CN110991616B - Method for predicting BOD of effluent based on pruning feedforward small-world neural network - Google Patents
Method for predicting BOD of effluent based on pruning feedforward small-world neural network Download PDFInfo
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- QVGXLLKOCUKJST-UHFFFAOYSA-N atomic oxygen Chemical compound [O] QVGXLLKOCUKJST-UHFFFAOYSA-N 0.000 description 3
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Abstract
The method for predicting BOD concentration of the effluent based on the pruned feedforward small world neural network is an important branch in the field of advanced manufacturing technology, and belongs to the field of control and water treatment. According to the invention, by designing the pruned feedforward small world neural network, the real-time accurate measurement of the BOD concentration is realized according to the data acquired in the sewage treatment process, the problem that the BOD concentration of the effluent in the sewage treatment process is difficult to measure in real time is solved, and the real-time monitoring level of the water quality of the urban sewage treatment plant is improved.
Description
Technical field:
the invention relates to a water outlet BOD prediction method based on a truncated feedforward small-world neural network. The realization of real-time prediction of BOD concentration is an important branch in the advanced manufacturing technology field, and belongs to the control field and the water treatment field.
The background technology is as follows:
the biochemical oxygen demand (Biochemical Oxygen Demand, BOD) refers to the amount of dissolved oxygen in water consumed by decomposing organic matters by microorganisms in a specified time, is an important index for evaluating the quality of sewage, and can be used for rapidly and accurately measuring the BOD concentration of the effluent so as to be beneficial to effectively controlling water pollution. The current BOD measurement methods include dilution and inoculation methods, microorganism sensor rapid measurement methods and the like, the BOD analysis measurement period is 5 days, the measurement period is long, and the change of the BOD concentration in sewage cannot be reflected in real time. Meanwhile, the microbial sensor has the defects of high manufacturing cost, short service life, poor stability and the like, and the universality of the microbial sensor is reduced. Therefore, how to detect the BOD concentration of the effluent with low cost and high efficiency is a difficult problem in the sewage treatment process.
The soft measurement method adopts an indirect measurement thought, utilizes an easily-measured variable, predicts a difficult-to-measure variable in real time by constructing a model, and provides a high-efficiency and quick solution for measuring key water quality parameters in the sewage treatment process. The invention designs a water outlet BOD prediction method based on a truncated feedforward small-world neural network, which is an effective model in a soft measurement method and has the strong generalization capability.
Disclosure of Invention
The invention obtains the BOD prediction method of the effluent based on the pruned feedforward small world neural network, realizes the real-time measurement of the BOD concentration according to the data acquired in the sewage treatment process, solves the problem that the BOD concentration of the effluent in the sewage treatment process is difficult to be measured in real time, and improves the real-time monitoring level of the water quality of the urban sewage treatment plant;
a method for soft measurement of BOD concentration of effluent based on a pruned feedforward small world neural network is characterized by comprising the following steps:
step 1: selecting auxiliary variables of a BOD prediction model of the effluent;
directly selecting given M auxiliary variables; normalizing the auxiliary variable to [ -1,1] according to formula (1), and normalizing the output variable BOD to [0,1] according to formula (2):
wherein F is m Represents the mth auxiliary variable, O represents the output variable, x m And y represents the mth auxiliary variable and the output variable after normalization respectively; min (F) m ) And max (F) m ) Respectively representing the minimum value and the maximum value in the mth auxiliary variable, and min (O) and max (O) respectively represent the minimum value and the maximum value in the output variable;
step 2: designing a feedforward small world neural network model;
step 2.1: designing a feedforward small-world neural network model wiring mode;
constructing a feedforward small-world neural network according to the Watts-Strogatz rewiring rule; the specific construction process is as follows: firstly, constructing an L-layer feedforward neural network with regular connection, then randomly selecting one connection from the model according to reconnection probability p, disconnecting from the tail end and reconnecting to another neuron in the model, wherein p is selected empirically, the value range is (0, 1), if the new connection exists, randomly selecting another new neuron to connect, and the neurons in the same layer cannot be connected with each other;
step 2.2: designing a topological structure of a feedforward small-world neural network model;
the designed feedforward small world neural network topological structure is L-layer in total and comprises an input layer, an hidden layer and an output layer; the calculation functions of each layer are as follows:
(1) input layer: the layer has M neurons, representing M input auxiliary variables, and the input of the input layer is x (1) ＝[x 1 (1) ,x 2 (1) ,…,x M (1) ]Wherein x is m (1) Representing the mth input auxiliary variable of the input layer, m=1, 2, …, M, the layer output is equal to the input, the output of the mth neuron of the input layer is:
(2) hidden layer: by adopting the sigmoid function as an activation function of the hidden layer, the input and output definitions of the j-th neuron of the first layer of the neural network are shown in formulas (4) and (5), respectively:
wherein n is u Representing the number of neurons in the u-th layer of the neural network,representing a connection weight between an ith neuron of a ith layer and a jth neuron of a first layer of the neural network;
(3) output layer: the output layer comprises a neuron, and the output of the output neuron is as follows:
wherein the method comprises the steps ofRepresenting the connection weight between the jth neuron of the first layer of the neural network and the output neuron, n l Representing the number of neurons of the first layer of the neural network;
step 3: designing a deletion algorithm of the feedforward small world neural network;
step 3.1: defining a performance index function:
wherein Q is the number of samples, d q For the desired output value of the q-th sample,a predicted output value for the q-th sample;
step 3.2: carrying out parameter correction by adopting a batch gradient descent algorithm;
(1) the output weight correction of the output layer is as shown in formulas (8) - (10):
wherein the method comprises the steps of
Wherein,and->Represents the connection weight between the j-th neuron and the output neuron of the first layer of the neural network at the moments t and t+1 respectively,/o>Representing the variation value of the connection weight between the jth nerve of the jth layer of the neural network at the moment t and the output nerve element, eta v Represents the learning rate, eta in the output weight correction process of the output layer v Selected empirically to have a value in the range of (0,0.1)]；
(2) The output weight correction of the hidden layer is as shown in formulas (11) - (13):
wherein the method comprises the steps of
Wherein,and->Representing the connection weight between the ith neuron of the s-th layer and the jth neuron of the first layer of the neural network at the moments t and t+1 respectively, < + >>Representing the variation value of the connection weight between the ith neuron of the s-th layer and the jth neuron of the first layer of the neural network at the moment of t, eta w Represents the learning rate, eta in the process of correcting the output weight of the hidden layer w Selected empirically to have a value in the range of (0,0.1)]；
Step 3.3: inputting training sample data, updating output weights of an implicit layer and an output layer according to formulas (8) - (13) in step 3.2, wherein Iter represents training iteration times, the iteration times are increased once every time the weights are updated, if the iteration times of the training process can be divided by a learning step length constant tau, step 3.4 is executed, otherwise, step 3.6 is skipped, wherein tau is selected according to experience, and the value range is an integer in the range of [10,100 ];
step 3.4: calculating the Katz centrality and the normalized Katz centrality of all hidden layer neurons; katz centrality is defined as shown in equation (14):
wherein the method comprises the steps ofRepresents the k power of the connection weight between the neuron g and the neuron h, alpha represents the attenuation factor, and the set value of alpha needs to satisfy 0<α<1/λ max Alpha is selected empirically to have a value in the range of (0,0.1)]，λ max A value representing the maximum eigenvalue of the network adjacency matrix with greater Katz centrality indicates that the node is more important, and vice versa;
the normalized Katz centrality definition is shown in equation (15):
wherein the method comprises the steps ofKatz centrality of jth neuron representing the s-th layer of the neural network, +.>Katz centrality normalized by the jth neuron representing the s-th layer of the neural network; is provided with->Average value of normalized Katz centrality of all neurons representing the s-th layer of the neural network, wherein θ is a preset threshold parameter, and is selected empirically within a range of [0.9,1 ]]If the Katz centrality of neurons satisfies +.>The neuron is considered as an unimportant neuron, and the unimportant hidden layer neuron set in the s layer of the neural network is marked as A s The remaining set of neurons in layer s is denoted B s ；
Step 3.5: computing set A s And set B s The correlation coefficient between hidden layer neurons in (2) is defined as shown in formula (16):
wherein,and->Respectively representing the output values of the ith neuron and the jth neuron of the ith layer of the neural network when the qth training sample is input; />And->Respectively representing the input of all samples +.>And->Average value of (2); sigma (sigma) i Sum sigma j Respectively representing the input of all samples +.>And->Standard deviation of (2); will set A s Hidden layer neurons in (denoted as neuron a, a e A) s ) The neuron with the highest correlation coefficient (named neuron B, b.epsilon.B) s ) Combining to generate a new neuron c, wherein the connection weight between the neuron c and the neuron of the forward network layer is constructed according to the reconnection rule of Watt-Strogatz and the reconnection probability p, wherein p is selected empirically and has the value range of (0, 1) so as to ensure the small worldwide performance of the network, and the output of the neuron c is shown as a formula (17):
wherein the method comprises the steps ofRepresenting a connection weight between an ith neuron in an nth layer of the neural network and a neuron c in an s-th layer;
the connection weights between neurons c and neurons of the backward network layer according to the pruning algorithm are as shown in equations (18) - (19):
wherein the method comprises the steps ofAnd->Connection weights between neurons a, b and c representing the s-th layer of the neural network and neuron j in the backward hidden layer, respectively, +.>And->Connection weights between neurons a, b and c, respectively representing the s-th layer of the neural network, and the output neurons,/for>And->Output values of neurons a, b and c respectively representing the s-th layer of the neural network are combined according to formulas (17) - (19), and then step 3.3 is skipped;
step 3.6: calculate training RMSE, if it is satisfied that RMSE is less than the desired training RMSE (RMSE d ) Or the number of iterations reaches the maximum number of iterations (Iter max ) Stopping the calculation at the time, wherein Iter max Selected according to experience, the value range is [5000,10000 ]]Otherwise, jumping to step 3.3, wherein the definition of the RMSE is shown in a formula (20);
step 4: predicting BOD of the effluent;
and taking the test sample data as the input of the trained pruned feedforward small-world neural network, and performing inverse normalization on the output of the neural network to obtain the predicted value of the BOD of the output water.
Compared with the prior art, the invention has the following obvious advantages and beneficial effects:
(1) Aiming at the problems that the current sewage treatment process has long measurement period of key water quality parameters BOD and a mathematical model is difficult to determine, the invention provides a truncated feedforward small-world neural network model for realizing real-time measurement of the effluent BOD, and the method has the characteristics of good instantaneity, high precision, good stability and strong generalization capability.
(2) Aiming at the problems that the traditional small world neural network structure is large and the structure is easy to be overlarge and time-consuming due to fixation, the importance of hidden layer neurons is measured by adopting Katz centrality, and a pruning algorithm is provided to determine the number of the hidden layer neurons of the neural network, so that the condition that the network is overlarge in scale and more calculation time and storage space are needed is avoided.
Drawings
FIG. 1 is a diagram of the topology of a neural network of the present invention;
FIG. 2 is a graph of training Root Mean Square Error (RMSE) variation for the BOD concentration prediction method of the present invention;
FIGS. 3 and 4 are views of hidden layer node changes in the training process of the BOD concentration prediction method of the invention;
FIG. 5 is a graph showing the result of predicting BOD concentration of the effluent of the present invention;
FIG. 6 is a graph showing the BOD concentration prediction error of the effluent of the present invention.
Detailed Description
The invention obtains the BOD prediction method of the effluent based on the pruned feedforward small world neural network, realizes the real-time measurement of the BOD concentration according to the data acquired in the sewage treatment process, solves the problem that the BOD concentration of the effluent in the sewage treatment process is difficult to be measured in real time, and improves the real-time monitoring level of the water quality of the urban sewage treatment plant;
experimental data comes from 2011 water quality analysis data of a sewage plant, which contains 365 groups of data, ten water quality variables, including: (1) total nitrogen concentration of effluent; (2) ammonia nitrogen concentration of effluent; (3) total nitrogen concentration in the feed water; (4) BOD concentration of the incoming water; (5) ammonia nitrogen concentration of the inlet water; (6) effluent phosphate concentration; (7) biochemical MLSS concentration; (8) biochemical pool DO concentration; (9) influent phosphate concentration; (10) COD concentration of the inlet water. All 365 sets of samples were divided into two parts: 265 groups of data are used as training samples, and the other 100 groups of data are used as measurement samples;
a soft measurement method of BOD concentration of effluent based on a truncated feedforward small world neural network comprises the following steps:
step 1: selecting auxiliary variables of a BOD prediction model of the effluent;
directly selecting given M auxiliary variables; normalizing the auxiliary variable to [ -1,1] according to formula (1), and normalizing the output variable BOD to [0,1] according to formula (2):
wherein F is m Represents the mth auxiliary variable, O represents the output variable, x m And y represents the mth auxiliary variable and the output variable after normalization respectively; min (F) m ) And max (F) m ) Respectively representing the minimum value and the maximum value in the mth auxiliary variable, and min (O) and max (O) respectively represent the minimum value and the maximum value in the output variable;
in this embodiment, given 10 auxiliary variables, i.e., m=10, are directly selected; the 10 auxiliary variables comprise (1) total nitrogen concentration of effluent; (2) ammonia nitrogen concentration of effluent; (3) total nitrogen concentration in the feed water; (4) BOD concentration of the incoming water; (5) ammonia nitrogen concentration of the inlet water; (6) effluent phosphate concentration; (7) biochemical MLSS concentration; (8) biochemical pool DO concentration; (9) influent phosphate concentration; (10) COD concentration of the inlet water;
step 2: designing a feedforward small world neural network model;
step 2.1: designing a feedforward small-world neural network model wiring mode;
constructing a feedforward small-world neural network according to the Watts-Strogatz rewiring rule; the specific construction process is as follows: firstly, constructing a regularly connected L-layer feedforward neural network, wherein the random generation range of initial weights is [ -1,1], then randomly selecting one connection from a model according to reconnection probability p, disconnecting from the tail end and reconnecting to another neuron in the model, wherein p is selected empirically, the value range is (0, 1), if the new connection exists, randomly selecting another new neuron for connection, the newly generated weight range is [ -1,1], and the neurons in the same layer cannot be connected with each other, and p is 0.5 in the embodiment;
step 2.2: designing a topological structure of a feedforward small-world neural network model;
the designed feedforward small world neural network topological structure is L-layer in total and comprises an input layer, an hidden layer and an output layer; the calculation functions of each layer are as follows:
(1) input layer: the layer has M neurons, representing M input auxiliary variables, and the input of the input layer is x (1) ＝[x 1 (1) ,x 2 (1) ,…,x M (1) ]Wherein x is m (1) Representing the mth input auxiliary variable of the input layer, m=1, 2, …, M, the layer output is equal to the input, the output of the mth neuron of the input layer is:
(2) hidden layer: by adopting the sigmoid function as an activation function of the hidden layer, the input and output definitions of the j-th neuron of the first layer of the neural network are shown in formulas (4) and (5), respectively:
wherein n is u Representing the number of neurons in the u-th layer of the neural network,representing a connection weight between an ith neuron of a ith layer and a jth neuron of a first layer of the neural network;
(3) output layer: the output layer comprises a neuron, and the output of the output neuron is as follows:
wherein the method comprises the steps ofRepresenting the connection weight between the jth neuron of the first layer of the neural network and the output neuron, n l Representing the number of neurons of the first layer of the neural network;
in this embodiment, the hidden layers are two layers, and the number of initial neurons contained in each hidden layer is 40;
step 3: designing a deletion algorithm of the feedforward small world neural network;
step 3.1: defining a performance index function:
wherein Q is the number of samples, d q For the desired output value of the q-th sample,a predicted output value for the q-th sample;
step 3.2: carrying out parameter correction by adopting a batch gradient descent algorithm;
(1) the output weight correction of the output layer is as shown in formulas (8) - (10):
wherein the method comprises the steps of
Wherein,and->Represents the connection weight between the j-th neuron and the output neuron of the first layer of the neural network at the moments t and t+1 respectively,/o>Representing the variation value of the connection weight between the jth nerve of the jth layer of the neural network at the moment t and the output nerve element, eta v Represents the learning rate, eta in the output weight correction process of the output layer v Selected empirically to have a value in the range of (0,0.1)]η in the present embodiment v Taking 0.0003;
(2) the output weight correction of the hidden layer is as shown in formulas (11) - (13):
wherein the method comprises the steps of
Wherein,and->Representing the connection weight between the ith neuron of the s-th layer and the jth neuron of the first layer of the neural network at the moments t and t+1 respectively, < + >>Representing the variation value of the connection weight between the ith neuron of the s-th layer and the jth neuron of the first layer of the neural network at the moment of t, eta w Represents the learning rate, eta in the process of correcting the output weight of the hidden layer w Selected empirically to have a value in the range of (0,0.1)]η in the present embodiment w Taking 0.0003;
step 3.3: inputting training sample data, updating output weights of an implicit layer and an output layer according to formulas (8) - (13) in step 3.2, wherein Iter represents training iteration times, the iteration times are increased once every time the weights are updated, if the iteration times in the training process can be divided by a learning step length constant tau, executing step 3.4, otherwise, jumping to step 3.6, wherein tau is selected empirically, the value range is an integer in the range of [10,100], and tau is 20 in the embodiment;
step 3.4: calculating the Katz centrality and the normalized Katz centrality of all hidden layer neurons; katz centrality is defined as shown in equation (14):
wherein the method comprises the steps ofRepresents the k power of the connection weight between the neuron g and the neuron h, alpha represents the attenuation factor, and the set value of alpha needs to satisfy 0<α<1/λ max Alpha is selected empirically to have a value in the range of (0,0.1)]In this embodiment, α is 0.01, λ max A value representing the maximum eigenvalue of the network adjacency matrix with greater Katz centrality indicates that the node is more important, and vice versa;
the normalized Katz centrality definition is shown in equation (15):
wherein the method comprises the steps ofKatz centrality of jth neuron representing the s-th layer of the neural network, +.>Katz centrality normalized by the jth neuron representing the s-th layer of the neural network; is provided with->Average value of normalized Katz centrality of all neurons representing the s-th layer of the neural network, wherein θ is a preset threshold parameter, and is selected empirically within a range of [0.9,1 ]]In this example, θ is 0.93 if the Katz centrality of the neuron satisfies +.>The neuron is considered as an unimportant neuron, and the unimportant hidden layer neuron set in the s layer of the neural network is marked as A s The remaining set of neurons in layer s is denoted B s ；
Step 3.5: computing set A s And set B s The correlation coefficient between hidden layer neurons in (2) is defined as shown in formula (16):
wherein,and->Respectively representing the output values of the ith neuron and the jth neuron of the ith layer of the neural network when the qth training sample is input; />And->Respectively representing the input of all samples +.>And->Average value of (2); sigma (sigma) i Sum sigma j Respectively representing the input of all samples +.>And->Standard deviation of (2); will set A s Hidden layer neurons in (denoted as neuron a, a e A) s ) The neuron with the highest correlation coefficient (named neuron B, b.epsilon.B) s ) Combining to generate a new neuron c, wherein the connection weight between the neuron c and the neuron of the forward network layer is constructed according to the reconnection rule of Watt-Strogatz and the reconnection probability p to ensure the small worldwide property of the network, wherein p is selected according to experience and has a value range of (0, 1), in the embodiment, p takes 0.5, and the output of the neuron c is shown as a formula (17):
wherein the method comprises the steps ofRepresenting a connection weight between an ith neuron in an nth layer of the neural network and a neuron c in an s-th layer;
the connection weights between neurons c and neurons of the backward network layer according to the pruning algorithm are as shown in equations (18) - (19):
wherein the method comprises the steps ofAnd->Connection weights between neurons a, b and c representing the s-th layer of the neural network and neuron j in the backward hidden layer, respectively, +.>And->Connection weights between neurons a, b and c, respectively representing the s-th layer of the neural network, and the output neurons,/for>And->Output values of neurons a, b and c respectively representing the s-th layer of the neural network are combined according to formulas (17) - (19), and then step 3.3 is skipped;
step 3.6: calculate training RMSE, if it is satisfied that RMSE is less than the desired training RMSE (RMSE d ) Or the number of iterations reaches the maximum number of iterations (Iter max ) Stopping the calculation at the time, wherein Iter max Selected according to experience, the value range is [5000,10000 ]]Otherwise, jumping to step 3.3, wherein the definition of the RMSE is shown in a formula (20);
iter in this embodiment max 10000 rmse d Taking 0.05; training RMSE change graph as shown in fig. 2, X-axis: training steps, Y axis: training RMSE in mg/L; in the training process, hidden layer node change diagrams are respectively shown in fig. 3 and 4, and the X axis is as follows: training steps, Y axis: the number of neurons in the hidden layer is one;
step 4: predicting BOD of the effluent;
and taking the test sample data as the input of the trained pruned feedforward small-world neural network, and performing inverse normalization on the output of the neural network to obtain the predicted value of the BOD of the output water.
In this embodiment, the prediction result is shown in fig. 5, and the X axis: sample number, in units of number/sample, Y-axis: the BOD concentration of the effluent is in mg/L, the solid line is the actual output value of the BOD concentration of the effluent, and the dotted line is the predicted output value of the BOD concentration of the effluent; the error between the actual output value of the BOD concentration of the effluent and the predicted output value of the BOD concentration of the effluent is shown in FIG. 6, and the X axis is: sample number, in units of number/sample, Y-axis: predicting the BOD concentration of the effluent in mg/L; the result shows that the method for predicting the BOD concentration of the effluent based on the truncated neural network in the world is effective.
Tables 1-23 are experimental data for the present invention, wherein tables 1-11 are training samples: total nitrogen concentration of effluent, ammonia nitrogen concentration of effluent, total nitrogen concentration of influent, BOD concentration of influent, ammonia nitrogen concentration of influent, phosphate concentration of effluent, biochemical MLSS concentration, DO concentration of biochemical pool, phosphate concentration of influent, COD concentration of influent and BOD concentration of actual effluent, and tables 12-22 are training samples: the total nitrogen concentration of the effluent, the ammonia nitrogen concentration of the effluent, the total nitrogen concentration of the inlet water, the BOD concentration of the inlet water, the ammonia nitrogen concentration of the inlet water, the phosphate concentration of the outlet water, the biochemical MLSS concentration, the DO concentration of the biochemical pool, the phosphate concentration of the inlet water, the COD concentration of the inlet water and the BOD concentration of the outlet water actually measured are shown in Table 23, and the BOD predicted value of the outlet water of the invention is shown in Table 23.
Training samples:
TABLE 1 auxiliary variable total nitrogen in effluent (mg/L)
TABLE 2 auxiliary variable ammonia nitrogen in effluent (mg/L)
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TABLE 3 auxiliary variable total nitrogen in water (mg/L)
| 7.1360 | 10.5635 | 10.3759 | 10.3069 | 8.4245 | 12.8941 | 8.5735 | 8.6006 | 9.8132 | 7.6567 |
| 12.1750 | 7.8706 | 14.5415 | 7.5145 | 7.3283 | 8.7793 | 8.5030 | 8.5518 | 6.7060 | 10.1227 |
| 5.9246 | 7.9102 | 6.5293 | 8.7373 | 9.8952 | 8.5680 | 8.6127 | 11.4471 | 7.7494 | 7.7176 |
| 12.1330 | 14.2435 | 10.1667 | 13.8488 | 5.5685 | 7.9894 | 10.1884 | 7.2206 | 8.4340 | 8.6818 |
| 7.1448 | 6.9728 | 8.3270 | 8.7468 | 8.7197 | 8.1306 | 9.3014 | 11.4911 | 8.6276 | 9.0000 |
| 7.4244 | 8.9107 | 8.3771 | 8.5234 | 11.1708 | 6.6688 | 10.4897 | 11.9712 | 8.5254 | 8.6466 |
| 6.8428 | 7.0879 | 10.3610 | 10.4612 | 6.1020 | 10.2473 | 9.1104 | 6.2571 | 10.6915 | 9.7604 |
| 11.8737 | 8.9134 | 6.9315 | 10.9244 | 8.5139 | 7.8625 | 5.2353 | 9.7828 | 12.2380 | 11.5514 |
| 11.3794 | 6.1758 | 8.0685 | 7.7519 | 8.2715 | 8.5166 | 8.5992 | 8.2986 | 11.4078 | 8.6832 |
| 10.4342 | 10.2615 | 8.5748 | 10.0035 | 8.1293 | 10.0008 | 8.8321 | 8.6561 | 5.9693 | 7.9878 |
| 6.6877 | 10.1105 | 7.2490 | 7.1069 | 7.1840 | 7.4149 | 8.8186 | 14.4670 | 6.1210 | 7.1827 |
| 7.1339 | 7.0141 | 10.3245 | 9.2817 | 8.6628 | 7.1637 | 8.0074 | 11.5487 | 7.1380 | 11.1262 |
| 6.3824 | 10.7436 | 7.6377 | 8.8660 | 11.9976 | 8.2742 | 10.4267 | 8.0846 | 10.5093 | 10.0177 |
| 7.1380 | 14.3180 | 10.2520 | 7.8842 | 13.7865 | 10.1464 | 11.2074 | 8.8308 | 9.8965 | 14.7744 |
| 10.4308 | 7.0554 | 9.5959 | 7.2017 | 10.5195 | 6.2909 | 8.6046 | 11.6895 | 14.6160 | 10.5865 |
| 7.0967 | 10.0604 | 9.9155 | 9.2695 | 8.8890 | 10.3475 | 11.6313 | 8.4482 | 9.2729 | 9.9785 |
| 7.0513 | 5.2123 | 6.9301 | 10.0543 | 10.3353 | 12.1540 | 6.8089 | 8.3060 | 10.1755 | 8.6913 |
| 8.7895 | 9.5146 | 10.2791 | 8.1477 | 9.9175 | 13.4635 | 6.7182 | 8.2269 | 8.1990 | 8.0683 |
| 8.2904 | 9.3433 | 6.4799 | 10.4700 | 7.3540 | 14.1691 | 13.8294 | 8.3548 | 8.9391 | 11.0097 |
| 7.4278 | 11.7234 | 8.9743 | 13.0214 | 7.7636 | 5.9707 | 8.6019 | 10.3719 | 7.1163 | 10.1200 |
| 10.6156 | 4.8677 | 14.2970 | 7.2321 | 11.9245 | 8.9459 | 11.2555 | 10.2039 | 7.0019 | 7.2084 |
| 7.3289 | 6.6654 | 8.4055 | 7.8625 | 11.0354 | 8.1374 | 10.1850 | 9.9981 | 8.5884 | 7.1258 |
| 8.6981 | 8.2992 | 8.5559 | 14.7647 | 10.7700 | 9.1957 | 7.9654 | 8.6574 | 10.3915 | 6.2808 |
| 8.4394 | 6.8015 | 12.4264 | 9.0122 | 8.2160 | 6.9376 | 7.7765 | 6.8509 | 8.7549 | 8.7928 |
| 9.0481 | 6.5665 | 11.2629 | 12.2590 | 8.6493 | 8.6689 | 8.5200 | 6.9931 | 7.1793 | 5.6030 |
| 14.6904 | 10.4111 | 9.4909 | 8.8470 | 7.7596 | 7.2937 | 12.2170 | 4.5000 | 10.4335 | 9.5891 |
| 11.8303 | 7.5936 | 6.9735 | 8.6615 | 10.3096 |
TABLE 4 auxiliary variable BOD (mg/L) of incoming water
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TABLE 5 auxiliary variable intake ammonia nitrogen (mg/L)
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TABLE 6 auxiliary variable out-water phosphate (mg/L)
| 11.1500 | 9.2000 | 8.0250 | 11.4750 | 14.6500 | 13.8333 | 14.3750 | 13.5250 | 4.5000 | 13.0750 |
| 11.2750 | 12.9750 | 15.4000 | 14.6500 | 11.8500 | 9.5500 | 13.1250 | 13.3250 | 14.0250 | 11.5250 |
| 13.6250 | 14.0375 | 13.8250 | 14.6500 | 11.4250 | 13.7250 | 13.8250 | 11.4500 | 14.1750 | 14.1500 |
| 11.0750 | 14.8000 | 7.0750 | 11.6000 | 13.5750 | 14.1250 | 6.8500 | 10.4500 | 13.4250 | 14.0250 |
| 10.7500 | 11.6500 | 12.0250 | 14.1250 | 14.5250 | 13.7750 | 14.4750 | 14.3000 | 14.0000 | 8.9500 |
| 12.2000 | 14.3500 | 13.4250 | 13.6250 | 11.7500 | 12.7000 | 9.3750 | 11.1250 | 14.3250 | 14.4500 |
| 13.7000 | 10.9750 | 11.5750 | 11.5500 | 13.0750 | 11.7250 | 8.6500 | 13.6750 | 9.2000 | 11.2750 |
| 14.3500 | 14.4000 | 11.8500 | 11.7750 | 13.8250 | 14.2250 | 13.6250 | 14.2000 | 11.5750 | 10.8250 |
| 10.5000 | 13.4000 | 14.2125 | 13.8625 | 12.3750 | 13.6250 | 13.8250 | 12.7750 | 11.7250 | 14.5000 |
| 12.0250 | 7.5250 | 14.2000 | 6.1750 | 14.1000 | 5.6750 | 13.5000 | 14.4250 | 13.2500 | 11.6000 |
| 13.7000 | 6.6250 | 14.4500 | 10.9000 | 11.3250 | 14.3000 | 13.5500 | 15.2500 | 13.6000 | 10.6000 |
| 13.8500 | 11.4500 | 8.1250 | 10.6250 | 14.3750 | 10.6750 | 13.9500 | 11.5750 | 10.8500 | 12.0250 |
| 13.5500 | 11.8000 | 14.4750 | 14.6750 | 10.3000 | 13.3750 | 8.8500 | 12.8750 | 9.4750 | 5.9500 |
| 13.8500 | 14.9500 | 6.5750 | 14.1250 | 14.6000 | 9.0000 | 10.1750 | 10.6750 | 11.3250 | 12.2500 |
| 9.0750 | 11.2500 | 11.1250 | 10.5250 | 8.8750 | 12.9500 | 14.0000 | 11.4250 | 15.5500 | 7.4250 |
| 11.0500 | 11.4250 | 5.2250 | 10.6750 | 13.3750 | 7.9750 | 11.2500 | 10.4500 | 9.4750 | 11.3500 |
| 14.0000 | 13.5250 | 13.9000 | 6.1750 | 8.4250 | 11.1750 | 13.8000 | 14.4750 | 7.0750 | 13.8500 |
| 11.9750 | 8.2250 | 7.9750 | 14.3000 | 5.7250 | 15.3250 | 14.0500 | 14.3875 | 11.5500 | 13.2750 |
| 13.0000 | 10.7750 | 12.8250 | 9.2750 | 11.7500 | 14.6500 | 13.5222 | 11.8500 | 14.7500 | 14.5750 |
| 14.0000 | 11.1500 | 10.2750 | 14.5000 | 13.0250 | 13.8250 | 14.2750 | 11.9250 | 14.3500 | 6.6750 |
| 11.6250 | 13.5250 | 13.3667 | 11.5000 | 10.8250 | 13.2500 | 9.1250 | 11.6750 | 14.0000 | 14.1500 |
| 13.4500 | 13.9000 | 12.7250 | 13.1750 | 9.8500 | 14.2500 | 11.6250 | 5.7250 | 14.1000 | 10.8250 |
| 13.9250 | 12.2000 | 13.8250 | 13.2111 | 11.7000 | 10.5750 | 13.2250 | 13.9250 | 8.8750 | 13.6750 |
| 13.5500 | 13.9750 | 13.9889 | 14.8250 | 12.7250 | 14.0500 | 11.6500 | 14.1500 | 13.7000 | 14.6000 |
| 10.3750 | 13.6000 | 11.6500 | 11.6750 | 13.7250 | 9.8500 | 14.2250 | 13.7750 | 10.6500 | 13.7250 |
| 15.7000 | 8.9750 | 10.9750 | 14.4250 | 13.1250 | 14.2000 | 11.4750 | 13.4250 | 8.4250 | 10.9750 |
| 11.2750 | 13.6875 | 14.0250 | 13.8750 | 11.8250 |
TABLE 7 auxiliary variable Biochemical MLSS (mg/L)
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TABLE 8 auxiliary variable Biochemical pool DO (mg/L)
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TABLE 9 auxiliary variable intake phosphate (mg/L)
| 5.3779 | 6.3867 | 6.1319 | 7.9230 | 6.0115 | 6.6805 | 5.7779 | 5.4381 | 5.3673 | 4.8221 |
| 7.7000 | 4.9212 | 6.3088 | 5.6150 | 5.6186 | 9.3142 | 5.3035 | 5.3708 | 5.6469 | 7.5761 |
| 4.8823 | 13.4080 | 5.7389 | 5.8381 | 6.9956 | 5.6681 | 5.6611 | 9.1584 | 5.5513 | 5.3637 |
| 7.6150 | 6.2097 | 6.7018 | 8.7761 | 4.7867 | 13.8664 | 5.8770 | 5.6398 | 5.6894 | 5.7637 |
| 5.4558 | 5.0664 | 5.5867 | 5.6398 | 5.8062 | 5.3496 | 6.0611 | 5.9867 | 5.6575 | 9.9301 |
| 5.7389 | 5.9018 | 5.2434 | 5.6752 | 8.5637 | 5.4133 | 6.5248 | 7.8133 | 5.8062 | 5.7920 |
| 4.9035 | 5.3177 | 9.0097 | 8.0115 | 5.1690 | 7.4062 | 10.2381 | 5.6894 | 8.7973 | 6.9566 |
| 6.0150 | 5.9336 | 4.9708 | 8.2770 | 5.7000 | 5.4027 | 4.8823 | 5.6788 | 7.8274 | 9.5938 |
| 9.4345 | 4.6204 | 14.3248 | 12.4912 | 5.5513 | 5.7142 | 5.7389 | 5.1195 | 8.7761 | 5.7460 |
| 7.1513 | 9.8062 | 5.5619 | 10.9743 | 5.5832 | 5.6221 | 5.4274 | 5.7000 | 4.5000 | 5.4487 |
| 5.0027 | 6.7195 | 5.4876 | 5.3637 | 5.4381 | 5.3425 | 5.4982 | 6.2841 | 5.6646 | 5.5478 |
| 5.0381 | 5.1619 | 6.1637 | 6.3867 | 5.7920 | 5.5018 | 5.4027 | 8.5354 | 5.4487 | 9.2575 |
| 4.7407 | 9.1336 | 5.5619 | 5.8345 | 9.4841 | 5.1832 | 6.3230 | 5.0204 | 6.5035 | 5.6858 |
| 5.2080 | 6.2345 | 10.8646 | 5.4558 | 6.1566 | 10.1920 | 9.2752 | 8.5142 | 6.6416 | 8.4221 |
| 6.5885 | 5.2575 | 8.7619 | 5.5938 | 8.6381 | 5.2504 | 5.4982 | 8.2947 | 6.3336 | 10.3655 |
| 5.3531 | 7.6611 | 5.5265 | 6.6133 | 5.4097 | 9.4168 | 9.5549 | 8.3903 | 10.5885 | 8.8858 |
| 5.1726 | 4.6912 | 5.1159 | 6.7372 | 6.6487 | 7.6575 | 5.0593 | 15.7000 | 10.1956 | 5.6044 |
| 13.1442 | 9.3106 | 6.6664 | 14.7832 | 11.3637 | 6.1885 | 5.2327 | 15.2416 | 5.5442 | 5.0628 |
| 5.5619 | 6.7018 | 5.3319 | 6.5460 | 5.1619 | 6.1850 | 7.1431 | 5.6044 | 5.8487 | 5.8487 |
| 5.2858 | 9.7531 | 6.2593 | 6.1000 | 4.8717 | 5.2646 | 5.6080 | 7.2363 | 5.4239 | 5.8451 |
| 8.1000 | 4.6912 | 7.3743 | 5.4982 | 8.3690 | 5.3920 | 9.3673 | 6.7690 | 5.1018 | 5.2221 |
| 5.3389 | 5.7637 | 5.1690 | 4.9425 | 9.1159 | 5.6788 | 7.4912 | 6.7549 | 5.7460 | 5.4097 |
| 5.5726 | 5.5690 | 5.5088 | 7.6056 | 8.1885 | 6.5248 | 5.0027 | 5.6540 | 6.6310 | 4.9779 |
| 5.6540 | 5.7885 | 6.4493 | 5.8628 | 5.5159 | 5.8133 | 5.3531 | 5.2965 | 5.5513 | 5.8204 |
| 6.3478 | 4.9460 | 8.7619 | 7.8699 | 5.5053 | 9.0062 | 5.7425 | 5.1690 | 5.5442 | 5.0735 |
| 6.3584 | 6.6097 | 6.8788 | 5.8133 | 4.8823 | 5.2858 | 7.7850 | 4.5000 | 9.0274 | 6.5142 |
| 8.0540 | 11.5743 | 5.2965 | 5.4805 | 7.3212 |
TABLE 10 auxiliary variable inflow COD (mg/L)
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Table 11. Found BOD concentration of effluent (mg/L)
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Test sample:
table 12. Auxiliary variable total Nitrogen in effluent (mg/L)
| 12.5450 | 5.8739 | 8.4295 | 6.3286 | 7.2489 | 5.8362 | 13.0872 | 13.1711 | 6.5681 | 13.0872 |
| 13.2356 | 5.8812 | 9.5222 | 4.5000 | 4.5815 | 13.0021 | 6.4052 | 12.5061 | 11.5140 | 12.5134 |
| 11.7036 | 13.3936 | 13.0483 | 12.6666 | 12.4271 | 5.5237 | 7.4605 | 7.0447 | 8.8781 | 12.4307 |
| 6.9426 | 7.0863 | 12.0343 | 12.9280 | 12.2313 | 6.5085 | 11.8422 | 5.6331 | 9.5091 | 7.4690 |
| 7.9152 | 12.3869 | 5.5298 | 11.7620 | 6.6544 | 9.8702 | 6.2641 | 6.4708 | 11.4106 | 12.4988 |
| 5.0702 | 11.6489 | 13.7340 | 12.9204 | 12.5766 | 6.3225 | 7.0787 | 12.6362 | 15.7000 | 12.1170 |
| 6.1912 | 5.9930 | 7.6283 | 13.2047 | 13.0483 | 6.2033 | 8.1303 | 12.6119 | 12.9936 | 8.2021 |
| 7.1237 | 12.9280 | 5.9359 | 13.0495 | 12.7541 | 12.9632 | 5.7693 | 8.8708 | 5.4143 | 9.0702 |
| 12.6362 | 11.1857 | 10.8502 | 13.0775 | 7.4775 | 12.6301 | 12.7128 | 8.5401 | 13.4398 | 11.4726 |
| 13.2647 | 12.8951 | 12.2872 | 12.4404 | 12.5729 | 13.3863 | 8.7711 | 13.4605 | 5.8508 | 6.7578 |
Table 13 auxiliary variable ammonia nitrogen (mg/L) in the effluent
| 12.4818 | 5.0091 | 4.9000 | 6.2636 | 7.2909 | 4.8273 | 8.7273 | 12.8091 | 5.3909 | 12.1273 |
| 12.4091 | 5.9091 | 5.5182 | 5.7636 | 5.6636 | 12.8091 | 6.2455 | 11.6273 | 12.0182 | 12.3545 |
| 8.4273 | 15.6273 | 12.1000 | 14.6091 | 10.2636 | 6.3000 | 7.0455 | 6.5818 | 7.5182 | 14.7182 |
| 6.9273 | 6.3455 | 13.4091 | 11.6818 | 12.1273 | 4.5000 | 12.0818 | 7.9545 | 7.2636 | 4.5636 |
| 7.4727 | 12.4091 | 5.6182 | 13.2636 | 5.3909 | 6.1091 | 6.0909 | 5.6636 | 8.0364 | 11.6273 |
| 4.7818 | 8.7273 | 9.5545 | 9.6545 | 12.4909 | 6.5727 | 5.8364 | 9.5727 | 13.5182 | 9.4636 |
| 4.8455 | 5.6818 | 7.6545 | 11.1273 | 12.5182 | 5.4818 | 6.7727 | 12.0273 | 11.5909 | 7.9364 |
| 4.7727 | 11.1091 | 6.6455 | 11.5364 | 13.4455 | 11.8091 | 5.8636 | 6.3182 | 6.8545 | 6.5000 |
| 11.9000 | 7.1364 | 12.5091 | 11.3000 | 4.8364 | 11.5273 | 11.6000 | 7.8636 | 11.7182 | 9.1000 |
| 13.3727 | 11.2636 | 12.8636 | 11.5909 | 11.3545 | 12.3091 | 6.9909 | 12.3000 | 5.4091 | 5.9455 |
Table 14. Auxiliary variable total Nitrogen in Water (mg/L)
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TABLE 15 auxiliary variable BOD (mg/L) of incoming water
| 5.8200 | 5.7800 | 9.7800 | 11.6200 | 8.9000 | 11.1400 | 7.5800 | 6.2200 | 8.7000 | 6.1000 |
| 6.6200 | 9.8600 | 12.4644 | 9.5400 | 9.1400 | 6.4600 | 11.2200 | 6.2200 | 5.7400 | 5.2600 |
| 5.7800 | 5.7800 | 4.8600 | 5.2600 | 5.5800 | 7.5400 | 9.1800 | 11.6200 | 10.7000 | 7.7400 |
| 12.9800 | 15.2956 | 5.1400 | 4.9000 | 7.2200 | 6.7400 | 5.2600 | 7.7000 | 5.1800 | 8.2600 |
| 9.3000 | 6.4200 | 9.3800 | 5.8600 | 8.5800 | 12.0600 | 7.8600 | 7.3000 | 9.0600 | 5.6600 |
| 6.7400 | 9.5400 | 6.4600 | 7.9160 | 8.1400 | 8.3400 | 9.8200 | 8.4200 | 4.7800 | 9.2600 |
| 6.2600 | 10.3000 | 8.7800 | 7.4120 | 6.0600 | 8.6200 | 14.0822 | 9.0200 | 5.9800 | 9.8200 |
| 9.3000 | 6.3800 | 8.0200 | 6.3000 | 6.3400 | 9.1000 | 9.4200 | 5.8600 | 7.3800 | 5.0600 |
| 4.5000 | 10.9800 | 6.9400 | 4.9000 | 8.8600 | 5.5000 | 5.6200 | 10.2600 | 5.8600 | 9.1800 |
| 6.1000 | 5.2600 | 6.5800 | 7.5800 | 7.4200 | 8.4600 | 6.2600 | 7.6200 | 6.4200 | 11.3400 |
Table 16 auxiliary variable intake ammonia nitrogen (mg/L)
| 8.1356 | 10.8022 | 8.7933 | 10.3933 | 11.1222 | 10.7889 | 10.6333 | 8.4378 | 10.8644 | 8.1356 |
| 8.0378 | 11.8689 | 15.7000 | 10.3800 | 9.5844 | 8.7578 | 9.8244 | 8.5356 | 7.1133 | 7.8244 |
| 8.1533 | 7.2200 | 7.7444 | 7.6778 | 8.3933 | 7.7089 | 15.0556 | 12.0378 | 12.0867 | 10.2111 |
| 11.7533 | 12.6333 | 8.7133 | 8.1089 | 7.6022 | 9.8733 | 7.1933 | 7.9044 | 7.8600 | 11.5756 |
| 12.7311 | 7.6022 | 10.6644 | 8.3311 | 12.6067 | 14.0556 | 12.6244 | 12.4778 | 11.0689 | 7.6244 |
| 12.1356 | 10.8867 | 10.2467 | 9.5844 | 8.6467 | 7.8022 | 12.5222 | 8.1533 | 7.3444 | 9.9933 |
| 8.7133 | 11.0289 | 10.4733 | 9.8156 | 7.6289 | 10.3222 | 12.7000 | 7.6467 | 7.5978 | 12.6778 |
| 10.6867 | 7.7667 | 7.2778 | 7.3711 | 8.5533 | 8.1356 | 11.8111 | 7.7622 | 7.3400 | 7.5933 |
| 7.6022 | 10.3044 | 7.9578 | 8.0911 | 10.1711 | 6.8289 | 7.8867 | 13.8778 | 7.4244 | 10.4956 |
| 10.4200 | 8.0378 | 10.6556 | 7.8511 | 7.7756 | 8.7756 | 8.4333 | 8.1667 | 8.7933 | 12.6067 |
Table 17 auxiliary variable out-water phosphate (mg/L)
| 14.6250 | 10.1500 | 10.9500 | 11.1250 | 14.5500 | 11.3500 | 10.7250 | 13.9750 | 11.3750 | 13.6250 |
| 13.4000 | 11.7250 | 13.0556 | 11.3250 | 11.4000 | 14.3000 | 10.8750 | 14.0000 | 13.4000 | 13.5250 |
| 12.5500 | 13.7250 | 14.0250 | 14.1000 | 13.2750 | 12.4500 | 15.1000 | 14.4500 | 10.3750 | 13.9250 |
| 14.4000 | 14.1444 | 13.8500 | 14.1000 | 12.9250 | 8.3500 | 13.7000 | 12.3250 | 11.6750 | 9.6500 |
| 12.3750 | 12.8250 | 11.5500 | 13.5500 | 9.9750 | 12.9000 | 10.4750 | 9.5250 | 9.1750 | 14.2250 |
| 11.8500 | 9.5750 | 13.2250 | 13.7750 | 14.1250 | 11.6750 | 10.9750 | 13.6000 | 15.5250 | 6.9750 |
| 9.2500 | 11.5750 | 12.0250 | 13.9500 | 14.2500 | 11.8750 | 13.6778 | 13.3250 | 13.7500 | 12.7250 |
| 8.2750 | 13.7250 | 12.0250 | 13.8500 | 14.1500 | 13.6000 | 11.8750 | 11.7000 | 12.5750 | 11.5000 |
| 13.9500 | 7.4000 | 13.7500 | 14.5750 | 6.6250 | 12.9250 | 14.1250 | 11.5500 | 13.9250 | 7.5250 |
| 14.0750 | 14.3500 | 13.4750 | 14.3000 | 14.1500 | 14.3000 | 11.8000 | 14.5500 | 10.9000 | 14.9500 |
TABLE 18 auxiliary variable Biochemical MLSS (mg/L)
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TABLE 19 auxiliary variable biochemical pool DO (mg/L)
| 11.4597 | 9.0630 | 7.4959 | 7.9568 | 13.4877 | 8.8786 | 6.4358 | 10.0309 | 8.7403 | 9.4778 |
| 13.5337 | 9.1551 | 9.2012 | 8.9708 | 7.6342 | 8.2794 | 8.1872 | 9.9387 | 9.4317 | 10.9527 |
| 13.4416 | 12.2432 | 12.3354 | 10.4918 | 13.2572 | 13.1189 | 11.0449 | 10.6761 | 7.5420 | 13.5337 |
| 13.2111 | 10.1691 | 10.9527 | 12.0128 | 8.6481 | 8.5560 | 10.4918 | 12.5658 | 12.7502 | 9.0630 |
| 7.9568 | 12.5658 | 6.4358 | 13.0267 | 8.4177 | 9.3856 | 8.6481 | 8.0490 | 8.0951 | 13.2572 |
| 8.0490 | 8.8786 | 8.0029 | 8.9708 | 12.0588 | 13.1189 | 6.7584 | 8.9708 | 13.6259 | 8.0490 |
| 9.1551 | 9.1091 | 11.5058 | 8.8786 | 12.4737 | 7.4498 | 9.1551 | 11.4136 | 11.3214 | 7.9568 |
| 8.7403 | 11.5979 | 12.1049 | 10.1230 | 12.9807 | 11.2292 | 9.4778 | 12.1049 | 10.7222 | 14.0407 |
| 12.6119 | 7.8646 | 12.0588 | 13.8564 | 8.6481 | 13.2111 | 11.8284 | 8.4177 | 11.0449 | 9.0169 |
| 10.3074 | 13.7181 | 11.5519 | 13.3033 | 13.2111 | 11.1831 | 14.0868 | 12.3815 | 8.2333 | 13.3033 |
TABLE 20 auxiliary variable intake phosphate (mg/L)
| 5.8381 | 8.6982 | 9.1301 | 6.9177 | 6.1283 | 9.3566 | 11.8664 | 5.4239 | 7.7425 | 5.4451 |
| 5.4451 | 7.0735 | 7.8369 | 7.7460 | 7.8345 | 5.4416 | 6.7903 | 5.1726 | 4.6345 | 5.6823 |
| 5.5336 | 5.0735 | 5.6469 | 5.2292 | 5.6080 | 5.5761 | 6.2593 | 6.0717 | 6.6416 | 5.4558 |
| 6.0434 | 6.2180 | 4.9814 | 5.4239 | 4.9708 | 10.5460 | 4.8611 | 5.6575 | 5.6221 | 9.8381 |
| 7.0239 | 5.0699 | 8.9602 | 4.7690 | 8.8681 | 8.0681 | 6.4363 | 8.9566 | 6.5673 | 5.7885 |
| 8.3655 | 6.4823 | 14.4221 | 12.0327 | 5.8381 | 5.5584 | 7.5726 | 11.1159 | 6.3584 | 8.0327 |
| 9.6221 | 7.0345 | 6.8965 | 12.9496 | 5.3602 | 9.0168 | 6.9118 | 5.1230 | 5.4628 | 7.1513 |
| 9.8664 | 5.1053 | 5.6788 | 5.2646 | 5.3071 | 5.2965 | 7.1124 | 5.2575 | 5.4947 | 5.6398 |
| 5.1690 | 6.0044 | 5.7142 | 5.7920 | 10.5850 | 5.2363 | 5.5159 | 6.8965 | 5.2469 | 6.6841 |
| 5.3248 | 5.6540 | 4.5956 | 5.7637 | 5.7106 | 5.5088 | 5.0664 | 5.5513 | 8.6381 | 6.0186 |
Table 21. Auxiliary variable COD (mg/L) of inflow
| 9.2032 | 7.3698 | 5.6559 | 10.2794 | 13.1093 | 9.2431 | 10.8374 | 10.4388 | 9.7214 | 9.3626 |
| 9.6018 | 12.8701 | 9.8808 | 10.0801 | 9.1633 | 13.1890 | 11.1164 | 8.8843 | 6.0544 | 7.5292 |
| 9.2032 | 9.8808 | 8.0075 | 6.7719 | 9.3228 | 8.1270 | 10.4388 | 8.8445 | 11.8737 | 11.1164 |
| 10.3192 | 10.5982 | 12.1128 | 10.2794 | 10.5584 | 10.1199 | 7.9278 | 8.8843 | 9.2032 | 7.1306 |
| 8.7648 | 8.1669 | 11.9534 | 9.1633 | 10.8772 | 15.7000 | 13.2687 | 11.9135 | 9.8409 | 7.7683 |
| 11.0765 | 10.6779 | 10.5185 | 11.6744 | 9.8011 | 8.8843 | 10.8772 | 7.7683 | 4.8587 | 10.6779 |
| 9.6815 | 11.5548 | 8.9242 | 10.5584 | 8.6053 | 10.0801 | 10.3591 | 8.8046 | 7.2103 | 13.6274 |
| 9.9206 | 9.6018 | 9.1633 | 8.9242 | 12.1527 | 12.0331 | 14.4644 | 6.6125 | 7.0907 | 7.7683 |
| 7.5690 | 8.8046 | 9.8409 | 8.4459 | 8.5256 | 8.8445 | 7.7683 | 14.7833 | 8.5256 | 9.8409 |
| 12.3918 | 10.0004 | 9.3228 | 9.1633 | 8.2865 | 10.9968 | 8.5655 | 9.3626 | 8.0473 | 10.7178 |
Table 22. Actual measurement of BOD concentration (mg/L) of effluent
| 11.1429 | 11.6714 | 13.1286 | 12.8571 | 13.8429 | 14.5429 | 12.3143 | 10.9000 | 13.3857 | 10.9143 |
| 10.8000 | 12.6857 | 14.1000 | 13.8000 | 13.8143 | 10.3000 | 12.7429 | 10.2429 | 10.1286 | 10.2857 |
| 11.4286 | 11.0429 | 10.7143 | 10.7714 | 11.5143 | 11.4857 | 12.6714 | 14.5857 | 13.0857 | 12.2286 |
| 14.9571 | 15.5000 | 10.3857 | 10.2857 | 11.0286 | 12.1000 | 10.3143 | 11.4429 | 11.5714 | 12.6143 |
| 13.0000 | 11.1143 | 14.2857 | 10.1571 | 14.0000 | 13.9000 | 12.1143 | 14.0857 | 12.7286 | 10.8286 |
| 13.9000 | 12.5000 | 12.1714 | 12.6600 | 12.6000 | 10.8857 | 13.1000 | 12.8000 | 11.9000 | 12.5286 |
| 11.8857 | 12.7286 | 12.8000 | 12.5200 | 10.8000 | 12.9286 | 14.9000 | 10.6143 | 10.9857 | 13.2000 |
| 14.4000 | 11.1000 | 11.2286 | 11.0000 | 10.2714 | 10.6571 | 12.6429 | 11.7714 | 11.5286 | 11.6000 |
| 10.2000 | 12.6286 | 12.2429 | 11.7143 | 14.6571 | 11.1429 | 11.2000 | 13.1429 | 10.8000 | 12.7714 |
| 10.6000 | 11.4571 | 11.2571 | 11.4000 | 11.3000 | 11.2857 | 11.8571 | 11.4000 | 11.9714 | 11.9857 |
TABLE 23 prediction of BOD concentration (mg/L) of the water by the soft measurement method of the invention
| 10.9587 | 12.4779 | 13.3882 | 13.5493 | 12.9277 | 14.3140 | 12.4474 | 10.8082 | 13.2564 | 10.9683 |
| 11.1841 | 13.2436 | 13.8942 | 13.2937 | 13.2559 | 10.8312 | 13.4366 | 10.8581 | 10.6723 | 10.7437 |
| 11.4622 | 10.6948 | 10.7798 | 10.5419 | 11.2111 | 11.5180 | 13.4736 | 13.7765 | 13.2609 | 11.4760 |
| 14.1395 | 14.4762 | 10.6665 | 10.8532 | 10.7924 | 13.1953 | 10.6136 | 11.5691 | 11.4409 | 13.3273 |
| 12.6630 | 10.8584 | 14.0394 | 10.7962 | 13.8171 | 14.1084 | 12.5939 | 13.3541 | 12.3291 | 11.0501 |
| 13.0923 | 12.3768 | 12.2838 | 12.6805 | 11.4128 | 11.7002 | 13.9789 | 12.4416 | 10.7326 | 12.5194 |
| 12.7084 | 13.2320 | 12.1601 | 12.6555 | 10.9330 | 13.6297 | 14.0882 | 11.3101 | 10.9983 | 13.1563 |
| 14.0852 | 10.9683 | 11.7059 | 10.8075 | 11.0923 | 11.5350 | 13.2023 | 11.6394 | 11.4022 | 11.6044 |
| 10.7674 | 12.8016 | 11.1348 | 11.0762 | 14.2194 | 10.8501 | 11.0021 | 13.4111 | 10.8473 | 12.2415 |
| 10.9474 | 11.1059 | 10.9798 | 11.5382 | 11.4708 | 11.3660 | 11.3216 | 11.2816 | 12.5421 | 12.6423 |
Claims (1)
1. A method for soft measurement of BOD concentration of effluent based on a pruned feedforward small world neural network is characterized by comprising the following steps:
step 1: selecting auxiliary variables of a BOD prediction model of the effluent;
directly selecting given M auxiliary variables; normalizing the auxiliary variable to [ -1,1] according to formula (1), and normalizing the output variable BOD to [0,1] according to formula (2):
wherein F is m Represents the mth auxiliary variable, O represents the output variable，x m And y represents the mth auxiliary variable and the output variable after normalization respectively; min (F) m ) And max (F) m ) Respectively representing the minimum value and the maximum value in the mth auxiliary variable, and min (O) and max (O) respectively represent the minimum value and the maximum value in the output variable;
step 2: designing a feedforward small world neural network model;
step 2.1: designing a feedforward small-world neural network model wiring mode;
constructing a feedforward small-world neural network according to the Watts-Strogatz rewiring rule; the specific construction process is as follows: firstly, constructing an L-layer feedforward neural network with regular connection, then randomly selecting one connection from the model according to reconnection probability p, disconnecting from the tail end and reconnecting to another neuron in the model, wherein the value range of p is (0, 1), if the new connection exists, randomly selecting another new neuron for connection, and the neurons in the same layer cannot be connected with each other;
step 2.2: designing a topological structure of a feedforward small-world neural network model;
the designed feedforward small world neural network topological structure is L-layer in total and comprises an input layer, an hidden layer and an output layer; the calculation functions of each layer are as follows:
(1) input layer: the layer has M neurons, representing M input auxiliary variables, and the input of the input layer is x (1) ＝[x 1 (1) ,x 2 (1) ,…,x M (1) ]Wherein x is m (1) Representing the mth input auxiliary variable of the input layer, m=1, 2, …, M, the layer output is equal to the input, the output of the mth neuron of the input layer is:
(2) hidden layer: by adopting the sigmoid function as an activation function of the hidden layer, the input and output definitions of the j-th neuron of the first layer of the neural network are shown in formulas (4) and (5), respectively:
wherein n is u Representing the number of neurons in the u-th layer of the neural network,representing a connection weight between an ith neuron of a ith layer and a jth neuron of a first layer of the neural network, wherein f () is a sigmoid function;
(3) output layer: the output layer comprises a neuron, and the output of the output neuron is as follows:
wherein the method comprises the steps ofRepresenting the connection weight between the jth neuron of the first layer of the neural network and the output neuron, n l Representing the number of neurons of the first layer of the neural network;
step 3: designing a deletion algorithm of the feedforward small world neural network;
step 3.1: defining a performance index function:
wherein Q is the number of samples, d q For the desired output value of the q-th sample,a predicted output value for the q-th sample;
step 3.2: carrying out parameter correction by adopting a batch gradient descent algorithm;
(1) the output weight correction of the output layer is as shown in formulas (8) - (10):
wherein the method comprises the steps of
Wherein,and->Represents the connection weight between the j-th neuron and the output neuron of the first layer of the neural network at the moments t and t+1 respectively,/o>Representing the variation value of the connection weight between the jth nerve of the jth layer of the neural network at the moment t and the output nerve element, eta v Represents the learning rate, eta in the output weight correction process of the output layer v The value range is (0,0.1)]；
(2) The output weight correction of the hidden layer is as shown in formulas (11) - (13):
wherein the method comprises the steps of
Wherein,and->Representing the connection weight between the ith neuron of the s-th layer and the jth neuron of the first layer of the neural network at the moments t and t+1 respectively, < + >>Representing the variation value of the connection weight between the ith neuron of the s-th layer and the jth neuron of the first layer of the neural network at the moment of t, eta w Represents the learning rate, eta in the process of correcting the output weight of the hidden layer w The value range is (0,0.1)]；
Step 3.3: inputting training sample data, updating output weights of an implicit layer and an output layer according to formulas (8) - (13) in step 3.2, wherein Iter represents training iteration times, the iteration times are increased once every time the weights are updated, if the iteration times in the training process can be divided by a learning step length constant tau, executing step 3.4, otherwise, jumping to step 3.6, wherein the tau value range is an integer in a [10,100] range;
step 3.4: calculating the Katz centrality and the normalized Katz centrality of all hidden layer neurons; katz centrality is defined as shown in equation (14):
wherein the method comprises the steps ofRepresents the k power of the connection weight between the neuron g and the neuron h, alpha represents the attenuation factor, and the set value of alpha needs to satisfy 0<α<1/λ max Alpha has a value in the range of (0,0.1)]，λ max A value representing the maximum eigenvalue of the network adjacency matrix with greater Katz centrality indicates that the node is more important, and vice versa;
the normalized Katz centrality definition is shown in equation (15):
wherein the method comprises the steps ofKatz centrality of jth neuron representing the s-th layer of the neural network, +.>Katz centrality normalized by the jth neuron representing the s-th layer of the neural network; is provided with->Average value of normalized Katz centrality of all neurons representing the s-th layer of the neural network, wherein theta is a preset threshold parameter, and the value range of theta is [0.9,1 ]]If the Katz centrality of neurons satisfies +.>The neuron is considered as an unimportant neuron, and the unimportant hidden layer neuron set in the s layer of the neural network is marked as A s The remaining set of neurons in layer s is denoted B s ；
Step 3.5: computing set A s And set B s The correlation coefficient between hidden layer neurons in (2) is defined as shown in formula (16):
wherein,and->Respectively representing the output values of the ith neuron and the jth neuron of the ith layer of the neural network when the qth training sample is input; />And->Respectively representing the input of all samples +.>And->Average value of (2); sigma (sigma) i Sum sigma j Respectively representing the input of all samples +.>And->Standard deviation of (2); will set A s Combining the hidden layer neuron a and the neuron b with the highest correlation coefficient to generate a new neuron c, wherein the connection weight between the neuron c and the neuron of the forward network layer is constructed according to the reconnection rule of Watt-Strogatz and the reconnection probability p, wherein the value range of p is (0, 1) so as to ensure the small worldwide of the network, and the output of the neuron c is shown as a formula (17):
wherein the method comprises the steps ofRepresenting a connection weight between an ith neuron in an nth layer of the neural network and a neuron c in an s-th layer;
the connection weights between neurons c and neurons of the backward network layer according to the pruning algorithm are as shown in equations (18) - (19):
wherein the method comprises the steps ofAnd->Connection weights between neurons a, b and c representing the s-th layer of the neural network and neuron j in the backward hidden layer, respectively, +.>And->Connection weights between neurons a, b and c, respectively representing the s-th layer of the neural network, and the output neurons,/for>And->Output values of neurons a, b and c respectively representing the s-th layer of the neural network are combined according to formulas (17) - (19), and then step 3.3 is skipped;
step 3.6: calculating a training RMSE if the RMSE is satisfied to be less than the expected training rmsemse d Or the iteration number reaches the maximum iteration number Iter max Stopping the calculation at the time, wherein Iter max The value range is [5000,10000 ]]Otherwise, jumping to step 3.3, wherein the definition of the RMSE is shown in a formula (20);
step 4: predicting BOD of the effluent;
and taking the test sample data as the input of the trained pruned feedforward small-world neural network, and performing inverse normalization on the output of the neural network to obtain the predicted value of the BOD of the output water.
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