To minimise the drawbacks of analytical models and black box models, grey-box models are developed. There are various ways of formulating a grey-box model, e.g. a serial approach or a parallel approach (Estrada-Flores et al. 2006). A serial approach means an output of the black box model can be used as an input to the white box model (or vice versa). On the other hand, for the parallel approach, the same input data are sent to both the black and white box models at the same time. The model outputs were first calculated by the white box model, while the black box model compared the analytical results with the real fault free measurements and generated a correction signal to improve the overall accuracy.
The serial approach is often applied in the modelling of refrigeration systems (Estrada-Flores et al. 2006). When building up a component level physical model for a system, a black box sub-model can be applied for the complex components such as the compressor (Browne and Bansal 2002). Other examples of serial approach grey-box models for FDD applications include the thermodynamic models Gordon and Ng (2000), Lee (2004), Ng (2008), and the characteristic models of Jia and Reddy (2003).
The thermodynamic grey-box model of Gordon and Ng (2000), developed for centrifugal and reciprocating chillers, was based on the First and Second Laws of Thermodynamics, describing the energy and entropy balances of the chillers, as given below: 𝑇e,in 𝑇c,in 1 + 1 𝐶𝑂𝑃 − 1 = 𝑇e,in 𝑄e ∆𝑆t+ 𝑄leak 𝑇c,in−𝑇e,in 𝑇c,in×𝑄e + 𝑅×𝑄e 𝑇c,in 1 + 1 𝐶𝑂𝑃 2.7
where the internal entropy generation rate in the chiller due to internal irreversibility (ΔST, kW/K), the equivalent heat loss from the chiller (Qleak, kW) and the thermal resistance of the heat exchangers (R, K/kW) were the regression coefficients all of which had physical meanings, and they were obtained using a regression model. All the other parameters in the equation could be obtained from measurement. The training of this model was to use a set of the above measured parameters under fault free condition to determine ΔST, Qleak and R by multiple linear regressions. The model could then be used to predict system COP. In this case, ΔST, Qleak and R were the model parameters of a first principle model that cannot be measured directly. The
48 model was first transformed into a regression function to obtain the above three coefficients. Then they could be applied as constant inputs to Equation 2.7 to calculate the model output.
One advantage of this kind of grey-box model is that as the regression coefficients of the grey-box models are physically meaningful and it is possible to carry out fault detection and diagnosis by examining their variations during faulty conditions. Saththasivam and Ng (2008) applied the above model for faults detection. Instead of relying upon the model output COP for FDD, they claimed that the values of ΔST and R could be used to indicate the health of the chillers. ΔST could be linked to the operation of the compressor and the expansion valve (e.g. an increased ΔST may suggest excessive oil in the compressor), whereas the thermal resistance R was related to heat transfer rates in the evaporator and condenser (e.g. an increased R could indicate reduced cooling water flow or condenser fouling). For a given chiller system, those coefficients were considered as constants under normal operating conditions.
Equivalent heat loss Qleak was less likely to be influenced by operation conditions when compared with the other two, hence less useful as a FDD detection parameter. During the fault detection process, 12 fault free data sets from a 90-ton centrifugal chiller were applied to determine ΔST and R by the regression method. Input parameters based on measured values included: cooling load, secondary fluid inlet temperatures for the condenser and evaporator and COP, and model outputs were
ΔST and R, and they were compared with the constant values for FDD.
Another example of a serial hybrid model was the characteristic model developed by Jia and Reddy (2003). It combined refrigeration cycle analysis with regression correlations. Simplified lumped physical models were developed first to calculate the characteristic parameters of the components, that describe the performance of the components and were linked to certain types of faults (e.g. motor efficiency and polytrophic efficiency for the compressor, overall heat conductance (i.e.
UA values) of the evaporator and condenser, the product of the fluid friction
coefficient (a function of refrigerant velocity) and the cross-sectional area of the orifice for the expansion valve, and COP for the overall system performance).
49 When using the models for FDDs, the above characteristic parameters were first calculated by using a set of fault free measured data under different working conditions, and then they were fitted into regression functions taking cooling water inlet, chilled water inlet and chilled water outlet temperatures as inputs. The fitted regression functions were then used as baseline models for fault detection. This method was considered as an inversed serial grey-box model because the using of white box model is prior to the regression model.
The three main types of refrigeration system models each have their strengths and limitations. Comparisons of different models, mainly based on the accuracy, data requirement and computational requirement, had been done by various researches (Peitsman and Bakker 1996; Sreedharan and Haves 2001; Swider 2003). Swider (2003) compared four regression models, including a grey box model, and two ANN models when applying them to a single-circuit centrifugal chiller and a twin-circuit twin-screw chiller. The four regression methods included linear regression, bi-quadratic (Yik and Lam 1998) regression, multivariable regression and the Gordon and Ng’s (2000) thermodynamic grey box model; the two ANN models were a RBF model from Swider et al. (2001) and a MLP model using tanh activation function. All, except the bi-quadratic model, used the same three input variables: cooling capacity, condenser water and evaporator water inlet temperatures; the bi-quadratic model only needed the first two, and they all were trained with the same set of data. The output of all the models was COP.
In terms of modelling accuracy, all the above models provided accurate predictions for the centrifugal chiller, but the Gordon and Ng’s thermodynamic model had the advantage of requiring less training data due to its physical meaningful equations. However, for the more complicated twin-screw twin-circuit chiller, only the two ANN models could produce acceptable predictions, the four regression models could only provide accurate predictions when they were improved by setting up separate models for each circuit, hence reducing the system complexity in modelling sense. Generally speaking, when there were enough training data, the ANN models gave the highest accuracy, followed by the three regression models, and the thermodynamic model was the least accurate. The results also showed that although MLP gave slightly better prediction results than RBF, they required longer training
50 time. The thermodynamic model showed its advantage when the training data were highly limited.
In conclusion, the selection of a prediction model for FDD should consider the complexity of the target system and the selected faults, the availability of the measurements and the accuracy of the prediction.