# PID Control Assessment Using L-Moment Ratio Diagrams
## 4 Industrial Validation  
We used data from the ammonia production installation located at Grupa Azoty Zakdady Azotowe \"Pulawy\" SA in Poland. The production utilizes technology of the autothermal reforming of methane \(CH_{4}\) with the utilization of oxygen: pure and in the form of processed air [45]. Figure 19 presents a schematic diagram of the considered ammonia production plant.  
The plant is equipped with a modern control system that uses a hierarchical advanced process control strategy with a basic layer utilizing PID control algorithms, which are supervised using model predictive control [46]. All PID loops are well-tuned and maintained. The supervisory dynamic optimization allows optimal system operation.  
As the data selection was crucial for the subsequent analysis, much attention was given to using comparable and credible time series from normal operating regimes. We used a dataset from 14 consecutive months (July 2020-August 2021), satisfying a 1 min sampling time. To allow data comparability assumptions, it was ensured that the natural gas consumption within these periods varied at most \(\pm 1.5\%\). The load was constant and its fluctuations were process related. Time series were taken from the plant information system. The dataset comprised control error records from 22 PID loops: 12 flow, 2 level, 1 pressure, and 7 temperature controls. Detailed information about these loops is limited by the Grupa Azoty security regulations.  
Such construction of data files, i.e., 22 loops collected in 14 months, allowed two analytical perspectives. First, we could compare the homogeneity between the loops. Therefore, we used a single month (July 2020) and plot LMRDs for 22 loops. Two LMRDs  
Figure 18: The LMRD(\(l_{2}\), \(\tau_{4}\)) for \(G_{5}(s)\) related to overshoot.  
were plot. Figure 20 shows the L-skewness versus L-Cv LMRD(\(\tau_{2},\tau_{3}\)) diagram, while all L-kurtosis versus L-skewness LMRD(\(\tau_{3},\tau_{4}\)) plot is sketched in Figure 21.  
Figure 19: Schematic diagram of the ammonia production plant.  
Figure 20: The LMRD(\(LcV,\tau_{3}\)) for July 2020.  
The second perspective, i.e., a single loop #15 but a showing a comparison on a monthly basis is presented in the next plots. Figure 22 presents the LMRD(\(\tau_{2},\tau_{3}\)) and Figure 23 the LMRD(\(\tau_{3},\tau_{4}\)) plot.  
Figure 21: The LMRD(\(\tau_{3},\tau_{4}\)) for July 2020.  
Figure 22: The LMRD(\(LcV,\tau_{3}\)) for loop #15.  
A review of the above plots allows making observations. First, all the loops worked properly and they were well-tuned, as the LMRD(\(LcV,\tau_{3}\)) and LMRD(\(\tau_{3},\tau_{4}\)) show a concentration of points around L-skewness equal to zero. Loop #15, which was selected for the time perspective, seems to be the most challenging. This observation is confirmed with two other diagrams, which show large variations, which were most probably related to process nonlinearities.  
Industrial validation justified the applicability of LMRDs in practice. In this project, they were used to confirm the positive results of the control system rehabilitation, which were associated with the comprehensive tuning activities and the implementation of the APC system. Simulations allowed combining the loop control performance with the point location in the diagram. As a consequence, this knowledge allowed the interpretation of the industrial data.