DYNAMIC MULTI-OBJECTIVE PARTICLE SWARM OPTIMIZATION-BASED OPTIMAL CONTROL METHOD FOR WASTEWATER TREATMENT PROCESS

Abstract

A dynamic multi-objective particle swarm optimization based optimal control method is provided to realize the control of dissolved oxygen (SO) and the nitrate nitrogen (SNO) in wastewater treatment process. In this method, dynamic multi-objective particle swarm optimization was used to optimize the operation objectives of WWTP, and the optimal solutions of SO and SNO can be calculated. Then PID controller was introduced to trace the dynamic optimal solutions of SO and SNO. The results demonstrated that the proposed optimal control strategy can address the dynamic optimal control problem, and guarantee the efficient and stable operation. In addition, this proposed optimal control method in this present invention can guarantee the effluent qualities and reduce the energy consumption.

Description

CROSS REFERENCE TO RELATED PATENT APPLICATIONS

This application claims priority to Chinese Patent Application No. 201910495404.5, filed on Jun. 10, 2019, entitled “Dynamic multi-objective particle swarm optimization-based optimal control method for wastewater treatment process,” which is hereby incorporated by reference in its entirety.

TECHNICAL FIELD

In this present invention, a dynamic multi-objective particle swarm optimization (DMPSO) algorithm is used to solve the optimization objectives in wastewater treatment process (WWTP), and then the optimal solutions of dissolved oxygen (S_O) and nitrate nitrogen (S_NO) can be obtained. Both S_O and S_NO have an important influence on the energy consumption and effluent quality of WWTP, and determine the treatment effect of WWTP. It is necessary to design DMOPSO-based optimal control method to control S_O and S_NO for WWTP, which can guarantee the effluent qualities and reduce the energy consumption.

BACKGROUND

WWTP refers to the physical, chemical and biological purification process of wastewater, so as to meet the requirements of discharge or reuse. Currently, natural freshwater resources have been fully exploited and natural disasters are increasingly occurred. Water shortage has posed a very serious threat to the economy and citizens' lives of many cities around the world. Water shortage crisis has been the reality we are facing. An important way to solve this problem is to turn the municipal wastewater into water supply source. Municipal wastewater is available in the vicinity, with the features of stable sources and easy collection. Stable and efficient wastewater treatment system is the key to the recycling of water resources, which has good environmental and social benefits. Therefore, the research results of the present invention have broad application prospects.

WWTP is a complex dynamic system. Its biochemical reaction cycle is long and pollutant composition is complex. Influent flow rate and influent qualities are passively accepted. And the primary performance indices pumping energy (PE), aeration energy (AE) and effluent quality (EQ) are strongly nonlinear and coupling. The essence of WWTP is dynamic multi-objective optimization control problem. It is significant to establish appropriate optimization objectives based to the dynamic operation characteristics of WWTP. Since the optimization objectives of WWTP are interactional, how to balance the relationship of the optimization objectives is of great significance to ensure the quality of effluent organisms and reduce energy consumption. In addition, S_O and S_NO, as the main control variables, have great influence on the operation efficiency and the operation performance. Therefore, it is necessary to establish a dynamic optimal control method, which can establish the optimization objectives and realize the tracking control according to the different operation conditions and the dynamic operation environments. The dynamic optimal control method can efficiently guarantee the effluent organisms in the limits and reduce the operation cost as much as possible.

In this present invention, a DMOPSO-based optimal control method is designed for WWTP, where the models of the performance indices can be established based on the dynamics and the operation data of WWTP. DMOPSO algorithm is designed to optimize the established optimization objectives and derive the optimal solutions of S_O and S_NO. Then multivariable PID controller is introduced to trace the optimal solutions S_O and S_NO.

SUMMARY

In this present invention, a dynamic multi-objective particle swarm optimization-based optimal control method is designed for wastewater treatment process (WWTP), the steps are:

(1) design the models of performance indices for wastewater treatment process:

{circle around (1)} analysis the dynamic characteristics and the operation data of WWTP, obtain the related process variables of the key performance indices pumping energy (PE), aeration energy (AE) and effluent quality (EQ): influent flow rate (Q_in), dissolved oxygen (S_O), nitrate nitrogen (S_NO), ammonia nitrogen (S_NH), suspended solids (SS);

{circle around (2)} establish the models of the performance indices based on the operation time of S_O and S_NO, the operation time of S_O is thirty minutes, and the operation time of S_NO is two hours, the established models of the performance indices are adjusted per thirty minutes; if the operation time meets the operation time of S_NO, then the models are designed as:

$\begin{array}{cc}\{\begin{array}{c}{f}_1\left(t\right)=\sum _{r=1}^{10}W_{1r}\left(t\right)\times e^{-{x\left(t\right)-c_{1r}\left(t\right)}^2/2{b_{1r}\left(t\right)}^2}+W_1\left(t\right)\\ f_2\left(t\right)=\sum _{r=1}^{10}W_{2r}\left(t\right)\times e^{-{x\left(t\right)-c_{2r}\left(t\right)}^2/2{b_{2r}\left(t\right)}^2}+W_2\left(t\right)\end{array}& \left(1\right)\end{array}$

where f₁(t) is PE model at the tth time, f₂(t) is EQ model at the tth time, e^{−∥x(t)−c1r(t)∥2/2b1r(t)2} and e^{−∥1(t)−c2r(t)∥2/2b2r(t)2} are the rth radial basis function of f₁(t) and f₂(t) at the tth time, r=1, 2, . . . , 10, x(t)=[Q_in(t), S_O(t), S_NO(t), S_NH(t), SS(t)] is the input vector of PE model and EQ model, c_1r(t) and c_2r(t) are the centers of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and the ranges of c_1r(t) and c_2r(t) are [−1, 1] respectively; b_1r(t) and b_2r(t) are the widths of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and the ranges of b_1r(t) and b_2r(t) are [0, 2] respectively, W_1r(t) and W_2r(t) are the weights of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and the ranges of W_1r(t) and W_2r(t) are [−3, 3] respectively; W₁(t) and W₂(t) are the output offsets of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and the ranges of W₁(t) and W₂(t) are [−2, 2] respectively; if the operation time meets the time of S_O, then the models are designed as:

$\begin{array}{cc}\{\begin{array}{c}{f}_1\left(t\right)=\sum _{r=1}^{10}W_{1r}\times e^{-{x\left(t\right)-c_{1r}\left(t\right)}^2/2{b_{1r}\left(t\right)}^2}+W_1\left(t\right)\\ f_2\left(t\right)=\sum _{r=1}^{10}W_{2r}\times e^{-{x\left(t\right)-c_{2r}\left(t\right)}^2/2{b_{2r}\left(t\right)}^2}+W_2\left(t\right)\\ f_3\left(t\right)=\sum _{r=1}^{10}W_{3r}\times e^{-{x\left(t\right)-c_{3r}\left(t\right)}^2/2{b_{3r}\left(t\right)}^2}+W_3\left(t\right)\end{array}& \left(2\right)\end{array}$

where f₃(t) is AE model at the tth time, e^{−∥s(t)−c3r(t)∥2/2b3r(t)2} is the rth radial basis function of f₃(t) at the tth time, c_3r(t) is the center of the rth radial basis function of f₃(t) at the tth time, and the range of c_3r(t) is [−1, 1]; b_3r(t) is the widths of the rth radial basis function of f₃(t) at the tth time, and the range of b_3r(t) is [0, 2]; W_3r(t) is the weights of the rth radial basis function of f₃(t) at the tth time, and the range of W_3r(t) is [−3, 3]; W₃(t) is the output offset of the rth radial basis function of f₃(t), and the range of W₃(t) is [−2, 2];

(2) dynamic optimization of the control variables for WWTP

(2)-1 set the maximum iterative numbers of the optimization process T_max;

(2)-2 take the established models of performance indices as optimization objectives;

(2)-3 regard the inputs of the optimization objectives x(t)=[Q_in(t), S_O(t), S_NO(t), S_NH(t), SS(t)] as the position of the particles, calculate values of the optimization objectives, update the personal optimal position (pBest_k,i(t)) and the position and the velocity of the particles, the update process are:

x _k,i(t+1)=x_k,i(t)+v_k,i(t+1)   (3)

v _k,i(t+1)=ω(t)v_k,i(t)+c₁α₁(pBest_k,i(t)−x_k,i(t))+c₂α₂(gBest_k(t)−x_k,i(t))   (4)

where x_k,i(t+1) is the position of the ith particle in the kth iteration at the t+1th time, v_k,i(t+1) is the velocity of the ith particle in the kth iteration at the t+1th time, ω is the inertia weight, the range of ω is [0, 1], c₁ and c₂ are the learning parameters, α₁ and α₂ are the uniformly distributed random numbers, pBest_k,i(t) is the personal optimal position of the ith particle in the kth iteration at the tth time, and gBest_k(t) is the personal optimal position in the kth iteration at the tth time;

(2)-4 design the diversity index and the convergence index based on the Chebyshev distance, diversity index is designed to measure the distribution quality of the non-dominated solutions,

$\begin{array}{cc}U\left(t\right)=\sqrt{\frac{1}{\mathrm{NS}\left(t\right)-1}\sum _{m=1}^{\mathrm{NS}\left(t\right)}{\left(\stackrel{⋓}{D}\left(t\right)-D_m\left(t\right)\right)}^2}& \left(5\right)\end{array}$

where U(t) is the diversity of the optimal solutions at the tth time, m=1, 2, . . . , NS(t), NS(t) is the number of non-dominated solutions at time t, Ď(t) is the average distance of all the Chebyshev distance D_m(t), D_m(t) is the Chebyshev distance of consecutive solutions of the mth solution; and convergence index is developed to obtain the degree of proximity, which are calculated as

$\begin{array}{cc}A\left(t\right)=\frac{1}{\mathrm{NS}\left(t\right)}\sqrt{\sum _{l=1}^{\mathrm{NS}\left(t\right)}d_l\left(t\right)}& \left(6\right)\end{array}$

where A(t) is the convergence of the optimal solutions at the tth time, d_l(t) is the Chebyshev distance of the lth solution between the kth iteration and the k−1th iteration;

(2)-5 judge the changes of the optimization objectives, if the number of the objectives is changed, return to step (2)-6; otherwise, return to (2)-7;

(2)-6 when the number of the objectives is increased, some particles will be changed to enhance the diversity performance, the update process of the population size is

$\begin{array}{cc}{N}_{k+1}\left(t\right)=\{\begin{array}{cc}{N}_k\left(t\right)& {\alpha }_k\left(t\right)=0\\ N_k\left(t\right)-\left(N_k\left(t\right)-{\mathrm{NS}}_k\left(t\right)\right)·{\alpha }_k\left(t\right)& {\alpha }_k\left(t\right)<0\\ N_k\left(t\right)+{\mathrm{NS}}_k\left(t\right)·{\alpha }_k\left(t\right)& {\alpha }_k\left(t\right)>0\end{array}& \left(7\right)\end{array}$

where N_k+1(t) and N_k(t) are the population size in the kth iteration and in the k+1th iteration at the tth time respectively, α_k(t) is the gradient of diversity in the kth iteration at the tth time, which is calculated as

$\begin{array}{cc}{\alpha }_k\left(t\right)=\frac{U_k\left(t\right)-U_k\left(t-\varepsilon \right)}{\varepsilon }& \left(8\right)\end{array}$

where ε is the adjusted frequency of population size, and the range of ε is [1, T_max]; if the number of the objectives is decreased, some particles will be changed to improve the convergence performance, the update process of the population size is

$\begin{array}{cc}{N}_{k+1}\left(t\right)=\{\begin{array}{cc}{N}_k\left(t\right)& {\beta }_k\left(t\right)=0\\ N_k\left(t\right)+{\mathrm{NS}}_k\left(t\right)·{\beta }_k\left(t\right)& {\beta }_k\left(t\right)<0\\ N_k\left(t\right)-\left(N_k\left(t\right)-{\mathrm{NS}}_k\left(t\right)\right)·{\beta }_k\left(t\right)& {\beta }_k\left(t\right)>0\end{array}& \left(9\right)\end{array}$

where β_k(t) is the gradient of convergence in the kth iteration at the tth time, which is calculated as

$\begin{array}{cc}{\beta }_k\left(t\right)=\frac{A_k\left(t\right)-A_k\left(t-\varepsilon \right)}{\varepsilon }& \left(10\right)\end{array}$

(2)-7 compare pBest_k(t) with the solutions Φ_k−1(t) in the archive, where Φ_k−1(t)=[yφ_k−1,1(t), φ_k−1,2(t), . . . , φ_k−1,ι(t)], φ_k−1,ι(t) is the ιth optimal solutions in the k−1th iteration at the tth time of the archive, the archive Φ_k(t) is updated by the dominated relationship, and the calculation process of the dominated relationship can be shown as

Φ_k(t)=Φ_k−1(t)∪p_k−1(t), if f_h(a_k−ι(t))≥f_h(p_k(t)), h=1, 2, 3   (11)

where ∪ is the relationship of combine, if the value of pBest_k−1(t) is lower than the objective value of a_k−1,ι(t), then the pBest_k−1(t) will be saved in the archive, otherwise, a_k−1,ι(t) will be saved, then gBest_k(t) will be selected from the archive according to the density method;

(2)-8 if the current iteration is greater than the preset T_max, then return to step (2)-9, otherwise, return to step (2)-3;

(2)-9 select a set of global optimal solutions gBest_Tmax(t) from the archive randomly, and gBest_Tmax(t)=[Q_in,Tmax*(t), S_O,Tmax*(t), S_NO,Tmax*(t), S_NH,Tmax*(t), SS_Tmax*(t)], where Q_in,Tmax*(t) is the optimal solution of influent flow rate, S_O,Tmax*(t) is the optimal solution of dissolved oxygen, S_NO,Tmax*(t) is the optimal solution of nitrate nitrogen, S_NH,Tmax*(0 is the optimal solution of ammonia nitrogen, SS_Tmax*(t) is the optimal solution of suspend solid;

(3) tracking control of the optimal solutions in WWTP

(3)-1 design the multivariable PID controller, the output of PID controller is shown as

$\begin{array}{cc}\Delta u\left(t\right)=K_p\left[e\left(t\right)+H_{\tau }{\int }_0^te\left(t\right)\mathrm{dt}+H_d\frac{\mathrm{de}\left(t\right)}{\mathrm{dt}}\right]& \left(12\right)\end{array}$

where Δu(t)=[ΔK_La₅(t), ΔQ_a(t)]^T, ΔK_La₅(t) is the error of the oxygen transfer coefficient in the fifth unit at time t, ΔQ_a(t) is the error of the internal recycle flow rate at time t, K_p is the proportionality coefficient; H_τ is the integral time constants; H_d is the differential time constants; e(t) is the error between the real output and the optimal solution

e(t)=z(t)−y(t)   (13)

where e(t)=[e₁(t), e₂(t)]^T, e₁(t) and e₂(t) are the errors of S_O and S_NO, respectively; z(t)=[z₁(t), z₂(t)]^T, z₁(t) is the optimal set-point concentration of S_O at time t, z₂(t) is the optimal set-point concentration of S_NO at time t, y(t)=[y₁(t), y₂(t)]^T, y₁(t) is the concentration of S_O at time t, y₂(t) is the concentration of S_NO at time t;

(3)-2 the outputs of PID controller are the variation of the manipulated variables oxygen transfer coefficient (ΔK_La) and internal circulation return flow (ΔQ_a);

(4) take ΔK_La and ΔQ_a as the input of the control system of WWTP, and then control S_O and S_NO by the calculated ΔK_La and ΔQ_a, and the outputs of the control system in WWTP are the real concentration of S_O and S_NO.

The Novelties of this Present Disclosure Contain

(1) In this present invention, the multiple time-scale optimization problem is designed based on the dynamic characteristics the operation data of WWTP. DMOPSO algorithm is designed to optimize the multi-time-scale optimization objectives and calculate the optimal solutions.

(2) DMOPSO-based optimal control method is proposed to realize the optimization of the operation performance. The dynamic optimal solutions of S_O and S_NO can be derived and traced by the proposed optimal control method.

Attention: for the convenient description, multi-time-scale optimization objectives, DMOPSO algorithm and multivariable PID controller are used to realize the optimal control of WWTP. The other optimal control methods based on different multi-time-scale optimization objectives, dynamic optimization algorithm and control strategy also belong to the scope of this present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the results of S_O in DMOPSP-based method.

FIG. 2 shows the results of S_NO in DMOPSP-based method.

DETAILED DESCRIPTION

A dynamic multi-objective particle swarm optimization-based optimal control method is designed for wastewater treatment process (WWTP), the steps are:

(1) design the models of performance indices for wastewater treatment process:

{circle around (1)} analysis the dynamic characteristics and the operation data of WWTP, obtain the related process variables of the key performance indices pumping energy (PE), aeration energy (AE) and effluent quality (EQ): influent flow rate (Q_in), dissolved oxygen (S_O), nitrate nitrogen (S_NO), ammonia nitrogen (S_NH), suspended solids (SS);

{circle around (2)} establish the models of the performance indices based on the operation time of S_O and S_NO, the operation time of S_O is thirty minutes, and the operation time of S_NO is two hours, the established models of the performance indices are adjusted per thirty minutes; if the operation time meets the operation time of S_NO, then the models are designed as:

$\begin{array}{cc}\{\begin{array}{c}{f}_1\left(t\right)=\sum _{r=1}^{10}W_{1r}\left(t\right)\times e^{-{x\left(t\right)-c_{1r}\left(t\right)}^2/2{b_{1r}\left(t\right)}^2}+W_1\left(t\right)\\ f_2\left(t\right)=\sum _{r=1}^{10}W_{2r}\left(t\right)\times e^{-{x\left(t\right)-c_{2r}\left(t\right)}^2/2{b_{2r}\left(t\right)}^2}+W_2\left(t\right)\end{array}& \left(1\right)\end{array}$

where f₁(t) is PE model at the tth time, f₂(t) is EQ model at the tth time, e^{−∥x(t)−c1r(t)∥2/2b1r(t)2} and e^{−∥1(t)−c2r(t)∥2/2b2r(t)2} are the rth radial basis function of f₁(t) and f₂(t) at the tth time, r=1, 2, . . . , 10, x(t)=[Q_in(t), S_O(t), S_NO(t), S_NH(t), SS(t)] is the input vector of PE model and EQ model, c_1r(t) and c_2r(t) are the centers of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and the ranges of c_1r(t) and c_2r(t) are [−1, 1] respectively; b_1r(t) and b_2r(t) are the widths of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and b_1r(t)=1.2, b_2r(t)=1.2; W_1r(t) and W_2r(t) are the weights of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and W_1r(t)=2.4, W_2r(t)=1.8; W₁(t) and W₂(t) are the output offsets of the rth radial basis function of f₁(t) and f₂(t) at the tth time, and W₁(t)=0.8, W₂(t)=−0.2; if the operation time meets the time of S_O, then the models are designed as:

$\begin{array}{cc}\{\begin{array}{c}{f}_1\left(t\right)=\sum _{r=1}^{10}W_{1r}\left(t\right)\times e^{-{x\left(t\right)-c_{1r}\left(t\right)}^2/2{b_{1r}\left(t\right)}^2}+W_1\left(t\right)\\ f_2\left(t\right)=\sum _{r=1}^{10}W_{2r}\left(t\right)\times e^{-{x\left(t\right)-c_{2r}\left(t\right)}^2/2{b_{2r}\left(t\right)}^2}+W_2\left(t\right)\\ f_3\left(t\right)=\sum _{r=1}^{10}W_{3r}\left(t\right)\times e^{-{x\left(t\right)-c_{3r}\left(t\right)}^2/2{b_{3r}\left(t\right)}^2}+W_3\left(t\right)\end{array}& \left(2\right)\end{array}$

where f₃(t) is AE model at the tth time, e^{−∥s(t)−c3r(t)∥2/2b3r(t)2} is the rth radial basis function of f₃(t) at the tth time, c_3r(t) is the center of the rth radial basis function of f₃(t) at the tth time, and the range of c_3r(t) is [−1, 1]; b_3r(t) is the widths of the rth radial basis function of f₃(t) at the tth time, and b_3r(t)=0.5, W_3r(t) is the weights of the rth radial basis function of f₃(t) at the tth time, and W_3r=1.4; W₃(t) is the output offset of the rth radial basis function of f₃(t), and W₂(t)=−1.2;

(2) dynamic optimization of the control variables for WWTP

(2)-1 set the maximum iterative numbers of the optimization process T_max, T_max=200;

(2)-2 take the established models of performance indices as optimization objectives;

(2)-3 regard the inputs of the optimization objectives x(t)=[Q_in(t), S_O(t), S_NO(t), S_NH(t), SS(t)] as the position of the particles, calculate values of the optimization objectives, update the personal optimal position (pBest_k,i(t)) and the position and the velocity of the particles, the update process are:

x _k,i(t+1)=x_k,i(t)+v_k,i(t+1)   (3)

v _k,i(t+1)=ω(t)v_k,i(t)+c₁α₁(pBest_k,i(t)−x_k,i(t))+c₂α₂(gBest_k(t)−x_k,i(t))   (4)

where x_k,i(t+1) is the position of the ith particle in the kth iteration at the t+1th time, v_k,i(t+1) is the velocity of the ith particle in the kth iteration at the t+1th time, ω is the inertia weight, ω=0.8, c₁ and c₂ are the learning parameters, c₁=0.4, c₂=0.4, α₁ and α₂ are the uniformly distributed random numbers, α₁=0.2, α₂=0.2, pBest_k,i(t) is the personal optimal position of the ith particle in the kth iteration at the tth time, and gBest_k(t) is the personal optimal position in the kth iteration at the tth time;

(2)-4 design the diversity index and the convergence index based on the Chebyshev distance, diversity index is designed to measure the distribution quality of the non-dominated solutions,

$\begin{array}{cc}U\left(t\right)=\sqrt{\frac{1}{\mathrm{NS}\left(t\right)-1}\sum _{m=1}^{\mathrm{NS}\left(t\right)}{\left(\stackrel{⋓}{D}\left(t\right)-D_m\left(t\right)\right)}^2}& \left(5\right)\end{array}$

where U(t) is the diversity of the optimal solutions at the tth time, m=1, 2, . . . , NS(t), NS(t) is the number of non-dominated solutions at time t, Ď(t) is the average distance of all the Chebyshev distance D_m(t), D_m(t) is the Chebyshev distance of consecutive solutions of the mth solution; and convergence index is developed to obtain the degree of proximity, which are calculated as

$\begin{array}{cc}A\left(t\right)=\frac{1}{\mathrm{NS}\left(t\right)}\sqrt{\sum _{l=1}^{\mathrm{NS}\left(t\right)}d_l\left(t\right)}& \left(6\right)\end{array}$

where A(t) is the convergence of the optimal solutions at the tth time, d_l(t) is the

Chebyshev distance of the lth solution between the kth iteration and the k−1th iteration;

(2)-5 judge the changes of the optimization objectives, if the number of the objectives is changed, return to step (2)-6; otherwise, return to (2)-7;

(2)-6 when the number of the objectives is increased, some particles will be changed to enhance the diversity performance, the update process of the population size is

$\begin{array}{cc}{N}_{k+1}\left(t\right)=\{\begin{array}{cc}{N}_k\left(t\right)& {\alpha }_k\left(t\right)=0\\ N_k\left(t\right)-\left(N_k\left(t\right)-{\mathrm{NS}}_k\left(t\right)\right)·{\alpha }_k\left(t\right)& {\alpha }_k\left(t\right)<0\\ N_k\left(t\right)+{\mathrm{NS}}_k\left(t\right) ·{\alpha }_k\left(t\right)& {\alpha }_k\left(t\right)>0\end{array}& \left(7\right)\end{array}$

where N_k+1(t) and N_k(t) are the population size in the kth iteration and in the k+1th iteration at the tth time respectively, α_k(t) is the gradient of diversity in the kth iteration at the tth time, which is calculated as

$\begin{array}{cc}{\alpha }_k\left(t\right)=\frac{U_k\left(t\right)-U_k\left(t-\varepsilon \right)}{\varepsilon }& \left(8\right)\end{array}$

where ε is the adjusted frequency of population size, and the range of ε is [1, 200]; if the number of the objectives is decreased, some particles will be changed to improve the convergence performance, the update process of the population size is

$\begin{array}{cc}{N}_{k+1}\left(t\right)=\{\begin{array}{cc}{N}_k\left(t\right)& {\beta }_k\left(t\right)=0\\ N_k\left(t\right)+{\mathrm{NS}}_k\left(t\right)·{\beta }_k\left(t\right)& {\beta }_k\left(t\right)<0\\ N_k\left(t\right)-\left(N_k\left(t\right)-{\mathrm{NS}}_k\left(t\right)\right)·{\beta }_k\left(t\right)& {\beta }_k\left(t\right)>0\end{array}& \left(9\right)\end{array}$

where β_k(t) is the gradient of convergence in the kth iteration at the tth time, which is calculated as

$\begin{array}{cc}{\beta }_k\left(t\right)=\frac{A_k\left(t\right)-A_k\left(t-\varepsilon \right)}{\varepsilon }& \left(10\right)\end{array}$

(2)-7 compare pBest_k(t) with the solutions Φ_k−1(t) in the archive, where Φ_k−1(t)=[φ_k−1,1(t), φ_k−1,2(t), . . . , φ_k−1,ι(t)], φ_k−1,ι(t) is the ιth optimal solutions in the k-−1th iteration at the tth time of the archive, the archive Φ_k(t) is updated by the dominated relationship, and the calculation process of the dominated relationship can be shown as

Φ_k(t)=Φ_k−1(t)∪p_k−1(t), if f_h(a_k−i(t))≥f_h(p_k(t)), h=1, 2, 3   (11)

where ∪ is the relationship of combine, if the value of pBest_k-−1(t) is lower than the objective value of a_k−1,ι(t), then the pBest_k−1(t) will be saved in the archive, otherwise, a_k−1,iι(t) will be saved, then gBest_k(t) will be selected from the archive according to the density method;

(2)-8 if the current iteration is greater than the preset T_max, then return to step (2)-9, otherwise, return to step (2)-3;

(2)-9 select a set of global optimal solutions gBest_Tmax(t) from the archive randomly, and gBest_Tmax(t)=[Q_in,Tmax*(t), S_O,Tmax*(t), S_NO,Tmax*(t), S_NH,Tmax*(t), SS_Tmax*(t)], where Q_in,Tmax* (t) is the optimal solution of influent flow rate, S_O,Tmax*(t) is the optimal solution of dissolved oxygen, S_NO,Tmax*(t) is the optimal solution of nitrate nitrogen, S_NH,Tmax*(t) is the optimal solution of ammonia nitrogen, SS_Tmax*(t) is the optimal solution of suspend solid;

(3) tracking control of the optimal solutions in WWTP

(3)-1 design the multivariable PID controller, the output of PID controller is shown as

$\begin{array}{cc}\Delta u\left(t\right)=K_p\left[e\left(t\right)+H_{\tau }{\int }_0^te\left(t\right)\mathrm{dt}+H_d\frac{\mathrm{de}\left(t\right)}{\mathrm{dt}}\right]& \left(12\right)\end{array}$

where Δu(t)=[ΔK_La₅(t), ΔQ_a(t)]^T, ΔK_La₅(t) is the error of the oxygen transfer coefficient in the fifth unit at time t, ΔQ_a(t) is the error of the internal recycle flow rate at time t, K_p is the proportionality coefficient; H_τ is the integral time constants; H_d is the differential time constants; e(t) is the error between the real output and the optimal solution

e(t)=z(t)−y(t)   (13)

where e(t)=[e₁(t), e₂(t)]^T, e₁(t) and e₂(t) are the errors of S_O and S_NO, respectively; z(t)=[z₁(t), z₂(t)]^T, z₁(t) is the optimal set-point concentration of S_O at time t, z₂(t) is the optimal set-point concentration of S_NO at time t, y(t)=[y₁(t), y₂(t)]^T, y₁(t) is the concentration of S_O at time t, y₂(t) is the concentration of S_NO at time t;

(3)-2 the outputs of PID controller are the variation of the manipulated variables oxygen transfer coefficient (ΔK_La) and internal circulation return flow (ΔQ_a);

(4) take ΔK_La and ΔQ_a as the input of the control system of WWTP, and then control S_O and S_NO by the calculated ΔK_La and ΔQ_a, and the outputs of the control system in WWTP are the real concentration of S_O and S_NO.

The outputs of the control system based on DMOPSO-based optimal control method is the concentration of S_O and S_NO. FIG. 1(a) gives S_O values, X axis shows the time, and the unit is day, Y axis is control results of S_O, and the unit is mg/L. FIG. 1(b) gives the control errors of S_O, X axis shows the time, and the unit is day, Y axis is control errors of S_O, and the unit is mg/L. FIG. 2(a) gives the S_NO values, X axis shows the time, and the unit is day, Y axis is control results of S_NO, and the unit is mg/L. FIG. 2(b) gives the control errors of S_NO, X axis shows the time, and the unit is day, Y axis is control errors of S_NO, and the unit is mg/L.

Claims

1. A dynamic multi-objective particle swarm optimization-based optimal control method for a wastewater treatment process (WWTP), comprising: (1) design models of performance indices for the wastewater treatment process:{circle around (1)} analysis dynamic characteristics and operation data of the WWTP, obtain process variables: influent flow rate (Qin) dissolved oxygen (SO), nitrate nitrogen (SNO), ammonia nitrogen (SNH), suspended solids (SS), which are related to the performance indices including pumping energy (PE), aeration energy (AE) and effluent quality (EQ);{circle around (2)} establish models of the performance indices based on the operation time of SO and SNO, the operation time of SO is thirty minutes, and the operation time of SNO is two hours, the established models of the performance indices are adjusted per thirty minutes; if an operation time meets the operation time of SNO only, then the models are designed as: