Neural Comput & Applic DOI 10.1007/s00521-013-1347-5
ORIGINAL ARTICLE
A hybrid artificial neural network—mechanistic model for centrifugal compressor Fei Chu • Fuli Wang • Xiaogang Wang Shuning Zhang
•
Received: 13 March 2012 / Accepted: 25 January 2013 Springer-Verlag London 2013
Abstract A mathematical model is an important tool for design and optimization of centrifugal compressor. However, owing to the varying compressor speeds and the complexity of the flow dynamics inside the impeller and diffuser, the currently available mechanistic models may yield inaccurate results. The purpose of this paper is to present a hybrid modeling approach for developing a quantitatively accurate model for centrifugal compressor. Two novel hybrid models, that is, additive and multiplicative hybrid models each of which consists of a threelayer back-propagation artificial neural network (ANN) component and a mechanistic component suitably modified to describe the performances of multistage centrifugal compressor, were constructed and compared with the welldeveloped ANN model. The results from the hybrid models showed better performance compared to the ANN model. Besides, the hybrid models demonstrated much better performance than the pure mechanistic model, and the multiplicative hybrid model, in general, showed better accuracy than that of the additive hybrid model in our case. Keywords Artificial neural network Centrifugal compressor Hybrid model Mechanistic model Back-propagation
F. Chu (&) F. Wang X. Wang S. Zhang State Key Laboratory of Integrated Automation for Process Industries, College of Information Science and Engineering, Northeastern University, Shenyang 110819, China e-mail:
[email protected]
1 Introduction Since the early twentieth century, centrifugal compressors, owing to their simplicity, high efficiency, reliable operation, and easy maintenance when working in a wide range of conditions, have been widely used for the pressurization of fluid. Applications include turbocharging of internal combustion engines, air compression in gas turbines used in power plants, and for aircraft and marine propulsion. Another important application is pressurization and transportation of gas in pipelines and in the process and chemical industries. Mathematical models play an important role in the design and optimization of centrifugal compressor or centrifugal compression system. During the last several decades, many mechanistic models, which are also called ‘‘deterministic’’ or ‘‘first principles’’ model, have been developed in order to examine the heat transfer, pressure rise and losses processes occurring throughout the stage [1–10]. However, due to the varying compressor speeds and the complexity of the flow dynamics inside the centrifugal compressor, several of these mechanistic models may yield inaccurate results; especially, some small losses, for example, inlet casing losses, mixing losses, and leakage losses cannot be reliably described by the mechanistic models. Thus, many previous studies have attempted to establish attractive alternative compressor models based on input–output relations to predict compressor performance [11–17]. Artificial neural network (ANN) has been proven to be one of the most effective techniques applied for approximating arbitrarily complex processes [18]. It can approximate a nonlinear relationship between input and output variables without the requirement of explicit mathematical representations. By using ANN, it is unnecessary to have
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Neural Comput & Applic
any prior knowledge about the relationships that exist among the states of the process [19, 20]. Although an ANN approach can be used to model the centrifugal compressor [21], a hybrid model constructed from a basic mechanistic model coupled with an ANN model should be considered since it can be applied to describe some physical processes within the compressor. It was stated that the combination of the mechanistic model with ANN can provide a good accuracy of process modeling with faster development [22–24]. In this paper, we will present a hybrid ANN mechanistic model for predicting the main performances of centrifugal compressor, that is, the pressure ratio and efficiency, under different operating conditions. More specifically, the mechanistic model, which is suitably modified to describe the characteristic of the variable speed centrifugal compressor with multistage, and ANN will be used to improve the accuracy of the hybrid model by estimating the unmodeled part in the mechanistic model. Two data sets from a real gas turbine power plant and a simulation study are used for training the ANN, respectively, in the current study. The reliability of the hybrid model is examined by comparing its prediction to those of ANN and mechanistic model with the test data sets. The remainder of this paper is organized as follows. Brief introduction to ANN and the formulation of the hybrid model will be given in Sect. 2. Section 3 will describe the construction of ANN and hybrid models. In Sect. 4, the results and discussion will be presented. Conclusions will be made in Sect. 5.
2 Method formulation 2.1 ANN ANNs are nonlinear mapping systems that have emerged as a result of simulation of biological nervous system on a computer. Their ability to approximate arbitrarily complex processes by experimental or historical data makes ANN very flexible and powerful than any other parameter approaches [15, 25]. The multilayered feed-forward (MLFF) neural network [26] is the most popular ANN in engineering application, and a typical MLFF network is shown in Fig. 1. It consists of massively interconnected simple processing elements, that is, the neurons or nodes as shown in Fig. 2, arranged in a layered structure, where the strength of each connection is given by an assigned weight; these weights are the internal parameters of the network. The input neurons are connected to the output neurons through layers of hidden nodes. Each neuron in the hidden or output layer receives information in the form of inputs from other neurons and
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Y1
Ym
YM
ωkio Z1
b2
Zk
ZK
ωijh
b1
X1
Xn
XN
Fig. 1 Typical multilayered feed-forward neural network
X1
W1
W2 X2
∑
F (∑)
Y
Wn Xn
Fig. 2 Artificial neural unit
processes it through transfer function. As shown in Fig. 2, the output from a neuron, in the hidden or output layer, can be defined as ! N X y¼w x j xj þ b ð1Þ j¼1
where xj is the weight, ‘‘b’’ is the bias added to the weighted sum, and w(sum) is the transfer function. The most widely used transfer function is the sigmoid function, a common form of which is defined as follows wðsumÞ ¼
1 1 þ easum
ð2Þ
where a is the slope parameter. The model equation for the entire neural network as shown in Fig. 1 can be expressed as follows ! K X 2 o yk ¼ w xki zi þ b2 i¼1 ! ! K N X X 2 o 1 h ¼w xki w xij xj þ b1 þ b2 ð3Þ i¼1
j¼1
where zi is the output signal from the ith hidden neuron and yk the output signal from the kth output neuron. The ANN in Fig. 1 only contains one hidden layer. Nevertheless, as determined by the complexity of the problem, more than
Neural Comput & Applic
one hidden layer can be chosen. The number of neurons in the hidden layer is also related to the complexity of the problem, which is often obtained by trial and error during training. More detailed ANN theory can be found in the literature [25, 26].
compressor. The modified mechanistic model is summarized below.
2.2 Hybrid model
The large centrifugal compressors in industry application are all multistage for obtaining high pressure rise. Each of the stage consists essentially of a rotating impeller which impacts a high velocity to the gas and a number of fixed diverging passages in which the gas is decelerated with a consequent rise in static pressure [6]. The gas is charged into the first stage by a stationary inlet casing. Then, the gas is compressed by each of the stage one by one. Finally, the high-pressure gas is discharged through the volute and outlet pipe as shown in Fig. 4. The performance of the multistage centrifugal compressor can be obtained through a stage stacking calculation [28], that is, the calculation process proceeds from the first stage to the last stage based on the characteristics of each stage, and the static pressure and temperature at the stage outlet are used as the inlet conditions for the next stage just as shown in Fig. 4. Then, the pressure ratio e, temperature ratio s, and efficiency g of the entire multistage centrifugal compressor can be calculated as n P0nþ1 P02 P03 P0nþ1 Y e¼ ¼ ... ¼ ei ð6Þ P01 P01 P02 P0n i¼1
In order to more accurately predict the performance of centrifugal compressor in the case when a mechanistic model with limited accuracy is available, we propose a hybrid model that combines a mechanistic element with an ANN element. The ANN is trained on the residual between the data and the mechanistic model to compensate for any uncertainties that arise from the inherent complexity of the centrifugal compressor. In other words, this approach combines the part of the model that is well known from the physics of the compressor, with another part that is poorly known but can be estimated quite effectively using ANN. The schematic of an additive hybrid model is shown in Fig. 3. The pressure ratio e (or efficiency g) can be described by an additive model emul(X), which has a mechanistic component and an ANN component as follows e ¼ eadd ðXÞ ¼ emech ðXÞ þ eANN ðXÞ
ð4Þ
where the vector X = [x1,x2, … ,xN] represents process variables. The function emech(X) indicates the mechanistic model, while the function eANN(X) is an ANN model. The net prediction e is the sum of the predictions from emech(X) and eANN(X). Similarly, a multiplicative model can be constructed as e ¼ emul ðXÞ ¼ emech ðXÞ eANN ðXÞ ð5Þ The mechanistic model used in the hybrid model is an analytical centrifugal compressor mechanistic model based on the model developed by Jiang [1]. This original model that is suitable for single-stage centrifugal compressor was validated against the experimental data from a laboratoryscale gas turbine installation [1, 5], which was erected in the Energy Technology Laboratory of Eindhoven University of Technology [27]. Appling the method described in [28], we extended the capability of the original model to predict the performance of multistage centrifugal Mechanistic model
2.3 Mechanistic model of multistage centrifugal compressor
s¼
n T0nþ1 T02 T03 T0nþ1 Y ¼ ... ¼ si T01 T01 T02 T0n i¼1
gðm; U1 Þ ¼
ð7Þ
ec 1=c 1 s1
ð8Þ
where P0i and T0i (i = 1,2,3,…,n) are the pressure and temperature of each of the stage, respectively, ei and si (i = 1,2,3,…,n) are the pressure ratio and temperature ratio of each of the stage, respectively, gi(m, U1) is efficiency, and c is the specific heat ratio of the gas. The pressure ratio and temperature ratio of each stage can be calculated by the model described in [1, 5, 6], details of which is list in Appendix 1.
Ymech
3 Development of ANN models and hybrid models
+
3.1 Preparation of data sets
X
Y
+ ANN model
Fig. 3 Schematic of additive hybrid model
YANN
Since tight and lower budget no longer allow an extensive compressor test program, a simulation study is applied in this paper to produce full-scale compressor data set for evaluating the performance of the hybrid models. The simulated data set is obtained from numerical simulation
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Neural Comput & Applic Fig. 4 Diagrammatic of multistage centrifugal compressor
Gas Inlet P01 T01
P02 T02 m02
Stage
m 01
1
Fig. 5 Gas turbine power plant unit and large three-stage centrifugal compressor
Stage
P03 T03
P0n T0n
m 03
m0n
2
Valve B
Stage P0n+1 T0n+1 m 0n+1 High Pressure Gas
N
Cooling Water
Bypass Valve C
Water Cooled Condenser Valve A
Scrubber Bypass
N
Gas Compressor
Gas holder
Generate
P T
using a detailed centrifugal compressor mechanistic model as mentioned in Sect. 2.3 (Eqs. (6)–(8)). As the mechanistic model is also used for the mechanistic component of the hybrid model, some difference is introduced to the model for the two different applications, as list in Appendix 2, to validate the rationale of the hybrid model that ANN can be used to improve the accuracy of compressor performance modeling by estimating the unmodeled part in the mechanistic model. Then, we obtain two different mechanistic models: type A mechanistic model, which is used for constructing the hybrid model; type B mechanistic model, which is employed to produce the data set for training ANN and validating the hybrid models. Table 4 shows the values of parameters used in the compressor mechanistic models: type A and type B mechanistic models. To generate the data set, the main operational variables of the centrifugal compressor, that is, inlet pressure, inlet temperature, flow rate, and speed are varied in the range of 106–146 kPa, 273.15–323.15 K, 4–97 kgs-1, and 4,000–5,500 rpm, respectively, and the values of the corresponding pressure ratio and efficiency are recorded as the outputs. There are a total of 130 samples produced. Furthermore, in order to simulate the real operation and measurement environment, noise (5 %) was added to the pressure ratio and efficiency, respectively. In order to further evaluate the capability of the hybrid models in performance modeling of centrifugal compressor, data set from a real case was collected. In this real case, the large three-stage centrifugal compressor under
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Gear box
Combustion Chamber
Power Output
Flow meter
Turbine Exhaust
Air Compressor
T P
Air
study is from the spare power plant of a steelwork, which is used to compress the coal gas burned by gas turbine (GT) power plant. The compressor is coupled to the GT through a gear and consumes a non-negligible fraction of the power generated as shown in Fig. 5. The power plant consisted of a gas turbine (includes of an air compressor, a combustion chamber, and a turbine), a fuel system (includes of a gas compressor, a condenser, a scrubber, and three valves), and a generator. As shown in Fig. 5, the setup included a Brayton cycle and a gas flow circuit. The gas from the gas holder flows through the scrubber and was compressed by the three-stage centrifugal gas compressor. Part of the gas with high pressure was injected into the combustion chamber to be burned for energy. Another part was cooled by condenser and flow back to the scrubber for the anti-surge control of centrifugal gas compressor. The required air by combustion chamber was provided by the air compressor. Both the air and gas compressors were driven by the turbine and consumed a non-negligible fraction of the power generate. It was noteworthy that the gas compressor was coupled to the turbine by a three-speed gear box. As shown in Fig. 5, to measure the steady-state performance of the centrifugal compressor, the power plant was equipped with several temperature probes and pressure transducers. Moreover, the compressor mass flow rate could be determined from the mass flow transducer installed at the location close to the compressor outlet, whereas the compressor speed was measured by
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tachometer. All the measured data were stored in the storage area of the distributed control system (DCS). We took 66 historical data samples of the compressor from DCS. The compressor input parameters are as follows: gas inlet temperature and pressure (Tgi, Pgi), mass flow rate (mg), and rotational speed (Ng). Unlike the data from experiments over a wide range of operating conditions, the range of variation of input parameters for the historical data is narrow: 273 K \ Tgi \ 318 K, 106 kPa \ Pgi \ 110 kPa, 37 kgs-1 \ mg \ 51 kgs-1 and, when the unit reaches the steady-state condition, the compressor speed keeps constant ng = 5,302 rpm. However, since the historical data contains a great deal of information which reflects the real compressor characteristics, it is still a good sample to test the performances of the models developed in this paper. The output parameters are outlet temperature (Tgo) and pressure (Pgo). Based on the output parameters, the compressor pressure ratio and efficiency can be calculated by the Eqs. (6) and (7). For optimal design of the ANN, each data set is divided into three different groups: training, test, and validation data. The training data were used to train the neural network to obtain the weights for the network. The test data were used to determine when training should be stopped. Finally, the validation data were employed to demonstrate the performance of the hybrid models. 3.2 Constructing ANN and hybrid models An optimal structure of ANN model that gives the minimum value of the MSE is considered to be an appropriate ANN model. From results, it was found that a three-layer feed-forward neural network with one hidden layer was adequate for modeling. In general, input and output variables of the ANN can be any process variables that measurable. In our studies, input variables of the ANN include Input variables
Output variables
Centrifugal compressor
X
+ +
ANN model
Training algorithm Fig. 6 Constructing the additive hybrid model
+ _
3.3 Training ANN Levenberg–Marquardt back-propagation algorithm with the early stopping mechanism is used to train the threelayer feed-forward ANNs [11]. The activation function in the hidden layer, an important feature influencing the network performance, was chosen as a tangent sigmoid function. To optimize the network structure, the number of hidden neurons in the hidden layer is altered during the training processes. Training of an ANN model is the process of adjusting the weights of links among the neurons. The weights are updated after processing the whole training data set (batch training). During training, a mean square error (MSE) function is reduced as the number of training epoch increases. Test data set is used to determine when to stop training process by monitoring the error of the test data. If the network provided a satisfied error (MSE B 1 9 10-4), the training process is finished. Once the ANN is trained, the hybrid model is ready to predict the compressor performances: the pressure ratio and efficiency.
4 Results and discussions
Y Mechanistic model
inlet temperature/pressure, flow rate, and speed; the output variables are the pressure ratio and efficiency. The procedure to create an additive hybrid model is illustrated in Fig. 6. From training data, we can obtain the input vector X and the output vector Y (Y = [e, g]). From the input vector, the mechanistic component of the hybrid model, type A mechanistic model, can calculate the output vector Ymech. The difference Y–Ymech is obtained by subtracting the results given by type A model from the output. The input vector and the difference vector then are used to train the ANN models. Finally, the outputs of the trained ANN model and the mechanistic model are combined to determine the total model outputs, corresponding to a given set of input variables. Similarly, the multiplicative hybrid model can be constructed as above.
In this section, the performances of the two hybrid models in modeling centrifugal compressor are compared to those of the ANN and mechanistic models using simulation study and real case data sets. All the algorithms are implemented in Matlab 7.6 and the back-propagation algorithm is solved by the neural network toolbox [30]. With validated data from the simulation study, Figs. 7 and 8 compare the performance of the additive hybrid model and multiplicative hybrid model with the pure ANN model, respectively. Each data point from the prediction in the Figs. 7 and 8 indicates an output from one of the models mentioned above, corresponding to a certain set of the input variables in the validated data set, and the outputs
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Neural Comput & Applic
(a) 5.5 Pressure ratio
Table 1 Summary of prediction errors from the models tested in the simulation study
Type B Type A Pure ANN Additive hybrid
5 4.5 4 3.5 3 2.5
Type A mechanistic model
2 1.5 1
5
10
15
20
25
30
Simulated data samples
(b) 0.8
Pressure ratio
Efficiency
MRE (%)
MSE (10-3)
MRE (%)
MSE (10-3)
20.6
279.2066
38.08
12.3654 0.5429
Pure ANN model
4.78
19.8246
5.52
Additive hybrid model
2.48
9.0060
5.16
0.3349
Multiplicative hybrid model
2.6
7.1000
3.01
0.1227
0.7
Efficiency
0.6 0.5 0.4 0.3 Type B Type A Pure ANN Additive hybrid
0.2 0.1 0
5
10
15
20
25
30
Simulated data samples
Fig. 7 Prediction results of additive hybrid model and pure ANN model for a pressure ratio and b efficiency in the simulation study
(a) 5.5 Type B Type A Pure ANN Multiplicative hybrid
Pressure ratio
5 4.5 4 3.5 3 2.5 2 1.5 1
5
10
15
20
25
30
Simulated data samples
(b) 0.8
Efficiency
0.7 0.6 0.5 0.4 0.3
Type B Type A Pure ANN Multiplicative hybrid
0.2 0.1 0
1
3
5
7
9
11
13
15
17
19
21
23
25
27 29 30
Simulated data samples
Fig. 8 Prediction results of multiplicative hybrid model and pure ANN model for a pressure ratio and b efficiency in the simulation study
of the validation data (outputs of the type A and II mechanistic models) are also plotted in the same figures. The data points are arranged according to the magnitude of the
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pressure ratio output of the type A mechanistic model. From Figs. 7 and 8, it can be seen that, since two kinds of difference listed in Appendix 2 is used in the simulation study, there is obviously deviation between the pressure ratio and efficiency outputs of the type A and type B mechanistic models. It can be also seen that the performances of the additive and multiplicative hybrid models can be well improved by the ANN components. Besides, the results from the pure ANN model are in good agreement with the validation data also. The corresponding prediction errors, the mean relative error (MRE) and mean square error (MSE) from all the models tested in the simulation study, are listed in Table 1. From Table 1, it can be seen that the additive hybrid model shows much better pressure ratio e predictions and slightly better efficiency g predictions than the pure ANN model. On the other hand, the multiplicative hybrid model shows much better both pressure ratio e and efficiency g predictions than the pure ANN model. The reason for the performance improvement in the hybrid model is that by design, the ANN component of the hybrid model only approximates the theoretically unknown parts, while the other parts are already described by the mechanistic component [29]. Nevertheless, in a pure ANN model, the full scale of the centrifugal compressor has to be handled by a single ANN model. The prediction results of the pure mechanistic model, pure ANN model, additive hybrid model, and multiplicative hybrid model for the real case data set are shown in Figs. 9 and 10. We can see that the performances of the additive hybrid model, the multiplicative hybrid model, and the pure ANN models are all better than that of the pure mechanistic model in two cases of pressure ratio and efficiency prediction, respectively. At present, for the available compressor mechanistic models, a certain number of aeration kinetics parameters, such as slip factor r, shock loss coefficient f, and reference area A, have to be adjusted beforehand to fit historical or experimental data (for the real case, based on the compressor historical data, the values of slip factor r, shock loss coefficient f, and
Neural Comput & Applic
(a) 3.9 Pressure ratio
Table 2 Parameters for the centrifugal compressor mechanistic model
Historical data Mechanistic model Pure ANN model Additive hybrid model
3.8 3.7
Parameter
3.6
Stage 1
3.5
Slip factor r
0.5870
0.6400
0.5908
3.2
Blade inlet angle b1b ()
33
33.5
32
3.1
Average diameters at impeller eye D1 (m)
0.5883
0.5803
0.5767
Impeller exit diameter D2 (m)
1.080
1.080
1.080
Mean impeller channel length li (m)
0.4250
0.4040
0.3720
Mean diffuser channel length ld (m)
1.0310
0.9860
0.4170
Impeller mean hydraulic channel diameter Di (m)
0.1158
0.1006
0.0904
Diffuser mean hydraulic channel diameter Dd (m)
0.0822
0.0677
0.0874
2
4
6
8
10
12
14
(b) 0.81 Historical data Mechanistic model Pure ANN model Additive hybrid model
0.8
Efficiency
Stage 3
3.3
Historical data samples
0.79 0.78 0.77 0.76 0.75 0.74
2
4
6
8
10
12
14
Historical data samples
Fig. 9 Prediction results of additive hybrid model, pure mechanistic model, and pure ANN model for a pressure ratio and b efficiency in the real case study
Average molecular weight of gas
27.68
27.68
27.68
Specific heat ratio
1.36
1.36
1.36
Specific heat capacity (Jkg-1 K-1)
1,118.50
1,118.50
1,118.50
Shock loss coefficient
1.0158
1.3907
1.1832
Reference area A (m2)
0.2717
0.2643
0.2612
Sum of clearance loss, volute loss, and back flow loss
0.055
0.055
0.055
Table 3 Summary of prediction errors from the models tested in the real case study
(a) 3.9
Historical data Mechanistic model Pure ANN model Multiplicative hybrid model
3.8
Pressure ratio
Stage 2
3.4
3
3.7
Pressure ratio
Efficiency
MRE (%)
MSE (10-3)
MRE (%)
MSE (10-3)
Pure mechanistic model
4.38
25.70
1.65
0.2
Pure ANN model
1.25
4.5
0.73
0.0533
Additive hybrid model
0.95
3.1
0.65
0.0423
Multiplicative hybrid model
0.7
1.0
0.48
0.0212
3.6 3.5 3.4 3.3 3.2 3.1 3
2
4
6
8
10
12
14
Historical data samples
(b) 0.81 Historical data Mechanistic model Pure ANN model Multiplicative hybrid model
0.8 0.79
Efficiency
Value
0.78 0.77 0.76 0.75 0.74
2
4
6
8
10
12
14
Historical data samples
Fig. 10 Prediction results of multiplicative hybrid model, pure mechanistic model, and pure ANN model for a pressure ratio and b efficiency in the real case study
reference area A of the mechanistic model are determined by genetic algorithm and shown in Table 2, in which the physical dimensions of some of the parameters of the large three-stage centrifugal compressor are listed. Even so, mechanistic models may yield inaccurate results due to the complexity of the flow dynamics inside the centrifugal compressor. When new operating conditions are used, the mechanistic models usually cannot offer quantitatively good predictions without new round of parameters adjustment. Table 3 summarizes the prediction error results of the four models tested in the real case study. We can see that as
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Neural Comput & Applic
different classes of experiments [1, 5]. Further, Jiang et al. [1] developed a modified version which is capable of system simulation in the virtual test bed (VTB) computational environment. For easy implementation in the VTB platform, the nonlinear governing equations are discretized in resistive companion form. The main compressor characteristics equation of the model can be described as follows:
the compressor historical data are concentrated in a small range, the accuracy of the pure mechanistic model is good and acceptable for industrial application. Nevertheless, the ANN components can make further refinements to the accuracy of the model and the additive hybrid model shows slightly better performances than the pure ANN model; meanwhile, there is a great improvement in accuracy in the multiplicative model in terms of MRE and MSE comparison. The reason for the performance improvement in the hybrid model is given in the above simulation study. In the two cases here, the data are sufficient to train ANN models with good accuracy in the prediction of compressor performances. In the cases when a well-trained ANN model is not available for inadequate data, it is expected greater improvement in the performance by the hybrid model over the pure ANN model.
gi ðm; UÞ ¼
5 Conclusions
Dhideal ¼
In this paper, two hybrid models, that is, the additive hybrid model and the multiplicative hybrid model each combining an ANN model with a mechanistic model were developed for the multistage centrifugal compressor. The motivation is that accuracies of the mechanistic model can be improved by the ANN model, which is a very flexible and powerful technique in approximating arbitrarily complex processes by experimental or historical data. In our tests, two data sets from a simulation study and a real case are used to train the ANN models and compare the performance of hybrid models with the pure ANN model and the pure mechanistic model, respectively. Both the additive and multiplicative hybrid models demonstrated much better performance than the pure mechanistic model, especially the multiplicative hybrid model. Additionally, the hybrid models showed modest improvement in accuracy, compared to the pure ANN model. Acknowledgments This work was financed by the National Nature Science Foundation of China (No. 61074074; No. 61174130; No. 61004083); Project 863 (No. 2011AA060204); Project 973 (No. 2009CB320601), China; the Fundamental Research Funds for the Central Universities (N100604008).
Appendix 1: Single-stage centrifugal compressor model for system simulation
x_ ¼
1 ðst sc Þ J
x sc ¼ Wc ¼ m Dhideal e¼
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ð10Þ
p02 g ðm; UÞ Dhideal c=c 1 ¼ ð1 þ i Þ p01 T01 cp
ð11Þ
Dhideal Dg Dhideal þ Dhloss
ð12Þ
Wc ¼ r U22 m
ð13Þ
where x is rotational speed (rpm), st is compressor torque (Nm), st is drive torque (Nm), J is spool moment of the inertia (kgm2), m is mass flow rate (kgs-1), Wc is power delivered to the fluid (J), Dhideal is ideal specific enthalpy delivered to the fluid (Jkg-1), T01 is inlet temperature (K), cp is gas velocity enter the impeller (ms-1), Dhloss is sum of the incidence loss and friction loss in impeller and diffuser, Dg represents others small losses, such as back flow loss and clearance loss. Ideally, we would have the same energy transfer Dhideal for all flow rates. However, due to various losses, the energy transfer is not constant and we now include this in the analysis. The two major losses, expressed as specific enthalpies, are incidence loss in impeller and diffuser, Dhii and Dhid; friction loss in the impeller and diffuser, Dhfi and Dhfd. 1 Dhii ¼ fðU1 Ch1 cot b1b Ca1 Þ2 2 1 cot b1b m ¼ f U1 2 q01A1 1 r D2 U1 cot a2b m Dhid ¼ f 2 q01 A1 D1 Dhfi ¼ Dhfd ¼
The model used for the mechanistic component of the hybrid model is an analytical multistage centrifugal compressor mechanistic model based on the model developed by Jiang [1]. This original model was originally developed for surge control design [5] and its response was compared to the measured response from the laboratory in three
ð9Þ
ð14Þ ð15Þ
2fli m2 sin2 b1b
ð16Þ
2fld m2 Dd q201 A21 sin2 a2b
ð17Þ
Di q201 A21
where f is the shock loss coefficient, U1 is the inlet tangential velocity of the impeller (ms-1), Ch1 is inlet tangential velocity of gas (ms-1), b1b is blade inlet angle (rad), Ca1 inlet radial velocity of gas (ms-1), q01 is constant stagnation inlet density (kgm-3), A1 is reference area (m2),
Neural Comput & Applic
r is slip factor, D1 is inlet reference diameter (m), D2 is diameter at the impeller tip (m), a2b is diffuser inlet angle (rad), f is friction loss, li is mean channel length of impeller (m), ld is mean channel length of diffuser (m), Di is mean hydraulic channel diameter at impeller (m), Dd is mean hydraulic channel diameter at diffuser (m), and ld is mean channel length of diffuser (m). Other losses, such as back flow loss Dgbf, clearance loss Dgc = 0.3(lcl/b), volute loss 0.02 B Dgv B 0.05, and losses 0.02 B Dgd B 0.07 due to inadequate diffusion, will be taken into account when computing the efficiency of the compressor. gi ðm; U1 Þ ¼
Dh02 Dgc Dgbf Dgv Dgd Dh02 þ Dhloss ð18Þ
where Dhloss = Dhii ? Dhid ? Dhfi ? Dhfd. More details of the compressor model can be found in the literature [1, 5, 6].
Appendix 2. Type A and type B mechanistic model In order to validate the rationale of the hybrid model that ANN can be used to improve the accuracy of compressor performance modeling by estimating the unmodeled part in the mechanistic model, some difference is introduced to the mechanistic model when it is used for the mechanistic component of the hybrid model and producing data sets for training ANN, respectively. Then, two different type mechanistic models, type A and type B, are obtained as described earlier in this paper. Mechanistic models may yield inaccuracy results owing to two major reasons: one is that some complexity physical phenomena in the compressor, such as small losses, cannot be reliably described by the mechanistic models; the other one is that some important aeration kinetics parameters, for example, shock loss coefficient f, are hard to be accurately determined by empirical method. Even these parameters can be estimated from experimental or historical data, they are likely to have substantial errors. For simulating the two major reasons above, two kinds of difference were introduced to the mechanistic model; first, when the mechanistic model (type A mechanistic model) is used for the mechanistic component of the hybrid model, the small losses: clearance loss, volute loss, and back flow loss are ignored; when the mechanistic model (type B mechanistic model) is used for producing data, all the losses listed in Eqs. (14) to (18) are considered. Second, the values of the important parameters,
Table 4 Parameters for the centrifugal compressor mechanistic model type A Parameter
Value Stage 1
Stage 2
Stage 3
Slip factor r
0.9
0.9
0.9
Blade inlet angle b1b ()
33
33.5
32
Average diameters at impeller eye D1 (m)
0.5883
0.5803
0.5767
Impeller exit diameter D2 (m)
1.080
1.080
1.080
Mean impeller channel length li (m)
0.4250
0.4040
0.3720
Mean diffuser channel length ld (m)
1.0310
0.9860
0.4170
Impeller mean hydraulic channel diameter Di (m)
0.1158
0.1006
0.0904
Diffuser mean hydraulic channel diameter Dd (m)
0.0822
0.0677
0.0874
Average molecular weight of gas
27.68
27.68
27.68
Specific heat ratio
1.36
1.36
1.36
Specific heat capacity (Jkg-1 K-1)
1,118.50
1,118.50
1,118.50
Shock loss coefficient
1.0
1.0
1.0
Reference area A (m2)
0.3262
0.3041
0.3004
Sum of clearance loss, volute loss, and back flow loss
0.0
0.0
0.0
Table 5 Parameters for the centrifugal compressor mechanistic model type B Parameter
Value Stage 1
Stage 2
Stage 3
Slip factor r
0.9
0.9
0.9
Blade inlet angle b1b ()
33
33.5
32
Average diameters at impeller eye D1 (m)
0.5883
0.5803
0.5767
Impeller exit diameter D2 (m)
1.080
1.080
1.080 0.3720
Mean impeller channel length li (m)
0.4250
0.4040
Mean diffuser channel length ld (m)
1.0310
0.9860
0.4170
Impeller mean hydraulic channel diameter Di (m)
0.1158
0.1006
0.0904
Diffuser mean hydraulic channel diameter Dd (m)
0.0822
0.0677
0.0874
Average molecular weight of gas
27.68
27.68
27.68
Specific heat ratio
1.36
1.36
1.36
Specific heat capacity (JKg-1 K-1)
1,118.50
1,118.50
1,118.50
Shock loss coefficient
1.1
1.0
1.1
Reference area A (m2)
0.2719
0.2645
0.2612
Sum of clearance loss, volute loss, and back flow loss
0.06
0.06
0.06
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that is, shock loss coefficient f and the reference area A1 are different between the type A and type B mechanistic models as shown in Tables 4 and 5.
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