Design and optimization of multidimensional control strategy based on optimization algorithm and modeling of wastewater treatment process

Based on the BSM1 platform using multi-objective optimal control, the essence is to optimize two key and contradictory indicators to achieve relative balance: water quality (I_EQ) and total energy consumption (I_OC). I_EQ is expressed as: (13)IEQ= 1T×1 000 ∫t0tf( 2STSS( t )+SCOD( t )+3SNKj( t ) + 10SNO( t )+2SBODs( t ) ) Qe( t )dt

Where STSS,SCOD,SNKj,SNO,SBOD5 and Q_e represent the concentration of suspended solids, chemical oxygen demand, Kjeldahl nitrogen concentration, nitrate nitrogen concentration, 5-day biochemical oxygen demand and clear water discharge, respectively.

The expression for I_oc is: (14)IOC=EA+EP+3EC

where E_A denotes aeration energy consumption, E_p denotes pumping energy consumption, and E_c denotes carbon source cost, which is expressed as: (15)EA=8T×1.8 × 1 000∫t0tf∑i = 15 ViKLai(t)dt (16)Ep=1T×1 000 ∫t0tf(4Qa(t)+8Qr(t)+50Qw(t))dt (17)EC= 400T∫t0tf∑k = 15qEC, jdt

Where T is the sampling period, t₀ and t_f denote the start time and end time respectively, V_i and K_Lai denote the volume and aeration of the ith biochemical reactor respectively, Q_a, Q_r and Q_w denote the internal return flow, the external return flow and the residual sludge flow respectively, and q_EC, j denote the flow of carbon source added to the jth reaction zone.

The optimization problem of wastewater treatment process can be described as: (18)min F(X) = {fOCI(X) ,fEQI(X) } s.t.SNh, e, avg<4 mg/L SNtot, e, avg<18 mg/L SBOD5, e, avg<10 mg/L SCOD, e, avg<100 mg/L STSS, e, avg<30 mg/L

Where f_OCl(X) and f_EQl(X) denote the optimization functions of I_OC and I_EQ, respectively; S_{Nh, e, avg} denotes the average concentration of ammonia nitrogen in the effluent; S_{Ntot, e, avg} denotes the average concentration of total nitrogen in the effluent; X is the decision vector, X = (x₁, x₂), x₁, and x₂ are S_{0, 5} and S_{NO, 2}, respectively; and l_i ≤ x_i ≤ u_i, u_i, and l_i denote the upper and lower bounds of each decision variable, i = 1, 2 respectively. The limitations are based on five water quality parameters given in the nationally established effluent discharge standards, the average concentration of which meets the standards before being allowed to discharge.

The setting values of S_{0, 5} and S_{NO, 2} directly affect S_{Nh, e} and S_{Ntot, e}, and K_Lai and Q_a affect the concentration setting, so choosing the correct setting values can effectively reduce energy consumption and improve the water quality. In this paper, the setpoints of S_{0, 5} and S_{N0, 2} are optimized by using the multi-objective evolutionary algorithm based on decomposition (DPMNMOEA/D) algorithm based on dynamic population multi-neighborhood MOEA/D (DPMNMOEA/D), and the PID controller achieves real-time tracking control of the setpoints by adjusting K_lai and Q_a, so as to achieve the goal of improving water quality and reducing energy consumption.

After the multi-objective evolutionary algorithm for S_{0, 5} and S_{NO, 2} set values for an optimization, the optimization process will produce a series of Pareto solutions, although the entire optimization and control process of water quality to meet the emission standards, but part of the optimization cycle there is an average of the phenomenon of water quality exceeding the standard, so for the cycle of exceeding the standard for the selection of the most suitable for the set values of S_{0, 5} and S_{N0, 2}, to ensure that in order to improve the quality of water on the basis of reducing energy consumption, the specific The process is as follows:

Step1: Take the Pareto solutions obtained from the preference search as S_{0, 5} and S_{N0, 2} set values in turn, substitute them into the prediction model, and keep the solutions that satisfy the constraints on the average effluent quality in the cycle and deposit them into the solution set P_A.

Step2: If P_A is not equal to the empty set, it means that the solutions in the set satisfy the condition of meeting the water quality standard, at this time, the reduction of energy consumption becomes the primary goal, so the preferred solution (X_select) selected from P_A should obtain the lowest energy consumption (f_OCImin).

Step3: If P_A is equal to the empty set, it indicates that all the solutions can not guarantee that the water quality can meet the standard, at this time to improve the water quality has become the main goal, so the selected preference solution should obtain the optimal water quality (f_EQImin).

The method of selecting setting values can prejudge whether the water quality is up to standard or not, and selecting the preferred solutions as S_{0, 5} and S_{N0, 2} setting values according to the water quality up to standard can minimize the energy consumption under the premise of ensuring that the water quality is up to standard. Although this method can improve the effluent water quality, but can not completely avoid the phenomenon of S_{Nh, e} and S_{Ntot, e} peaks exceeding the standard, so it is necessary to design a suppression strategy to deal with the problem of exceeding the standard. In order to verify the advantages and disadvantages of the suppression strategy, on the basis of the evaluation indexes of energy consumption and water quality, the percentage of water quality exceeding time P is selected as the evaluation index, i.e., the ratio of the total water quality exceeding time T_c to the total operation time T_z, and its expression is: (19)P=Tc/Tz

The set of good points was proposed by Hua Luogeng et al. Its basic definition is: Let G_s be the unit cube of a s-dimensional Euclidean space, i.e., x ∈ G_s, X = (x₁, x₂, x₃, ..., x_s), where the point r = (r₁, r₂, ..., r_s) in 0 ≤ x_i ≤ 1, i = 1, 2, ..., S, G_s, such that r ∈ G_s, and the deviation φ(n) of the form Pn(k)={(r1(k),…,rS(k)),k=1,2,…,n} satisfy φ(n) = C(r, ε)n^−1+ε, where C(r, ε) is a constant related only to r, ε(ε > 0), then P_n(k) is said to be the set of good points, and r is the point of goodness [28].

The optimization performance of the algorithm has a certain relationship with the diversity of the initial population, the better the diversity, the faster the algorithm converges and the higher the accuracy. In the basic sparrow algorithm, the initialization of the population adopts the random number initialization method, which is easy to fall into the local optimum due to the uneven distribution of the population and the poor diversity, while the initialization of the population based on the set of good points has an even distribution history and the initial population diversity is better, which can effectively improve the performance of the algorithm. The specific steps of initializing the population based on the good point set: (1) Generate the good point set X = {x₁, x₂, …x_i, …, x_k}(i = 1, 2, …, k), k for the population size; (2) The arbitrary dimensional component in x_i = {x_i1, x_i2, …x_ij, …, x_id} is xij=i×2cos(2πjp) , d for the spatial dimensions; (3) Map the good point set to the search space: x_ij = Lb_j + mod(x_ij, 1) × (Ub_j − Lb_j), where x_ij is the position of the sparrow, and Ub_j, Lb_j represents the upper and lower bounds of the jth dimension.

To verify the superiority of the good point set initialized population, it is used as a comparison with random number initialized population and tent mapping initialized population. The random number initialized population method generates an initial population by randomly generating individuals in the solution space. tent mapping initialized population method first generates a chaotic sequence by using tent mapping, and then generates an initial population by mapping the sequence into the solution space.

In the original discoverer position update strategy, when R₂ < ST, the discoverer is prone to converge to the origin in the late iteration, and the discoverer in turn guides the population in its search, which can easily cause the algorithm to fall into a local optimum. When the individual sparrows do not detect the danger, the original position update formula, which does not utilize the globally optimal position, lacks sufficient and effective communication between the discoverers. The improved formula, which incorporates an optimal bootstrapping strategy, allows optimal position information to be quickly passed between discoverers. Meanwhile, a decreasing inertia weight $w$ is introduced, which makes the algorithm have strong global search capability in the early iteration and strong local exploitation capability in the late iteration. The improved finder position update formula is: (20)Xi′+1={ Xi′⋅w+randn(0,1)⋅|Xi′−Xbest′| if R2<ST Xi′+Q⋅L if R2>ST (21)w=r1−r2⋅(t/MaxIt)2

where r₁ and r₂ are the two weight coefficients, t is the number of iterations, and MaxIt is the maximum number of iterations of the algorithm.

For the vigilant at the edge of the population, i.e., the individual of f_i ≠ f_g, decreasing inertia weight w₂ is used to bring it closer to the optimal position step by step to speed up the convergence of the algorithm; for the vigilant at the center of the population, increasing inertia weight w_i is used to increase the ability of the algorithm to jump out of the local optimum at the later stage of the iteration. Meanwhile, the fitness values f_i and f_w in the multi-objective optimization problem are multi-dimensional vectors, which are unable to calculate the position of the vigilantes, so their updating formulas are improved, and the updating formulas are: (22)Xzt+1={ Xbestt+w2⋅|Xzt−Xbestt| if fi≠fg Xzt+w1⋅|Xzt−Xworstt| if fi=fg (23)w1=r3+r4⋅(t / MaxIt)2 (24)w2=r5−r6⋅(t / MaxIt)2

where r₃, r₄, r₅, r₆ are all weighting coefficients.

The improved position update formulas for both the discoverer and the vigilant introduce a globally optimal position, which will cause the algorithm to fall into a local optimum if that optimal individual is a locally optimal individual due to the gradual convergence to the optimal individual in the later iterations of the algorithm [29]. To address this problem, this paper adopts a perturbation strategy to improve the ability of the algorithm to jump out of the local optimum by transforming the probability P_m and perturbing the optimal position. (25)Pm=1−tMaxIt

From Eq. (25), the transition probability decreases with the number of iterations. When P_m is larger than the generated random number r between [0, 1], then the optimal position is not perturbed, and in the algorithm in the pre iteration P_m is larger, the probability of perturbation is smaller, so that the algorithm in the early stage of the rapid search for the global optimal solution; conversely, the optimal position is perturbed, and in the late stage of the iteration P_m of the algorithm is smaller, the probability of perturbation is larger. The perturbation formula is: (26)ct=Xrand(i1)+Xrand(i2)2−Xbest (27)X′=rc⋅ct+Xbest,rc∈(0,1)

where Xrand(i1) and Xrand(i2) are two random sparrow individual positions, ct is the perturbation term, r_c is a random number between [0, 1], and X′ is the global optimal position after the perturbation.

Population renewal is performed by comparing the relationship between individuals in the previous generation population and individuals in the new population: 1)

If the individuals of the previous generation population dominate the individuals of the new population, replace the positions and fitness values of the corresponding individuals of the previous generation population with the positions and fitness values of the new population.

By this method of updating the population, the individuals with high adaptation can be updated in the next iteration, which not only improves the subgeneration utilization rate and algorithm convergence speed, but also improves the diversity of the population to a certain extent.

The updated population is used as the next-generation population and is non-dominated sorted with the optimal solution set archive to produce a new elite external archive that replaces the previous generation optimal solution set archive in the next generation update. The cycle continues until the termination condition is satisfied and the final elite archive is output.

The parameter information of six test functions selected from the CEC2019 series is shown in Table 1.

The modified sparrow algorithm (MSIMOSSA) is compared with several other optimization algorithms with specific parameter settings as shown in Table 2.

The IGD and rPSP values of MSIMOSSA with MOPSO, MODA, MOALO, MOMVO and MSSA algorithms in the test functions are shown in Table 3 and Table 4. From the test results, it can be seen that the overall IGD mean and variance of the MSIMOSSA algorithm proposed in this paper is better than that of MOEA/D, MODA, MOALO, MOMVO, and MSSA in MMF1, MMF2, MMF3, MMF8, and MMF12, and although there is a slight shortfall in MMF15 the difference is very small, with a difference of only 0.0284~0.0073 Range. Among the rPSP values, MSIMOSSA and MOEA/D show better solution coverage. However, the rPSP metrics of MOMPA are smaller and satisfy most of the test functions when compared to the two. The above analysis shows that the MOMPA algorithm is superior in convergence and diversity.

In order to verify the effectiveness of MOMPA, the algorithm was applied to the BSM1 simulation platform, and the total optimization period was set to be 14d, the performance index optimization period was set to be 2h, and each sampling interval was set to be 15 min.The simulation data were selected as the wastewater data under the sunny, rainy, and stormy days, respectively. The optimization ranges of S_O and S_NO are 0~2.7 mg/L and 0~2 mg/L, respectively. Fig. 12 shows the optimization control curves and error plots of MSIMOSSA algorithm for S_O and S_NO in sewage treatment under sunny day data.

Figure 12.

Optimized control curves and error plots of S_O and S_NO in sunny days

From the figure, it can be seen that under sunny weather, the change process of effluent water quality indexes is relatively smooth.MSIMOSSA algorithm presents good tracking control effect according to the actual operation status of the system, the tracking trajectory and the set value change process basically coincide, and the average value of the overall discharge concentration of the effluent water quality parameter is in the range of effluent discharge indexes of 0~2.7 mg/L, which meets the effluent treatment process The constraints of the wastewater treatment process are met, which verifies the superiority of MSIMOSSA in the optimization of wastewater treatment, and is a prerequisite to further ensure that the energy consumption achieves the best treatment results.