failure replacement are similar, and the maintenance effects of the two approaches are similar.
We know from the results that the expected cost of preventive replacement is higher than the expected cost of non-preventive replacement until the 144th cycle, and the expected cost of preventive replacement starts to be lower than the expected cost of non-preventive replacement at the 144th cycle, which is its best maintenance moment. Assuming that the logistics department requires a 10-cycle lead time for ordering spare parts, we can predict that the best time to order spare parts is the 134th cycle.
To better analyze the effectiveness of the proposed maintenance policy for a single component, we compare it with the traditional condition-based maintenance policy, periodic maintenance policy (PeM) based on historical reliability data, and ideal predicted maintenance policy (IPM) [
17]. The state-based maintenance focuses on instantaneous decisions, while the remaining two policies can accomplish future moment decisions like the policy proposed in this paper. We present two other policies in the following:
- (1)
Periodic preventive maintenance strategy, which is executed when based on the historical reliability data of components. Specifically, the average failure time
of components is first obtained using the historical reliability data of multiple components, and then periodic preventive maintenance with a cost of
is executed at time
as follows:
where
denotes taking the smallest integer greater than or equal to the real number
x.
Since maintenance activities are planned in advance, spare parts are available at this time. Conversely, if a part fails before moment
, the part cannot arrive immediately and cannot be used for fault maintenance. In this case, fault maintenance will be performed. Therefore, the maintenance cost rate for this strategy is given by the following equation:
- (2)
Ideal predictive maintenance strategy, which is executed based on the assumption of perfectly predicted failure times. In this case, we assume that the remaining life is accurately predicted; then, the maintenance execution moment
should be one cycle before the actual failure moment
of the component.
Then, the decision based on this perfect information will result in minimizing the value of the cost rate, which is given by the following equation:
The first step involves comparing the RUL-based predictive maintenance strategy proposed in this study with the traditional state-based maintenance approach. The decision results are summarized in
Table 6.
For the state-based maintenance, the decision is to perform no immediate maintenance but order parts at cycle 125 (current cycle). It is evident that the state-based approach only provides instant decisions without considering future predictions. In contrast, our proposed maintenance strategy predicts a scheduled maintenance time at cycle 144 and suggests ordering spare parts at cycle 134. Notably, for a specific component with failure expected at cycle 161, the predicted maintenance activity time aligns reasonably well.
This demonstrates that our predictive maintenance strategy accurately forecasts when preventive measures should be taken for individual components. This ability aids in the proactive planning of spare parts inventory and production activities.
Furthermore, we conducted RUL prediction using data from nine battery components and calculated the optimal moment for performing maintenance as well as the associated maintenance cost rate (MCR) for each component. We compared these results with both PeM and IPM strategies which also focus on long-term decisions.
Figure 12 illustrates the MCR of these three maintenance strategies specifically applied to nine instances of #1 cells.
The figure illustrates the outcomes of our analysis, juxtaposing the proposed predictive maintenance strategy (referred to as PdM) with the periodic preventive maintenance strategy (PeM). It emphasizes that, in the majority of battery performance cases, the PdM strategy demonstrates a reduced rate of maintenance cost (MCR) in comparison to PeM.
This can be attributed to the conservative and lagged execution nature of PeM, which often leads to delayed maintenance, resulting in economic inefficiency due to a higher number of failed batteries at the time of maintenance. In contrast, our proposed strategy allows for more timely and efficient component maintenance, significantly reducing costs.
While an ideal predictive maintenance strategy (IPM) boasts the lowest MCR in theory, it is based on perfect predictive information which is not practically achievable in production settings. However, it is worth noting that our proposed strategy closely aligns with IPM in terms of MCR. Specifically, we calculated average MCR values for each approach: 7.545 for PeM, 5.433 for PdM proposed in this study, and 5.039 for IPM. These findings affirm that our proposed strategy effectively reduces single-component maintenance costs.
To further evaluate the effectiveness of our approach at a system level, we assessed three different systems comprising nine battery components. The earliest instance requiring maintenance within each system served as an opportunity for system-level scheduled maintenance. Additionally, other components falling within their respective best opportunity windows were also given scheduled opportunities for maintenance—these are detailed in
Table 7.
Based on the data presented in the table, it is evident that implementing system-level opportunistic maintenance decisions yields substantial cost savings. Notably, Systems 1 and 3 demonstrate a remarkable outcome as all three components are grouped together and maintained simultaneously, resulting in respective savings of 670 and 440 units. In System 2, two out of the three components are included in the opportunistic maintenance combination, leading to a significant cost reduction in 260 units. This strategic approach effectively reduces overall system maintenance costs across all three systems, with System 1 showcasing the most prominent improvement.
The implementation of this strategy brings about a notable decrease in the system maintenance cost rate for System 1 from 16.40 to an impressive value of only 12.01. This reduction can be attributed to optimized prediction of maintenance timing, whereby all three components align closely within their respective opportunity maintenance windows.
When compared through calculations without opportunistically incorporating these maintenance measures, it is estimated that the average system maintenance cost amounts to approximately 2400 units at an average cost rate of around 16.30. However, by adhering to our proposed strategy involving opportunistic maintenance practices, this average cost significantly drops down to approximately 1943 units with an average cost rate reduced to only 13.43 units. This finding unequivocally demonstrates that our suggested strategy effectively minimizes fixed configuration-based maintenance costs at the system level.
Furthermore, our computations reveal that employing traditional periodic preventive maintenance would result in an average system maintenance expense of approximately 3600 units with an associated average cost rate reaching 22.63 units across all systems combined. By contrast, our novel model presents itself as highly advantageous as it slashes both the average system-wide maintenance expenses by up to 45% and diminishes the corresponding average cost rates by 40%.
In summary, based on these experimental outcomes and analyses conducted herein, this paper successfully establishes that our proposed multi-component system’s predictive RUL-based maintenance strategy proves to be a highly effective and efficient approach.