Modeling and Optimizing Boiler Design using Neural Network and Firefly Algorithm

1.1 Motivation and Contributions

Based on the description of the works related with modeling and controlling of the boiler design process, it can be asserted that the role of artificial intelligence is greater. The primary intelligent methodologies such as artificial neural network (ANN) and fuzzy inference system have been adopted in Refs. [5], [15], [19]. The second category of artificial intelligence, such as particle swarm optimization (PSO) and genetic algorithm, has been adopted in Refs. [13], [29], [30].

As the need of exploiting the intelligence concept is essential in accomplishing precise boiler modeling, we intended to work on it. However, such intelligent methodologies suffer from a few known issues, as described in Table 1. This research proposal attempts to solve the drawbacks of intelligent methodologies adopted in the boiler design and its control. The complete set of gaps that have been identified from the review can be categorized as converging experience, curse of dimensionality, and learning flexibility. In order to overcome these drawbacks, we intended to hybridize multiple intelligent methodologies. The intelligent methodologies include the ANN and firefly (FF) algorithm. The ANN is known for its learning flexibility, but suffers from the curse of dimensionality. This can be overcome by providing training by using robust optimization algorithms. Hence, the FF algorithm is used to perform the task. To ensure the robustness of the methodology, we attempt to hybridize in a sophisticated manner so that the precision of the boiler model can be improved. In another perspective, the boiler design process is separated in our proposal into two steps. In the first step, a blind boiler model is constructed with model parameters. The blind model is formulated based on the inference of the ANN. The ANN is optimized by using the FF algorithm to provide precise fitting of the model with realistic circumstances.

3.1 System Model

The system model of the intelligent process for boiler design is illustrated in Figure 1. In this system model, there are three inputs represented as u₁ for fuel flow, u₂ for governor valve input, and u₃ for feed water flow, and three output systems represented as y₁ for electric power, y₂ for steam pressure, and y₃ for separator outlet steam temperature. The input and output variables selected have a strong influence on the stability and quality of power plants. The neural model is developed and derived with the data obtained from the output and input variables. The acquired data are subjected to the model learning program and, thus, the neural network model is established.

Figure 1:

Intelligent Process for Boiler Design.

3.2 NLARX Model by Feedforward Neural Network

The architectural structure of the non-linear autoregressive exogenous input (NLARX) [9] is represented in Figure 2. Let us consider Y(k) as the output of NLARX for k^th sample and X(k) as the input to the NLARX model; thus, the NLARX model is given as

Figure 2:

Non-linear ARMAX Model for Modeling the Boiler Plant.

where F(.) refers to the functional model of NLARX architecture, and N and M indicate the number of past output and input terms, respectively. These past input terms are used for predicting the current output. The output predictions formed are synthesized from the NLARX model, and they are the results due to changing of the previous outputs and inputs that together form the regression functions with two types of blocks known as linear and non-linear blocks. Due to the delayed output or input variables, the conventional regressors are formed; however, the advanced regressors are formed in the arbitrary user-defined function form of delayed output and input variables. Thus, the issue of non-linear unconstrained optimization arise in NLARX model training, and it is represented as

where p, Z_T, Y_t(k), Y^t(k|ω), and ω indicate the number of weighing parameters, training library, desired output, output from NLARX, and weightage, respectively. The ||.||² mentioned in Eq. (2) refers to L2 – norm function, and the training library and the weights are represented as Z_T=[Y_t(k), X(k)]k=1, ….., T and ω=[ω₁, …, ω_i, …, ω_p], respectively.

In Eq. (2), an error metric known as the performance index of the network is needed to reduce the negative deviation of performance index to get rid of the metric error. The performance index indicates the network approximation for the provided training patterns, and the network parameters ω have to be changed to minimize the index e(ω, Z_T) above the complete trajectory for achieving less value.

4.1 Knowledge Base

For effective modeling of the boiler plant, the network has to understand the characteristics of the boiler plant. Understanding the characteristics of the boiler plant is nothing but determining the relationship between the output and the input of the boiler plants. In order to determine the relationship, a precise knowledge base is required. The knowledge base can be constructed using either an empirical model or a theoretical model.

1. Empirical model: The selected output and input variables are measured in the boiler plant by considering various parameters such as feed water flow, spray flow, steam pressure in drum, steam pressure in throttle, steam temperature, water level in drum, electrical power, steam pressure, and temperature outlet. With theoretical calculations, the dynamic behavior of the boiler plant cannot be evaluated due to the lack of physical properties, and the dynamic behavior depends on the alterations in the operating point. Therefore, the influence of enthalpy and electrical power on each input is estimated, and when the change in one input occurs, the other input is placed in constant value.

2. Theoretical model: Depending on the DIN (1942), the thermal analysis of the boiler BP-1150 is performed [25] and the balancing equation related to energy for the boiler can be represented as from Eqs. (3) to (19); all parameters are referred in Ref. [25].

The energy flux in the above equation is represented in the following forms:

Firstly, the input energy flux of a fuel is always proportional to the fuel flux and it is given as

where Wd∗ represents the fuel obtained at lower heating value and it is given as

Secondly, the input energy flux is always independent to the fuel flux, and this idea gives

Thirdly, the flux of the consumed fuel is proportional to the energy loss flux and so

The loss of energy is due to (a) the loss due to fuel gas and unburned combustibles, as given in Eq. (8); (b) loss because of unburned combustibles and enthalpy in slag, as given in Eq. (9); and (c) loss because of enthalpy and unburned combustibles in fuel dust, as given in Eq. (10):

The energy loss flux is independent of the fuel flux:

Depending upon Ref. [25],

Substituting Eqs. (4)–(11) in Eq. (3), we get

Thus, the boiler energy efficiency is the ratio of the useful heat flux per input energy of the boiler:

It can also be written as

where

It can be rewritten as

and the obtained formula is calculated based on the indirect method. The factors that influence the boiler efficiency include loss in fuel gas, L_rel, and loss because of unburned combustibles in slag and flue dust, Lrel3.

Though both the models are described here, our paper concentrates on collecting the empirical model based on the published results of Refs. [12], [19].

4.2 NLARX Learning by FF Algorithm

1. Learning process: In 1942, McCulloh and Pitts implemented the single artificial neuron mathematical description [7], [14]. The output from one neuron or the input from the other neuron generates signals that are passed to the dendrite of the neuron. The amount of signals reaching the dendrite can be evaluated by calculating their weights and then taking the sum, and the sum of all those signals, when multiplied with their corresponding weights, gives the neuron activation function argument. The net output obtained represents the functional activation of a neuron, and this output is passed to the other new neuron, where they act as the input signal.

   Considering a single neuron as a representation model is challenging because of its computational complexity. Thus, to make it easy, the single neuron is interconnected to a net, which is the organized form of many single neurons. In this organized form of neural network, the signals pass from one layer to the other layer, and the signal indicates the input signal. Commonly, in a neural network, there is one output signal, one input signal, and one or more hidden layers. The feed-forward neural network model, having the feature of multidimensional non-linear approximation, is implemented in the present work.

   A single training is needed for the ANNs, and this ability is unique and more effective than the traditional algorithms. In each technique, there is a need for multiple iterative trainings, which are required for finding the weight of neurons that provides the least difference between the desired output z and the network (model) output y.

   The objective function favors the mean squared error reduction of the answers:

   (20)E=12∑l=1N(zj−yj)2→min.

   Equation (20) consists of a delta factor that is indicated as δ=z−y, and it can be estimated using the expected values z. In hidden layers, the expected values z are not identifiable, and thus, it is not easy to find out the delta factor δ. Hence, the back-propagation delta rule [7], [14] can be used to identify the delta factor values, depending on the weights in between the connections of consecutive hidden layers and the delta factor δ values of the next hidden layer. The back-propagation method includes the evaluation of the output layer delta factor δ values, and then it turns backward leading to error propagation. If any problem in reducing the objective function arises, then it can be solved using the gradient method, and it is given as

   (21)w(s+1)=w(s)−η∇E(w(s))+αΔw(s−1).

   The above equation is used for identifying the weight of the neurons. In neural network training, the back-propagation neural network error technique is asserted as accurate. However, a non-linear system exhibits high non-linear relationship among the input and the output data. Hence, a step-size back-propagation algorithm is not competing. Hence, we propose the FF algorithm to replace the back-propagation, and hence the non-linearities can be precisely recognized. The detailed Architecture is illustrated in Figure 3.

2. FF-based learning:Depending upon the FF pattern of flashing and behavior, Yang and He [38], [39], [40] introduced the FF algorithm in the year 2007, and it follows three rules, as follows:

• The objective function landscape evaluates the FFs’ brightness.

• The brightness and the attractiveness of FFs are proportional to each other, and both the attractiveness and brightness decrease with increase in distance. The less brightness emitting FFs move toward the high brightness emitting FFs, and if there are no brighter ones, then they will take a random movement.

• The sexual character of FFs is unisex and, so, the FFs will attract each other commonly.

As the attractiveness of one FF is directly proportional to the intensity of the light emitted by the other FF, the changes in the attractiveness β with respect to the distance r can be calculated using

where β₀ represents the attractiveness when r=0. The FFs’ movement i is toward the brightness showing FF j, and it can be evaluated using

Pseudo code of the FF learning algorithm

where the second term is formed due to the attraction between the FFs and the third term is in random movement with α_t; ∈it represents the numbers selected in a random manner using the uniform or Gaussian distribution at a time period t; and α_t is the parameter for randomization. When β₀=0, the FF chooses the random movement and if γ=0, it is minimized to a variant of PSO [38]. In addition to this, the ∈it randomization can be moved to the other distribution such as Levy flights [38]. In the literature [24], it has been reported that the FF algorithm has been used for training pi-sigma neural network, multilayer perceptron, and back-propagation neural network. However, the NLARX network has not been trained by the FF algorithm as per our review. Nevertheless, the FF algorithm has been used in this paper to improve the performance of the NLARX network.

5.1 Experiments

The developed experimental model is simulated in MATLAB platform. Totally, 10 parameters are considered to study the developed model efficiency. They are steam flow, temperature outlet, electrical power, steam pressure, feed water flow, steam pressure in drum, spray water flow, steam pressure in throttle, water level in drum, and steam temperature. By supplying various inputs, the aforesaid parameters are acquired in Refs. [12], [19]. We refer to those data as our empirical model, and the experimentation is carried out in three test cases. In the first test case, steam flow, electrical power, and temperature outlet are considered as the boiler output, as given in Ref. [19]. The rest of the two test case data are acquired from Ref. [12]. The test case 2 has been segregated into sets A and B, and modeling has been done.

5.2 Design Performance

In test case 1, three parameters – electrical power, steam pressure, and temperature outlet – are taken, and correlations among the output from the plant and modeling techniques such as NM (refers to multilayer perceptron with back-propagation learning algorithm) and FF-NM are evaluated and shown in Figure 4. When the outlet temperature is simulated, the deviation is found at 385°C (during the start of sample iteration) and 365°C (at sample 400) between NM and FF-NM, respectively.

Figure 3:

NLARX Learning by FF Algorithm for Modeling of Boiler Design.

Figure 4:

Correlation between the Output of Different Models from Test Case 1.

The deviation between NM and FF-NM is found to be at 800 and 850 MW at a sample size of 300 and 1000, respectively, for the electrical power parameter simulation. For steam pressure, the deviation is found at 20 and 25 MPa with respect to the sample size of 300 and 250 for the NM and FF-NM models, respectively.

In test case 2 (for both sets A and B) and case 3, seven parameters are considered and their correlation is shown in Figures 5–7 , respectively.

Figure 5:

Correlation between the Output of Different Models from Test Case 2 – Set A.

Figure 6:

Correlation between the Output of Different Models from Test Case 2 – Set B.

Figure 7:

Correlation between the Output of Different Models from Test Case 3.

The deviation difference for NM and FF-NM is found to be 1.79% and 2.5%, 0.42% and 1.6%, and 1.21% and 0.85% for feed water flow, water level in drum, and steam flow, respectively. However, for steam pressure in throttle and steam pressure in drum, the plots are very much closer to each other with no deviation. In the case of steam temperature, the estimations are very close for FF-NM. A 0.12% deviation occurs for the NM model curve; for spray water flow, a smaller deviation is noticed. In test case 2 – set A, the curves for the parameters steam flow, steam temperature, steam pressure in drum, and steam pressure in throttle in the two models are closer; however, for water level in drum, feed water flow, and spray water flow, the deviations are found to be 6%, 5.65%, and 15%, respectively, for the FF-NM method, which are rather higher than the deviations between the NM method. In test case 2 – set B, for models NM and FF-NM with respect to the parameters water level in drum, feed water flow, spray water flow, steam flow, steam pressure in throttle, and steam temperature, the deviation is calculated as 4.03% and 1.94%, 2.57% and 7.04%, 6.41% and 3.22%, 1.68% and 4.85%, 0.6% and 4.6%, and 0.06% and 0.24%, respectively.

5.3 Impact of FF Algorithm Parameters on Design Performance

The error rate is analyzed for the proposed model and the graphs are plotted with β on x-axis, γ on y-axis with varied α values (0.1, 0.2, 0.3, 0.4, 0.5) for the parameters temperature outlet, steam pressure, and electrical power, as shown in Figures 8–10, respectively. The results are quantified and tabulated in Tables 2–4.

Figure 8:

Estimation of Steam Pressure under Varying β and γ with the Reference of α as (A) 0.1, (B) 0.2, (C) 0.3, (D) 0.4, and (E) 0.5.

Figure 9:

Estimation of electrical power under varying β and γ with the reference of α as (A) 0.1, (B) 0.2, (C) 0.3, (D) 0.4, and (E) 0.5.

Figure 10:

Estimation of Temperature Outlet under Varying β and γ with the Reference of α as (A) 0.1, (B) 0.2, (C) 0.3, (D) 0.4, and (E) 0.5.

For the electrical power parameter, the error rate is least when γ=0.2 and 0.5 with corresponding β=0.1 and 0.1 for α=0.1, and when the α value is set to 0.2, the error is least at γ=0.3 to 0.28 with corresponding β=0.32 to 0.44, for α=0.3, 0.4, and 0.5, γ=0.5, 0.3, 0.5 with corresponding β=0.1, 0.5, 0.3, γ=0.1, 0.5 to 0.4 to corresponding β=0.2, 0.21 to 0.3, γ=0.1 with β=0.1, respectively. In the case of steam pressure, for α=0.1, 0.2, 0.3, 0.4, 0.5, the error is less at γ=0.21, 0.1, 0.4, 0.1, and 0.5 with corresponding β=0.41, 0.2, 0.1, 0.4, and 0.42, respectively. For temperature outlet, when α=0.1, the error is minimized at γ=0.4 and β=0.3 and for α=0.2, γ=0.1 and 0.5 with corresponding β=0.3 and 0.43. When the α value is 0.5, the error is least at γ=0.3, 0.1, 0.4 with corresponding β=0.2, 0.4, and 0.4 and for α=0.3 and 0.4, γ=0.1 and 0.42 with corresponding β=0.2 and 0.5, respectively.

5.4 Degree of Training Information on Design Performance

The error deviation between the plant and the proposed model, and between NM and the plant is calculated and tabulated in Table 5. Three cases are taken – case 1, case 2, and case 3 – having different parameters, and the error deviation is calculated for all these parameters. From the results, it is found that from 25% to 75%, the error is decreased for the proposed model in all cases.