A New HWMA Dispersion Control Chart with an Application to Wind Farm Data †

: Recently, a homogeneously weighted moving average (HWMA) chart has been suggested for the e ﬃ cient detection of small shifts in the process mean. In this study, we have proposed a new one-sided HWMA chart to e ﬀ ectively detect small changes in the process dispersion. The run-length (RL) proﬁles like the average RL, the standard deviation RL, and the median RL are used as the performance measures. The RL proﬁle comparisons indicate that the proposed chart has a better performance than its existing counterpart’s charts for detecting small shifts in the process dispersion. An application related to the Dhahran wind farm data is also part of this study.


Introduction
In every industrial environment, product quality is affected by the variation in the manufacturing process. The presence of variation in the manufacturing processes is quite obvious, which blemishes the product's quality characteristics. To maintain the product quality, it is necessary to monitor the changes that occur due to assignable causes. Control charts are very popular statistical process monitoring (SPM) tools used to monitor the changes that occur due to assignable causes. There are two main types of control charts: memory-less and memory-type control charts. Memory-type control charts have received much attention in modern era industries to handle the unusual variations in the parameters of the distribution of manufacturing/service process characteristics. The cumulative sum (CUSUM) and the exponentially weighted moving average (EWMA) introduced by References [1,2] respectively, are frequently used memory-type charts, and the most commonly used memory-less charts are the Shewhart charts proposed by Reference [3].
Generally, assignable causes affect both the process mean and variance. A process can go to an out-of-control (OOC) process if the mean is shifted to another level. Likewise, an increase in variance can also cause inconsistency in the process [4]. In real-life applications, it is essential to monitor the process output for early detection of deviation in process parameters, i.e., mean and variance. However, we prefer to stabilize the variance first, as we know the mean structure depends on it, and also the decrease in variance leads to an enhancement in the process production [5].
for an IC process, and let σ 2 t σ 2 0 for an OOC process. Let τ = σ t /σ 0 be the ratio of the OOC and IC standard deviations. For an IC process, τ = 1, and for an OOC process, τ 1. Without loss of generality, we have assumed that µ t = 0. Let X t and S 2 t be the sample mean and sample variance at time t, respectively. Let Y t = ln S 2 t /σ 2 0 . The distribution of the random variable S 2 t /σ 2 0 is the gamma distribution with shape (n − 1)/2 and scale 2δ 2 t /(n − 1), and the parameters and the distribution of Y t is log-gamma distribution. Later on, the authors of Reference [19] showed that Y t is approximately normally distributed with mean µ Y and variance σ 2 Y , where Let the sample mean of Y t be defined as: . Motivated from Reference [10], the statistic of the suggested SJ H chart is defined as: where λ ∈ (0, 1] is the smoothing constant and Z t−1 , is the mean of all the previous Z t values. It is We consider Z 0 = 0. The statistic given in (1) can also be defined as: The SJ H chart triggers an OOC signal if H t is greater than and it indicates the increase in the process variance, where C t is the width of the control limits and can be chosen to achieve the desired ARL 0 . So many methods are available, like integral equations, Markov chains, and Monte Carlo (MC) simulations, to calculate the ARL. We have performed MC simulations to evaluate the ARL of the SJ H chart because this method is more accurate than the integral equations and Markov chain methods [13]. The flow chart for the computation of the ARL of the SJ H chart is given in Figure 1.
where ∈ (0, 1] is the smoothing constant and ̅ −1 , is the mean of all the previous values. It is . We consider ̅ 0 = 0. The statistic given in (1) can also be defined as: The SJ H chart triggers an OOC signal if is greater than and it indicates the increase in the process variance, where is the width of the control limits and can be chosen to achieve the desired ARL 0 . So many methods are available, like integral equations, Markov chains, and Monte Carlo (MC) simulations, to calculate the ARL. We have performed MC simulations to evaluate the ARL of the SJ H chart because this method is more accurate than the integral equations and Markov chain methods [13]. The flow chart for the computation of the ARL of the SJ H chart is given in Figure 1. The RL profiles' values of the proposed SJ H chart are provided in Table 1   The RL profiles' values of the proposed SJ H chart are provided in Table 1 for selective choices of λ by fixing ARL 0 ≈ 200. From Table 1, it is observed that if we increase the value of λ, the ARL 1 values of the SJ H chart decrease, and vice versa (for example, when δ t = 1.1, λ = 0.05, ARL 1 = 22.25 vs. δ t = 1.1, λ = 0.5, ARL 1 = 45.93). The MDRL values are smaller than ARL values which specify that the distribution of the RL is positively skewed ( Table 1). The SDRL values are decreased as the value of δ t increases (for example, when δ t = 0, λ = 0.05, SDRL = 246.86 vs. δ t = 1.2, λ = 0.5, SDRL = 9.61 (Table 1)). We have also checked the performance of the proposed SJ H chart for various choices of n and it is reported in Table 2. The ARL 1 values of the SJ H chart decrease as the value of n is increased for a fixed choice of δ t and λ (for example, when δ t = 1.1, λ = 0.05, n = 3, ARL 1 = 43.6 vs. δ t = 1.1, λ = 0.05, n = 15, ARL 1 = 16.12). Moreover, the RL curves of the proposed SJ H chart are provided in Figures 2 and 3 for various combinations of λ and n. It is noted that as the value of λ increases, the RL values of the SJ H chart are decreased for a fixed choice of n and δ t (Figure 2). From Figure 3, it is seen that as the value of n increases, the RL values of the SJ H chart decrease or are fixed values of λ and δ t .

Comparisons between Proposed and Existing Charts
The ARL 1 comparisons of the proposed SJ H chart with the existing charts are provided in this sub-section. For comparison purposes, the following existing charts are included: CH E , SJ E , and H-EWMA proposed by References [6,9,10], respectively. The performance comparisons between proposed and existing charts are also judged by using another measure called the percentage decrease in ARL, hereafter labeled as PD-ARL. The PD-ARL can be computed by using the formula × 100 [18]. A chart having a larger PD-ARL value is considered to be efficient. We have fixed the ARL 0 at 200 for valid comparisons.
To compare the ARL 1 performance of the SJ H chart with existing charts, we have found the following interesting points: i.
The ARL 1 performance of the SJ H chart is relatively improved compared to the CH E chart (for example, in the CH E chart with = 0.05, δ t = 1.

Graphical Comparisons between Proposed and Existing Charts
In this sub-section, we have also presented the ARL 1 -based graphical comparisons of the SJ H chart with the existing charts. From Figure 4a-d, it is noted that the performance of the proposed SJ H chart is relatively better than the CH E , SJ E , and H-EWMA charts for all selected choices of λ and δ t . It is found that as the value of λ increases the ARL 1 , differences between the proposed SJ H and existing  (Figure 4a-d). The performance of the SJ H is far better than the CH E and SJ E for selected choices of λ under selected choices of δ t (Figure 4a-d).
From tabulated and graphical comparisons, we have concluded that the SJ H chart performs well against the CH E , SJ E , and H-EWMA charts.

Graphical Comparisons between Proposed and Existing Charts
In this sub-section, we have also presented the 1 -based graphical comparisons of the SJ H chart with the existing charts. From Figure 4a-d, it is noted that the performance of the proposed SJ H chart is relatively better than the CH E , SJ E , and H-EWMA charts for all selected choices of and . It is found that as the value of increases the 1 , differences between the proposed SJ H and existing charts are decreased (Figure 4a-d). The performance of the SJ H is far better than the CH E and SJ E for selected choices of under selected choices of (Figure 4a-d). From tabulated and graphical comparisons, we have concluded that the SJ H chart performs well against the CH E , SJ E , and H-EWMA charts.

Application: Monitoring of Daily Power Generation at Dhahran Wind Farm
In this section, we present the application related to monitoring the daily power generated at the wind station located at the eastern coast of Dhahran (26°32′, 50°13′), Saudi Arabia. The daily energy generated was recorded during the winter period (15 November to 29 February 2020). The

Application: Monitoring of Daily Power Generation at Dhahran Wind Farm
In this section, we present the application related to monitoring the daily power generated at the wind station located at the eastern coast of Dhahran (26 • 32 , 50 • 13 ), Saudi Arabia. The daily energy generated was recorded during the winter period (15 November to 29 February 2020). The obtained data are given in Table 4 in the form of 21 subgroups, each of size 5, along with the plotting statistics of the charts considered for the application section.

Data Description
It is a tenable fact that the world is shifting focus from fossil fuels to renewable energy owing to CO 2 emission associated with fossil fuels during operation. Saudi Arabia subscribed to this initiative. Also, the increase in energy demand calls for the exploitation of other available cost-effective energy sources. Harnessing the readily available renewable energy sources such as wind and solar helps meet energy demand in a remote area and contributes significantly to the national grid. Extensive work has been carried out on wind data from various wind farms in the Kingdom; however, in the context of the control charts this is the first application. Wind speed data is available from different sources in the Kingdom. Among these are Saudi Aramco, Meteorology and Environmental Protection Administration (MEPA), and King Fahd University of Petroleum and Minerals (KFUPM) [20]. Wind power is the application of air flow through wind turbines to generate electric energy. The important parameter influencing the rate of energy generated in a wind farm are wind speed, wind direction, air temperature, and global solar radiation. Hourly metrological data was obtained from the meteorological monitoring station in the Eastern Province of Saudi Arabia [21]. The pictorial representation of the Dhahran wind farm is presented in Figure 5. We have also applied the Anderson Darling test to check the distribution of the Dhahran wind farm data, and from this test, it is observed that the distribution of the wind farm data is normal ( Figure 6). We have considered the SJ H , CH E , and SJ E charts to possibly examine the deviations in the process variance by fixing ARL 0 ≈ 200.  We have presented two examples based on wind farm data. In the first example, the first 10 subgroups are considered IC, and a shift of size 1.2 is introduced in the next 11 subgroups with = 0.05 (Figure 7a). In the second example, the first 16 subgroups are considered IC, and a shift of size 1.5 is introduced in the next 5 subgroups with = 0.2 (Figure 7b). The description of the wind farm data along with the plotting statistics of SJ H , CH E , and SJ E charts are reported in Table 4. From Figure 7a, it is seen that both the SJ H and SJ E charts trigger the OOC signal at the 17th subgroup point, whereas the CH E charts do not trigger any OOC signal. From Figure 7b, it is observed that  We have presented two examples based on wind farm data. In the first example, the first 10 subgroups are considered IC, and a shift of size 1.2 is introduced in the next 11 subgroups with = 0.05 (Figure 7a). In the second example, the first 16 subgroups are considered IC, and a shift of size 1.5 is introduced in the next 5 subgroups with = 0.2 (Figure 7b). The description of the wind farm data along with the plotting statistics of SJ H , CH E , and SJ E charts are reported in Table 4. From Figure 7a, it is seen that both the SJ H and SJ E charts trigger the OOC signal at the 17th subgroup point, whereas the CH E charts do not trigger any OOC signal. From Figure 7b, it is observed that We have presented two examples based on wind farm data. In the first example, the first 10 subgroups are considered IC, and a shift of size 1.2 is introduced in the next 11 subgroups with λ = 0.05 (Figure 7a). In the second example, the first 16 subgroups are considered IC, and a shift of size 1.5 is introduced in the next 5 subgroups with λ = 0.2 (Figure 7b). The description of the wind farm data along with the plotting statistics of SJ H , CH E , and SJ E charts are reported in Table 4. From Figure 7a, it is seen that both the SJ H and SJ E charts trigger the OOC signal at the 17th subgroup point, whereas the CH E charts do not trigger any OOC signal. From Figure 7b, it is observed that the SJ H chart has detected the OOC signals at the 17th subgroup point. However, SJ E and CH E charts trigger the OOC signal at the 19th and 20th subgroup points, respectively. These illustrative examples clearly show the superiority of the SJ H chart as compared to the CH E and SJ E charts. The real life application also supported the findings in Section 3.

Concluding Remarks
The increase in the process variance deteriorates the performance of the production processes under consideration. In this study, we have suggested the SJ H chart for quickly and efficiently monitoring the changes in the upward increase of the process variance. Monte Carlo simulations were used to compute the various RL profiles of the SJ H chart. The RL profiles of the SJ H chart have been compared with CH E , SJ E , and H-EWMA charts. The RL comparisons revealed that the SJ H chart shows superior performance compared with existing charts for monitoring upward shifts in the process dispersion. Hence, we recommend the practice of the SJ H chart to the SPM practitioners to monitor the upward shifts in the variance of a normally distributed process.
The scope of this study may be extended to develop efficient non-parametric and multivariate charts using the design structure of the SJ H chart.

Conflicts of Interest:
The authors declare no conflict of interest.