The system of ODEs can be written in the following general form [23] :

where, t is the time, is the state variables vector, and is the initial condition. is the boundary condition or the external forcing vector given for the entire simulation period. The first and second order BDFs can be represented considering a variable time-step in a given time-step (i) as [17] [24] :

where i indicates the time-step with non-constant step size, represents the discretized form of Equation (1) when and. In Equations (2) and (3), and are the first and second order BDFs considering non-constant step size at each time-step, respectively, which will be henceforth denoted as for the sake of generality. The conventional Newton-Raphson (NR) method for solving the system of non-linear to find can be formulated as:

where k is the iteration index beginning from 0, is the inverse of Jacobian matrix (Equations (5) and (6)). Due to the fact that inverting the Jacobian matrix is computationally intensive, especially as the size of the matrix grows, it is beneficial if the inverse of the Jacobian matrix can be reused and is recalculated only when necessary. Therefore, in the proposed method, in Equation (4) is a recycled Jacobian matrix evaluated at some previous time.

where n is the size of the ODEs, is the Kronecker Delta, and denote the Jacobian matrix for the first and second order, respectively, where is the K^th element of vector, and similarly, is the I^th element of vector at. In addition, to further speed up the NR convergence, the following initial guess of the state variables is used at the first iteration (initial guess) of the NR [25] :

where.

The adaptive time interval scheme adjusts the size of time-steps based on monitoring the convergence of the NR method [25] [26] . The time-step size keeps growing by the factor of 1 + ρ (ρ ≥ 0) until the NR’s iteration k in Equation (4) remains smaller than a threshold k_max, and this inverse of Jacobian matrix () will be used in Equation (4). The iteration continues until the number of iterations to achieve convergence in the NR method exceeds k_max. At this time, a new Jacobian matrix will be computed and/or the

time-step size will be reduced by the factor of (). Figure 1 depicts a de-

tailed flowchart of this algorithm, in which ρ, γ, and k_max are, respectively, the time-step inflation factor, the time-step depression factor, and the desired maximum iteration number of NR method that can be manipulated for each particular case of ODEs to achieve the minimum computation time.

The proposed algorithm was applied to solve an ASM based on a real, full-scale WWTP. Influent flow and its composition data were collected from nitrification-denitrification reactors at the Blue Plains wastewater treatment plant in Washington, DC over a period of 120 days. This system was modeled as three reactors in a series based on mass balance equation (Equation (8)) with a modified ASM1 reaction network consisting of 11 reactions and 15 constituents, as shown in Table 1 [11] . The total volume of the reactor was 17,500 m³, divided into three tanks in a series, as depicted in Figure 2. Figure 3 shows the influent flow and its ammonia concentration with return activated sludge (RAS) and waste activated sludge (WAS) flow. In addition, an alternative, hypothetical influent was created by applying a moving average filter with a seven-day mask to reduce noise and a high-order frequency to evaluate the effect of temporal fluctuations on the choice of the solver’s adjustable parameters. The time-series vector of flows and concentrations and the temperature (partly shown in Figure 3) represent the in Equation (1). Significant fluctuation can be observed in the input data, especially in the inflow rate, mainly because the Blue Plains WWTP receives combined sewer flow, and storm events can generate unpredictable fluctuations in both rate and composition of influent into the plant.

In ASMs, the activated sludge process is commonly represented as a series of attached, mixed reactors. These reactors hold a series of processes that can occur simultaneously, resulting in changes in the concentration of constituents. Figure 2 depicts a schematic of the activated sludge process of this study in the WWTP. A transient mass balance can be mathematically performed for each state’s variables, consisting of the concentration of each model constituent, resulting in a system of ODEs, as expressed in Equation (8). Therefore, Equation (8) expresses the changes in constituents’ concentrations over time as controlled by influents, effluents, reactions, and various mass-transfer processes [9] [23] :

where V (L³), C (M∙L⁻¹), Q (L³∙T⁻¹), and ω indicate the volume, concentration, flow, and flow-fraction factor (determining how much of the inflow and return flow enters each tank in a step-feed system), respectively, H is the Heaviside (unit step) function, R (M∙T⁻¹) is the reaction rate (Table 1), ∅ is the stoichiometric coefficient (Table 1), F (T⁻¹) is the mass transfer rate, (M∙L⁻¹) is the saturated concentration, and (M∙L⁻¹) is the external mass flow rate. And as for the indices, j indicates a constituent, m represents a tank, “ret” indicates a return flow, “in” shows conditions in influent, and r indicates the reaction. In addition, N_m is the total number of tanks, and N_r is the total number of reactions. is the flow rate from tank m' to tank m, due either to sequential stages or through feedback or bypass (a positive value means to flow into tank m). In Equation (8), the term on the left side is the rate of change in the total mass of the constituents j in tank m. The first term on the right side of Equation (8) is the mass inflow due to return flow; the second term is the mass inflow of the influent; the third term represents inflow from the preceding stage or feedback flow if it is present; the fourth term is the outflow of constituents; the fifth term is the production or disappearance of constituents due to reactions; the sixth term is the effect of rate-limited mass transfer (for example, aeration); and the last term is the direct addition of the constituents (for example, the addition of a carbon source for denitrification). To obtain solid-particle concentrations in the return flow (), a dynamic clarifier model [27] or a quasi-steady-state approximation (that is, performing mass balance while ignoring the solid storage changes in the clarifier) can be applied. The study described in this paper used the former approach.

To get the minimum computation time (running time) and acceptable accuracy, the user can assign the three adjustable parameters, including the inflation growth rate of the time-step (ρ), the depression rate of the time-step (γ), and the maximum NR’s iteration (k_max). A small initial time-step of h₀ = 10⁻⁵ day is chosen for all the cases. First, the effect of changing ρ and γ is investigated by varying ρ between 0.0001 and 0.1 and γ between 0.0001 and 1 at the fixed k_max of 4. Figure 5 shows the results for effect of ρ and γ on running time. In both scenarios, filtered and unfiltered, the running time is very sensitive to ρ, and the minimum running time occurs at ρ = 0.01. Figure 5 also shows that the sensitivity of the algorithm to γ is lower than to ρ in both the filtered and

unfiltered scenarios. For example, at ρ = 0.01, in the case of unfiltered influent, γ = 0.01 gives a minimal running time and γ = 0.1 gives a similar running time, showing the low sensitivity of the algorithm to γ. The effect of the maximum number of iteration k_max is examined by assigning the values of 4, 10, 40, and 100. Figure 6 shows the effect of k_max on the running time for two sets of inflation-depression parameters. In both scenarios, the algorithm shows a reasonably high sensitivity to k_max, and a minimum running time occurs at k_max = 10. In conclusion, for the selected system of ODEs, the minimum computation time is achieved at ρ = 0.01, γ = 0.01, and k_max = 10.

For a given set of ρ, γ, and k_max, the run-time is always smaller for the filtered than for the unfiltered, as expected, because the algorithm needs smaller time- steps to resolve the higher disturbance in the unfiltered. Nevertheless, the difference in the running time is not significant.

The adaptive time-step algorithm increases the size of the time-step in the case when the NR algorithm continues converging and decreases the size of the time-step when the NR fails to converge. This pattern can be seen in Figure 7, which shows the variation of the accepted time-steps during the course of the simulation for four sets of adjustable parameters. The results show that the pattern of variation in the time-step is substantially affected by different sets of adjustable parameters. For example, as Figure 7 shows, at k_max = 10, the time-step can reach a maximum value of 0.35 days, whereas, at k_max = 4, the maximum value is 0.14 days. In other words, a higher k_max allows larger time-steps and further reuse of the Jacobian matrix at the price of a higher number of NR iterations at each time-step. This indicates that the optimal value of k_max depends on

the effort needed in the Jacobian inverting process relative to the computational cost of NR iterations, which are mainly matrix-vector multiplications. Therefore, the optimal k_max can also depend on the size of the Jacobian matrix (that is, the number of state variables) and how well-posed the Jacobian matrix is.

As Figure 7 shows, at any given k_max, when γ is varied from 0.1 to 0.01, the pattern of the time-step variation differs significantly and shows more fluctuation at a lower γ. This indicates that the solver parameters ρ and γ essentially control how much the time-step is increased or decreased. A smaller ρ and γ lead to more conservative increases or decreases of the time-step and a larger ρ and γ lead to less conservative increases or decreases.

One way to evaluate the accuracy of the method is to compare the solution with the exact solution. However, the system of ODEs represented in Equation (1) cannot be solved using analytical mathematical methods to find the exact solution so that the

numerical error of the proposed adaptive BDF method can be evaluated. For this reason, to evaluate the accuracy of the method, the well-known Runge-Kutta 4^th order (RK4) method is used to solve the presented system of ODEs with a small constant time-step size. The selected time-step size is small enough that the truncation error remains below 10⁻⁸ [26] . Then, the numerical results obtained from the RK4 method are used as a reference to assess the maximum and average error of the proposed adaptive BDF algorithm.

Since the oxygen mass transfer rate in Tank 1 is set at = 190 day⁻¹ (Equation (8)), resulting in the highest order of stiffness in the system of ODEs, the dissolved oxygen concentration in Tank 1 is susceptible to showing higher sensitivity to the applied numerical method. Therefore, relative error is calculated based on the following equation:

where and show oxygen concentrations at Tank 1, obtained by the proposed algorithm and RK4, respectively. Table 2 lists the relative maximum (e_max) and average (e_avg) of for four sets of adjustable parameters. The accuracy assessment results show acceptable accuracy for the proposed algorithm. Specifically, for the four cases, the maximum and average errors contain less than 2 and 0.3 percent error, respectively.