6.2 Proposed Approach
6.3.2 Case Study 2
To compare the proposed framework with another state-of-the-art surrogate mod- elling technique, a more realistic engineering example is considered. The engineering problem considered in this chapter is a borehole function, which simulate the water flow through a borehole. This benchmark model has been discussed in [74]. The model consists of an eight-dimensional input vector x = (rw, r, Tu, Hu, Tl, Hl, L, Kw)T
[74]: h(x) = 2πTu(Hu − Hl) ln(r/rw)(1 + ln(r/r2LTw)ru2 wKw + Tu Tl) , (6.33)
where h(x) is the fluid water flow measured in m3/year, r
w is the radius of the
borehole, r the radius of influence, Tu the transmissivity of the upper aquifer, Hu the
potentiometric head of the upper aquifer, Tl the transmissivity of the lower aquifer,
Hl the potentiometric head of the lower aquifer, L the length of the borehole, and Kw
the hydraulic conductivity of the soil. The variability in the input vector are modelled as independent random variables whose properties are summarized in Table 6.9. For the log-normal distribution, the parameters are the mean and standard deviation of the natural logarithm of the variable. For the other variables, they describe the range of uniform distributions.
Table 6.9: Borehole Model Definition of the Probabilistic Model of the Input Param- eters
Parameters Units Distribution Parameters
rw [m] Uniform [0.05, 0.15] r [m] Log-normal [7.71, 1.0056] Tu [m2/year] Uniform [63070, 115600] Hu [m] Uniform [990, 1110] Tl [m2/year] Uniform [63.1, 116] Hl [m] Uniform [700, 820] L [m] Uniform [1120, 1680] Kw [m2/year] Uniform [9855, 12045]
For the purpose of this example, it is been assumed that the borehole function is expensive to run, thus, the developed framework in this chapter (i.e. robust ANN) and a Kriging model are adopted and compared in terms of their predictive capability and width of the confidence bounds.
6.3.2.1 Analysis
For the experimental settings, training data of size N = 200 have been generated from the expensive model and used to train a Kriging and a robust ANN in parallel. In particular, a two hidden layer ANN architecture of [8, 8, 5, 1] have been discovered by the GA, while in the Kriging model, an exponential kernel function have been used. For the robust ANN training procedure proposed, M = 20 have been chosen to for computational reasons.
6.3.2.2 Results
The statistics of interest is the water flow h(x) corresponding to combination of different input values. Figure 6.5 and 6.6 shows the prediction from the developed approach in this chapter and a Kriging model respectively. Comparing both figures (Figure 6.5 and 6.6), it is clearly shown that the prediction made by the robust ANN matches the target data with better precision compared to the Kriging model. In addition, the width of the confidence intervals from the robust ANN is tighter than that from the robust ANN.
Figure 6.6: Kriging Prediction
Table 6.10: Prediction Intervals Performance Comparison Between the Proposed Ap- proach and Classical Approach
Approach P ICP N M P IW
Kriging Model 0.93 0.42
Robust ANN 0.95 0.40
Furthermore, the P ICP and N M P IW value for the two surrogate models are shown in Table 6.10. Similarly, Table 6.10, shows that the robust ANN performs better in terms of the coverage probability and width. On the basis of these obtained results, we can adopt the proposed approach in this chapter to a more complex realistic engineering problem.
6.4
Chapter Summary
In this chapter, a novel approach for the quantification of ANN uncertainty as a result of the model structure and random initialization of the weight parameters in the model is proposed. The approach in this chapter extends the approach proposed in Chapter 5 by including optimization and an interval propagation approach. The approach in this chapter have been compared with the approach in Chapter 5 on the basis of simple numerical examples. Particularly, the intervals obtained from this approach tend to be smaller than the intervals obtained in Chapter 5. Although, the percentage of validation data that falls inside the interval calculated from this approach is lower, the RMSE is relatively smaller. Furthermore, the width of the confidence bounds obtained in this approach is relatively smaller than that of Chapter 5. Thus, we can conclude that from this approach the model uncertainty is reduced due to the tighter confidence bounds. On the other hand, adopting this approach requires a lot of computational resource, as a huge number of surrogate models are required to be trained. However, parallelization strategies can be adopted to reduce the computational cost.
Chapter 7
Uncertainty Quantification of the
Site Ion eXchange Plant Using
Robust Artificial Neural Networks
Nuclear decommissioning is a prime example of an uncertainty quantification task due to large uncertainties involved in dealing with nuclear materials. Thus, in this chapter, a nuclear decommissioning problem is set up and solved with the approaches developed in Chapter 4 and 5. In particular, the approaches are adopted for reliability and sensitivity analysis problem focusing on a process simulation model of a UK nuclear effluent treatment plant. This model has been extensively validated against plant and experimental data and used to support the UK effluent discharge strategy.
7.1
Overview
Currently, at Sellafield Nuclear decommissioning site, programmed activities are ong- ing to make a significant advance in the retrieval and decommissioning its legacy Ponds and silos. Particularly, the programme is scheduled for at least another 100 years, with an estimated cost in the tens of billions of pounds [1]. Subsequently, this wide ranging cost estimate reflects the uncertainties and contingencies within the pro- gramme due to the considerable technical, environmental and operational challenges. On the other hand, the wastes stored at the Sellafield Site have been generated over an extended time scale (1950s present). Interestingly, in the case of the legacy wastes generated up to the early 1980s there is some uncertainty about the condition of these wastes and the facilities they are stored in. This was due to poor record keeping at the time, and continued limited access due to high radiation hazards which poses challenges for sampling and detailed characterisation. Consequently, there is an un-
certainty of the future effluent arising that will be generated as part of the retrieval and decommissioning operations in terms of the volume generated and the chemical compositions. To mitigate this uncertainty, a key facility to support current decom- missioning operations being conducted at the Sellafield is the Site Ion eXchange Plant (SIXEP) [1]. A schematic of the working process of the SIXEP is illustrated in Figure