Topology generated from process knowledge

Groenewald (2014) provided a thorough hierarchical representation of the causality of each unit in this process from process knowledge. The basic causality maps for individual units generated by Groenewald are shown in Figure 7-4, Figure 7-5 and Figure 7-6. The general causality map focusing on the final tails grade is given in Figure 7-7. For any of the mills, the PSD of the mill product is a function of it power, load and the density of the contents. All these are determined by the throughput, the inlet water ratio, and the feed PSD. For the cyclone it’s mass split and cut-point are affected by the feed PSD and pressure, which are determined by the feed flow rate and feed density. For the flotation cells, the recovery and grade are functions of the mass pull, the feed and the reagents. The feed is in turn affected by the density, flow rate and PSD.

Figure 7-4: Mill performance basic causality map (Redrawn from Groenewald (2014))

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Figure 7-6: Flotation performance basic causality map (Redrawn from Groenewald (2014))

Figure 7-7: General concentrator causality map (Redrawn from Groenewald (2014))

7.3.2. Linear cross-correlation topology extraction

The number of lags, k, chosen for LC was 10. This value was selected since, this corresponded to 40 hours of operation, which is a long enough time period to incorporate the residence times and dead time associated with this process. Using the method described in section 4.3.4 for selection of the significance threshold, the sample size could be used in Equation 7-1 to determine a significance threshold.

𝜌_{𝑚𝑎𝑥,𝑡ℎ}𝐿𝐶 _{(𝑁) = 3𝑁}−0.452_{+ 0.11𝑁}−0.658 _{Equation 7-1}

𝜌_{𝑚𝑎𝑥,𝑡ℎ}𝐿𝐶 (150) = 0.316 Equation 7-2

Applying this threshold of 0.316, however, resulted in a connectivity graph with too many connections. It was decided to apply a stricter threshold, so the threshold was rounded up to 0.4. As with the previous case studies the threshold determined from the proposed method was too lax,

Chapter 7 -Case Study: Fault Diagnosis Applied to Concentrator Process Page 154 resulting in a connectivity graph with too many spurious connections. Figure 7-8 shows the resulting connectivity graph.

Considering the causality presented by Groenewald (2014) the following key connections are discussed for the LC graph shown in Figure 7-8:

 The primary mill water feed affects many key variables, including: mill load, power, the final tails PSD and the concentrate grade. This is accurate, since the feed is at the start of the process and would therefore have a substantial influence on downstream variables.

 The primary mill load affects the concentrate grade, as well as the mass split. This relationship is accurate from a fundamental perspective, since the ability of the mill to reduce the particle size is affected by the load in the mill.

 The flash floatation PSD affects the final tails PSD. Additionally, all the final floatation PSDs are connected.

 All the final tails PSD variables are connected to the same variables. This is accurate since each of the final tails PSD variables represents the same property.

 Mill23 power, availability and stops are all highly connected to Mill23s PSDs and the primary floatation PSDs (Ptails). Since the ability of these mills to reduce the particle size is affected by their operating variables, this is accurate.

 The Mill23s power has a strong connection on the final recovery, indicating that these mills have a strong influence on this KPI.

 The feed mass affects a large number of variables; especially the PSDs. This PSD affects primary mill performance, which, as seen in Figure 7-7, drives the performance of the subsequent units. It also strongly affects the mill power, which is accurate from a fundamental perspective.

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Figure 7-8: Linear cross-correlation connectivity graph on concentrator training data

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7.3.3. Partial cross-correlation topology extraction

The number of lags, k, chosen for PC was the same as was chose for LC. Using the method described in section 4.3.4 for selection of the significance threshold, the sample size could be used in Equation 7-3 to determine a value.

𝜌_{𝑚𝑎𝑥,𝑡ℎ}𝑃𝐶 (𝑁) = 1.647𝑁−0.428_{+ 3.864𝑁}−0.772 _{Equation 7-3}

𝜌_{𝑚𝑎𝑥,𝑡ℎ}𝑃𝐶 _{(150) = 0.274 Equation 7-4}

It was observed that this threshold of 0.274 was actually too low, giving too many connections. Upon inspection of the connectivity matrix it was observed that many of the entries were between 0.2 and 0.4. It was therefore decided to double the significance threshold to give a more accurate and useful connectivity graph. As with the previous case studies the threshold determined from the proposed method was too lax, resulting in a connectivity graph with too many spurious connections. Using this threshold the connectivity graph shown in Figure 7-9 was generated using partial cross-correlation. Considering the causality presented by Groenewald (2014) the following key connections are discussed for the PC graph shown in Figure 7-9:

 For the Mill23s their powers, stops availability and PSDs are all connected. The fact their powers affect their PSDs indicates accurate capture of process behaviour, since the operating conditions of these mills will have a strong influence on their ability to reduce size of the particles.

 The primary mill feed rate and its inlet water ratio are connected.

 The MassPull affects the recovery and grade of the tails, which is accurate, since a larger throughput would affect both grade and recovery.

 The final tails PSD variables are all connected, which is accurate since they all represent the particle size of the same stream.

 The final tails grade affects the recovery. These two KPIs are strongly linked, since recovery is dependent on grade.

 The Mill23s variables affect the downstream tails PSD.

 The Cyclone feed flow affects its pressure.

 The primary mill power affects its load. Unfortunately there is no connection between these variables and the mill feed rate and water inlet ratio.

 The cyclone split classification affects the Mill23s. The split classification is affected by the PSD of the primary floatation tails. This may seem like it’s in the wrong direction, but as stated in the beginning of the topology extraction section the classification is profoundly

Chapter 7 -Case Study: Fault Diagnosis Applied to Concentrator Process Page 157 affected by the feed PSD. Unfortunately no measurement is available for that PDS, and the closest estimate of this PSD is that of the primary floatation, so in fact it can be considered accurate. There are no connections between cyclone feed flow and pressure and its splitting unfortunately.

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Figure 7-9: Partial cross-correlation connectivity graph for concentrator training data

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7.3.4. Transfer entropy topology extraction

Using the method described in section 4.3.4 for selection of the significance threshold, the sample size could be used in Equation 7-5 to determine a value.

t_{𝐱→𝐲,th}(𝑁) = 0.0018𝑁0.465_{+ 0.0054𝑁}0.412 _{Equation 7-5}

t𝐱→𝐲,th(150) = 0.0611 Equation 7-6

Using this threshold of 0.0611 the connectivity graph resulted in a connectivity graph with too many connections. Doubling the threshold and then rounding it up to a value of 0.15 resulted in a more accurate connectivity graph, shown in Figure 7-10.

Considering the causality presented by Groenewald (2014) the following key connections are discussed for the TE graph shown in Figure 7-10:

 The recovery is affected by the flash floatation and Mill23 PSDs, indicating that the particle sizes have a substantial effect on the performance of the process.

 The feed grade is affected by many downstream variables, including the floatation PSDs and concentrate grade. This connection is contrary to the physical process behaviour, since the feed at the start of the process cannot be influenced by downstream variables.

 The cyclone pressure affects the mass pull.

 The mass split affects the tails Grade of Mill2 and Mill3.

 The Split2Cr affects a lot of variables, which makes sense since this is where the process split. It affects the Flash floatation and the Mill23s PSD.

 The Mill23s PSDs affect the mill power, as does the primary mill feed rate.

 In general, for the TE graph, it appears that many of the right connections exist, but possibly in the wrong direction. However, since the data used has a lot of complicated and unknown interactions that may be introduced by the operator, and not just by mass, energy balance and automatic control, this makes it difficult to define the proper direction for the connections.

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Figure 7-10: Transfer entropy connectivity graph for concentrator training data

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