The Flood Simulation of the Modified Muskingum Model with a Variable Exponent Based on the Artificial Rabbit Optimization Algorithm

Abstract

In order to further improve the accuracy of flood routing, this article uses the Variable Exponential Nonlinear Muskingum Model (VEP-NMM), combined with the Artificial Rabbit Optimization (ARO) algorithm for parameter calibration, to construct the ARO-VEP-NMM flood routing model. Taking Wilson’s (1974) flood as an example, the model calculation results were compared and analyzed with the Muskingum model constructed with seven optimization algorithms. At the same time, six measured floods in the Zishui Basin were selected for model applicability testing. The results show that the ARO algorithm exhibits stronger robustness and search ability compared with other optimization algorithms and can better solve the parameter optimization problem of the Muskingum model. The use of the ARO-VEP-NMM model for flood routing accurately reflects the movement patterns of floods. The Nash coefficient of the Wilson section reached 0.9983, and the average Nash coefficient during the flood validation period in the Zishui Basin was 0.9, further verifying the adaptability and feasibility of the ARO-VEP-NMM model in flood routing. The research results can provide certain references and a theoretical basis for improving the accuracy of flood forecasting.

1. Introduction

Flood disasters are currently the most significant natural calamities, causing substantial losses to humanity worldwide. With the impact of global climate change, frequent occurrences of intense rainfall events, tsunamis, and other disasters have led to an increasing number of extreme flooding incidents, highlighting the growing prominence of flood control and disaster reduction [1,2,3,4]. In order to mitigate the impact of flood disasters, using real-time and accurate flood forecasting is an effective and crucial measure. Among these, river flood simulation aims to predict the propagation and evolution of floods within river channels based on upstream inflow forecasts, thereby providing services for flood prediction and flood control scheduling [5,6]. At present, flood simulation is mainly divided into hydraulic methods and hydrological methods. The former, based on the Saint-Venant equations, requires a substantial amount of data and involves lengthy computation times. The latter, relying on principles of mass and momentum conservation, offers relatively simpler models and quicker computations.

The Muskingum model proposed by McCarthy, G.T., is a hydrological method widely applied in flood simulation problems [7,8,9]. It is based on the assumption of a one-dimensional model, mainly considering the flow in the downstream direction to predict the propagation and evolution process of floods in a river channel. Although the Muskingum model performs well in many applications, there are also some assumptions and limitations that need to be considered. Firstly, the Muskingum model is a one-dimensional model, which means that it only focuses on the longitudinal flow of the river during simulation. This results in the model being unable to consider lateral flow in areas such as floodplains and unable to capture the complexity of spatial changes. In some complex river systems, especially those with complex geometric shapes and flow characteristics, the Muskingum model may not perform as expected. However, in reality, rivers are numerous, and with human influences, the shapes of river cross-sections and hydraulic characteristics undergo constant changes, posing challenges to the application of the Muskingum model in flood simulation.

Over the past few decades, researchers have improved the model by introducing additional parameters [10,11,12,13]. These efforts aim to overcome some limitations of the Muskingum model, especially in its application under complex hydraulic conditions. The original model may perform well in some river systems, but as the complexity of the river increases, a model with more degrees of freedom and fitting ability is needed.

Among them, the Variable Exponent Nonlinear Muskingum Model (VEP-NMM) [12], with increased degrees of freedom and better fitting capabilities compared with the original model, has shown enhanced performance under more complex hydraulic conditions. However, in practical applications, the accuracy of the Muskingum model prediction relies on the definition of the model structure and the determination of model parameters. As the number of parameters increases, the estimation of these parameters becomes more complex. Therefore, determining the optimal parameter estimates for the model has become a crucial and challenging aspect of flood simulation.

With the deepening of research, various optimization algorithms are used to precisely calculate the parameters of the Muskingum model [14,15,16,17,18]. However, conventional optimization algorithms often yield similar non-optimal estimates, all falling into local optima. Therefore, the need for more robust algorithms arises to overcome the local optimum dilemma and obtain optimal parameter estimates [19,20,21].

The Artificial Rabbit Optimization algorithm (ARO), proposed by Liying Wang, is a metaheuristic algorithm inspired by the foraging behavior of rabbits [22]. In comparison with other classical algorithms, ARO has demonstrated outstanding performance in optimizing model parameters and handling nonlinear problems, exhibiting stronger global search capabilities and adaptability to complex engineering issues [23,24,25].

Therefore, this study uses the VEP-NMM model, utilizing the ARO algorithm for parameter calibration. The model is solved using various case studies, and comparative analyses with other methods are conducted to validate the model’s advanced and effective performance. The aim is to better simulate the evolution of riverine floods.

2. Materials and Methods

2.1. Variable Exponent Nonlinear Muskingum Model (VEP-NMM)

The Linear Muskingum Model (LMM) proposed by McCarthy, G.T. imposes relatively low requirements on channel roughness, slope, and morphology while still accurately simulating changes in river channel form [8]. The equation for the LMM is as follows:

$$S_t=K[X{I}_t+(1-X)Q_t]$$

(1)

$$\frac{dS}{dt}\approx \frac{\Delta S}{\Delta t}=I-Q$$

(2)

where $I_t$ and $Q_t$ represent the upstream and downstream flows at time t, respectively; $S_t$ is the channel storage; K is the slope of the relationship line between the storage capacity and the indicated storage flow, which has a time dimension and actually represents the propagation time of floods in the river section; and X is the proportion factor of flow, representing the relative weight of upper and lower section flow in the tank storage capacity, reflecting the role of wedge storage in flow calculation.

Research [26,27,28] has demonstrated that when there is a strong nonlinear relationship between channel storage and discharge, the accuracy of the Linear Muskingum Model in flood simulation is not satisfactory, necessitating the treatment of this nonlinear relationship. In this context, Gill [10] proposed the widely applied Nonlinear Muskingum Method (NMM), which introduces a nonlinear parameter m. The equation for NMM is as follows:

$$S_t=K{[X{I}_t+(1-X)Q_t]}^m$$

(3)

As research advanced, Easa et al. [12] suggested that the parameter m should be a continuous variable exponent parameter whose variation is related to the current inflow and the inflow process. Therefore, by introducing the dimensionless inflow variable $u_t$, the Variable Exponent Nonlinear Muskingum Model (VEP-NMM) was proposed, further enhancing the model’s performance, making the simulation results more consistent with natural river channel characteristics, and exhibiting time-varying properties. The specific computational equation is as follows:

$$\begin{array}{l}{m}_t=\beta (u_t)=a+b{e}^{-e^{c{u}_t}}\\ \left\{\begin{array}{l}{u}_t=I_t/I_{max}^{}\\ I_{max}=\underset{1\le t\le T}{max}{I}_t\end{array}\right.\end{array}$$

(4)

where a, b, and c are the model parameters to be optimized and $I_{max}$ is the maximum inflow to the river channel. Then, Equation (3) transforms into:

$$S_t=K{[X{I}_t+(1-X)Q_t]}^{\beta \left(u_t\right)}$$

(5)

After initializing the parameters, the value of $m_t$ for each time period is obtained. Assuming the initial outflow value is equal to the inflow value, Q₀ = I₀, the initial channel storage is then calculated using the following equation:

$$S_0=K{\left[X{I}_0+(1-X)I_0\right]}^{\beta \left(u_0\right)}$$

(6)

The equation for the cumulative channel storage in the next time period is expressed as follows:

$$S_{t+1}=S_t+\Delta t{q}_i$$

(7)

$$q_i=-(\frac{1}{1-X}){\left(\frac{S_i}{K}\right)}^{\frac{1}{\beta \left(u_{i-1}\right)}}+\left(\frac{1}{1-X}\right)I_i$$

(8)

The equation for the calculated outflow is then expressed as follows:

$$Q_{t+1}=\frac{1}{1-X}{\left(\frac{S_{t+1}}{K}\right)}^{1/\beta (u_t+1)}-\frac{X}{1-X}{I}_{t+1}$$

(9)

2.2. Artificial Rabbit Optimization Algorithm (ARO)

Originating from the survival strategies of rabbits in nature, the Artificial Rabbit Optimization Algorithm (ARO) is inspired by two distinct behaviors: the detour foraging strategy and the random hiding strategy, each associated with a transition between these strategies driven by their energy levels [22]. The detour foraging strategy is used by rabbits to avoid predators, involving foraging away from the burrow with the advantage of their broad field of vision, which facilitates the detection of food over a large area. This foraging strategy is considered the exploration phase of the algorithm. On the other hand, the random hiding strategy involves rabbits digging multiple burrows and randomly selecting one as a refuge. Rabbits may also execute sudden turns or abrupt stops while running, confusing predators and evading pursuit. This hiding strategy is seen as the exploitation phase of the algorithm.

Given the necessity for rabbits to run rapidly to escape danger, resulting in energy changes, they adaptively switch between foraging and evading danger by leveraging their energy variations. This adaptive switching enhances their chances of survival by balancing food acquisition and avoiding threats.

The ARO algorithm incorporates these survival traits of rabbits by simulating the transitions between detour foraging and random hiding to solve problems. This simulation corresponds to three behavioral phases: detour foraging, random hiding, and energy transformation. The mathematical model of the algorithm is represented as follows.

2.2.1. Detour Foraging Strategy (Exploration Phase)

In ARO, each rabbit in the population has a burrow area, and rabbits randomly visit the locations of other individuals for foraging. In reality, while foraging, rabbits often disrupt food sources to obtain an adequate amount of food. Therefore, the detour foraging behavior in ARO implies that each searching individual will randomly select another individual in the population to search and update its position, introducing a disturbance. Thus, the mathematical model for detour foraging is expressed as:

$${\overrightarrow{v}}_i\left(t+1\right)={\overrightarrow{x}}_j\left(t\right)+R\cdot \left({\overrightarrow{x}}_i\left(t\right)-{\overrightarrow{x}}_j\left(t\right)\right)+round\left(0.5\cdot \left(0.05+r_1\right)\right)\cdot n_1$$

(10)

$$R=L\times c\Leftarrow \left\{\begin{array}{l}L=\left(e-e^{{\left(\frac{t-1}{T}\right)}^2}\right)\cdot \sin (2\pi r_2)\\ c\left(k\right)=\left\{\begin{array}{l}1, if k==g\left(l\right)\leftarrow g=\mathrm{randperm}(D) \\ 0, else\end{array}\right.\end{array}\right.$$

(11)

where ${\overrightarrow{v}}_i\left(t+1\right)$ is the position of the i-th rabbit at time t + 1; i and j are both numbers from 1 to N, and N is the population size of rabbits; r₁, r₂, and r₃ are random numbers from 0 to 1; n₁ is a random number that follows a standard normal distribution; D is the dimension; T is the maximum number of iterations; t is the current number of iterations; and k = 1...D and l = 1... [r₃ *D].

2.2.2. Random Hiding Strategy (Exploitation Phase)

To evade predators, rabbits typically dig various burrows around their burrows. In each iteration of ARO, each rabbit generates holes along every dimension of the search space and randomly chooses one of these holes to hide in, reducing the likelihood of being preyed upon. The mathematical model for generating burrows is as follows:

$${\overrightarrow{b}}_{i,j}\left(t\right)={\overrightarrow{x}}_i\left(t\right)+H\cdot g\cdot {\overrightarrow{x}}_i\left(t\right), i=1,\dots ,n \mathrm{and} j=1,\dots ,D$$

(12)

$$H=\frac{T-t+1}{T}\cdot r_4$$

(13)

$$g\left(k\right)=\left\{\begin{array}{cc}1,& if k=j\\ 0,& else\end{array}\hspace{1em}k=1,\dots ,D\right.$$

(14)

where ${\overrightarrow{b}}_{i,j}\left(t\right)$ is the j-th cave generated by the i-th rabbit and H is a parameter that gradually decreases linearly with the number of algorithm iterations. Influenced by this parameter, the rabbit’s initial burrow occurs in a larger domain, which decreases as the iteration progresses.

As mentioned above, predators often chase and attack rabbits, so rabbits must find a safe hiding place to survive. The modeling of this random hiding strategy is as follows:

$${\overrightarrow{v}}_i\left(t+1\right)={\overrightarrow{x}}_i\left(t\right)+R\cdot \left(r_4\cdot {\overrightarrow{b}}_{i,r}\left(t\right)-{\overrightarrow{x}}_i\left(t\right)\right),i=1,\dots ,N$$

(15)

$${\overrightarrow{b}}_{i,r}\left(t\right)={\overrightarrow{x}}_i\left(t\right)+H\cdot g_r\cdot {\overrightarrow{x}}_i\left(t\right)$$

(16)

$$g_r\left(k\right)=\left\{\begin{array}{ccc}1,& if k==r_5\cdot D& \\ 0,& else& \end{array}k=1,\dots ,D\right.$$

(17)

where r₄ and r₅ are both random numbers from 0 to 1. The random selection of hidden caves is determined using Equation (17), and the meanings of other parameters are the same as before.

After selecting either the detour foraging strategy or the random hiding strategy, the update of the position for the i-th rabbit is as follows:

$${\overrightarrow{x}}_i\left(t+1\right)=\left\{\begin{array}{l}{\overrightarrow{x}}_i\left(t\right), if f\left({\overrightarrow{x}}_i\left(t\right)\right)\le f\left({\overrightarrow{v}}_i\left(t+1\right)\right)\\ {\overrightarrow{v}}_i\left(\mathrm{t}+1\right), if f\left({\overrightarrow{x}}_i\left(t\right)\right)>f\left({\overrightarrow{v}}_i\left(t+1\right)\right)\end{array}\right.$$

(18)

2.2.3. Energy Factor A

In ARO, rabbits tend to use the detour foraging strategy in the early iterations and switch to the random hiding strategy in the later iterations. This mechanism is related to the energy level within the rabbits. Therefore, factor A is designed to implement the transition from the exploration phase to the exploitation phase. The energy coefficient is defined as follows:

$$A\left(t\right)=4\left(1-\frac{t}{T}\right)\ln \frac{1}{r}$$

(19)

where the energy conversion factor A indicates the rabbit’s endurance and energy for detour foraging when higher and the need for random hiding when lower. In the equation, r is a random number between 0 and 1, and A decreases to zero with the growth in the oscillation amplitude during the iteration process.

In ARO, when A > 1, rabbits tend to randomly explore different areas for detour foraging; when A < 1, rabbits prefer the random hiding strategy. Thus, based on the energy conversion factor A, ARO transitions between detour foraging and random hiding strategies. To demonstrate the balance between exploration and exploitation in the algorithm, the probability of executing the detour foraging strategy is calculated using the following equation:

$$P\{A\left(t\right)>1\}=\frac{{{\displaystyle \int }}_0^2{{\displaystyle \int }}_0^{e^{-\frac{1}{2k}}}drd\theta }{1\cdot 2}=\frac{1}{4}\cdot {{\displaystyle \int }}_{-\infty }^{\frac{1}{4}}\frac{e^t}{t}dt+e^{-\frac{1}{4}}\approx 0.5177$$

(20)

Therefore, during the iteration process, the probability of ARO executing the detour foraging strategy is approximately 0.5. This means that the probabilities of executing detour foraging and random hiding are nearly equal, contributing to balanced exploration and exploitation in the algorithm.

Hence, compared with other optimization algorithms, the ARO algorithm is characterized by its simplicity in principles, fewer parameters, and effective balancing of exploration and exploitation capabilities with the energy transformation factor A. It uses corresponding detour foraging and random hiding strategies, achieving better global exploration and local exploitation to seek optimal global solutions.

2.3. ARO-VEP-NMM Flood Simulation Model

Based on the analysis of the VEP-NMM model in Section 2.1, given the inflow I from upstream and outflow Q from downstream of the river, the ARO algorithm is used to calibrate the model parameters K, X, a, b, and c. After calibration, these parameters are substituted into the model to obtain the simulated flow Q_s from upstream to downstream. The simulated flow is then compared with the observed flow, and if the criteria are not met, parameter calibration continues until satisfactory results are achieved. The workflow of the ARO-VEP-NMM flood simulation model is illustrated in Figure 1.

To achieve a more accurate simulated outflow, the optimization process aims to minimize the sum of squared differences (SSQ) between the observed and simulated outflow. The mathematical model for this optimization objective function is as follows:

$$SSQ={{\displaystyle \sum _{t+1}^T\left(Q(t)-Q_s(t)\right)}}^2$$

(21)

where $Q\left(t\right)$ is the measured flow rate at time t; and the flow rate is calculated at time t for $Q_s\left(t\right)$.

2.4. Model Evaluation Metrics

To assess the reliability of flood simulation results, multiple metrics are used. Tailoring the evaluation to the characteristics of different flood instances enhances the intuitiveness of the results. For case 1, the sum of squared differences (SSQ), the sum of absolute differences (SAD) between the observed and simulated flow, and the Nash coefficient (NSE) are used as evaluation metrics. For case 2, which involves multiple flood events, metrics such as peak flow, the time lag of peak flow, the peak flow relative error (QE), and flood forecasting accuracy based on peak flow and peak time are used. The Nash–Sutcliffe Efficiency (NSE) coefficient is used as an indicator to judge the fitting degree between forecasted and observed data. The mathematical models for each evaluation metric are as follows:

$$SAD={\displaystyle \sum _{t+1}^T\left|Q(t)-Q_s(t)\right|}$$

(22)

$$Q_E=\frac{\left|Q_c(i)-Q_0(i)\right|}{Q_0(i)}\times 100\%$$

(23)

$$NSE=1-\frac{{{\displaystyle \sum _{i=1}^N\left[Q_c\left(i\right)-Q_0\left(i\right)\right]}}^2}{{{\displaystyle \sum _{i=1}^N\left[Q_0\left(i\right)-\overline{Q_0}\right]}}^2}$$

(24)

where $Q_0\left(i\right)$ and $Q_c\left(i\right)$ are the measured and predicted flow data, respectively, and the definitions of other parameters are the same as before.

3. Results and Discussion

The establishment of the ARO-VEP-NMM model ultimately aims to effectively improve accuracy in river flood simulation. Building upon the analyses of the VEP-NMM model and the ARO algorithm in Section 2, an ARO-VEP-NMM flood simulation model is constructed. Two instances are considered for validation: Case 1 uses flood data from the Wilson River (1974) and Case 2 uses recurrent floods in the Zishui River basin. These instances are used to assess the flood simulation capabilities and practical effectiveness of the ARO-VEP-NMM model.

3.1. Case 1—Wilson River (1974)

The Wilson River (1974) data represent a smooth single-peak hydrograph with a pronounced non-linear relationship, making it widely used to validate the effectiveness of various parameter optimization estimation techniques in the Muskingum model [29,30].

To verify the feasibility and effectiveness of the ARO algorithm in parameter calibration for the VEP-NMM model, flood data from the Wilson River segment is chosen as an example for testing. A comprehensive comparison is conducted with a series of advanced optimization algorithms, including the Arithmetic Optimization Algorithm [31] (AOA), the Harris Hawks Optimization Algorithm [32] (HHO), and Aquila Optimize (AO) [33]. In order to thoroughly examine the performance of the ARO-VEP-NMM model, the simulation results of the ARO-VEP-NMM model are compared with a range of optimization algorithms used by past scholars, including the Genetic Algorithm (GA) [34], the Harmony Search Algorithm (HS) [35], Particle Swarm Optimization (PSO) [36], and Differential Evolution (DE) [37]. This aims to validate the adaptability and superiority of the ARO-VEP-NMM model in flood simulation.

The parameter calibration for the ARO-VEP-NMM model and other algorithms is conducted based on the mentioned theory, and the parameter settings for each algorithm are consistent with the references. Given the relatively fewer control parameters for algorithms like ARO, HHO, and AO, and their strong stochastic nature, each model is independently run 10 times to avoid falling into the local optimum dilemma and to seek the optimal simulation results.

Based on the introduction of the ARO-VEP-NMM model in the previous text, the ARO algorithm is used to calibrate the model parameters K, X, a, b, and c, given the inflow I upstream and outflow Q downstream of the Wilson River. After calibration, the parameters are input into the model to obtain the calculated flow rate Qs from upstream to downstream. The calculated flow rate is then evaluated and compared with the measured flow rate. If it does not meet the requirements, parameter calibration is continued until the results meet the evaluation criteria. Using parameter calibration, the optimal parameters for the model are determined as follows: K = 0.7091, X = 0.2750, a = 1.8057, b = 12.1784, and c = 8.9845.

To evaluate the performance of the model, the best value, average value, standard deviation, and solution time of the statistical model after 10 independent runs of the model are compiled as evaluation indicators. The comparative results are presented in

Table 1. Comparison of SSQ results after 10 independent runs of each model.

Index	ARO	AOA	HHO	AO
	20.47	21.46	47.34	77.13
	25.21	35.63	27.69	65.28
	20.47	22.81	43.92	83.04
	20.47	38.50	68.00	122.33
	20.47	21.06	104.86	106.29
	20.53	42.29	52.63	80.51
	23.14	20.55	72.21	47.51
	20.47	24.38	37.48	89.49
	20.47	20.74	153.53	124.75
	20.47	21.02	29.98	36.63
Best	20.47	20.55	27.69	36.63
Average	21.21	26.84	63.76	83.30
Standard deviation	1.63	8.48	39.10	29.13
Time (s)	6.35	2.54	2.97	3.98

. For a more intuitive understanding of the simulation effects of each model, Figure 2 provides box plots and normal distribution plots comparing the SSQ results after 10 runs for each model.

From Table 1, it can be observed that the optimal SSQ values for ARO-VEP-NMM, AOA-VEP-NMM, HHO-VEP-NMM, and AO-VEP-NMM are 20.47, 20.55, 27.69, and 36.63, respectively. Among these, ARO-VEP-NMM yields the optimal computation results, with the average value closely approaching the optimal value. Compared with the other algorithms, it demonstrates superior optimization accuracy and convergence capability. Additionally, the standard deviation of ARO-VEP-NMM is 1.63, indicating a relatively small degree of dispersion and higher stability compared with the other three algorithms. However, it is worth noting that compared with the other algorithms, the ARO algorithm requires a longer solving time.

As depicted in Figure 2, the SSQ values for HHO-VEP-NMM and AO-VEP-NMM fluctuate significantly across multiple runs, exhibiting substantial variations between the highest and lowest values, reflecting lower model stability. AOA-VEP-NMM exhibits relatively greater stability. In contrast, the ARO-VEP-NMM model maintains consistent SSQ values across multiple runs, indicating superior stability. Furthermore, its SSQ values are lower compared to the other three algorithms, highlighting its superior applicability.

Given that the calculations for the Genetic Algorithm (GA), Harmony Search Algorithm (HS), Particle Swarm Algorithm (PSO), and Differential Evolution algorithm (DE) were already conducted in reference [26], this paper directly references the numerical results from that study for a convenient comparison of the performance of the ARO-VEP-NMM model with other methods.

Table 2. Flow results calculated with different methods in the Wilson River (1974) case.

Time Period /(h)	Actual Inflow /(m3/s)	Actual Measured Outflow /(m3/s)	Flood Routing Calculates Flow Rate /(m3/s)
			VEP-NMM				NMM
			ARO	AOA	HHO	AO	DE	GA	HS	PSO
0	22	22	22	22	22	22	22	22	22	22
6	23	21	22.7	22.7	22.9	22.8	22	22	22.7	22
12	35	21	23.7	23.7	24.2	24	22.4	22.4	23.5	22.6
18	71	26	26.5	26.5	26.1	25.9	26.6	26.3	26.1	28.1
24	103	34	33.6	33.6	32.4	32.4	34.5	34.2	35.9	32.2
30	111	44	43.4	43.4	42.5	42.9	44.2	44.2	45.2	45
36	109	55	56.1	56.1	55.7	56.5	56.9	56.9	57	57
42	100	66	67.2	67.2	67.2	68.3	68.1	68.2	67.5	67.5
48	86	75	76.2	76.2	76.2	77.4	77.1	77.1	76.2	75.9
54	71	82	82.3	82.3	82.2	83.4	83.3	83.2	81.1	81.2
60	59	85	84.9	84.8	84.5	85.4	85.9	85.7	4.5	85.6
66	47	84	83.6	83.5	83	83.5	84.5	84.2	83.5	84.2
72	39	80	79.8	79.8	79	79.1	80.6	80.2	80.1	79.6
78	32	73	73.3	73.2	72.5	72.2	73.7	73.3	73.3	73.3
84	28	64	65.4	65.3	64.6	64	65.4	65	65.4	65
90	24	54	55.1	55.2	54.2	53.4	56	55.8	55.7	56.2
96	22	44	44.8	44.9	43.9	43.5	46.7	46.7	46.1	46.5
102	21	36	36.6	36.6	36.2	36	37.8	38	36.6	37.3
108	20	30	29.7	29.7	29.8	29.7	30.5	30.9	29.5	29.7
114	19	25	24.5	24.4	24.6	24.9	25.2	25.7	24.3	24.3
120	19	22	22.9	22.8	23.4	23.4	21.7	22.1	20.8	20.6
126	18	19	19.3	19.1	19.5	19.9	20	20.2	20	19.6
NSE			0.9983	0.9983	0.9978	0.9970	0.9969	0.9970	0.4669	0.9970

and

Table 3. Comparison of parameter calibration results for various methods in the Wilson River (1974) case.

Model	ARO-VEP-NMM	AOA-VEP-NMM	HHO-VEP-NMM	AO-VEP-NMM	DE-NMM	GA-NMM	HS-NMM	PSO-NMM
SSQ	20.47	20.55	27.69	36.63	36.77	38.23	36.78	36.89
SAD	16.44	16.64	18.23	21.55	23.46	23	23.4	24.1

present the outflow process and calibration results for various methods in the Wilson case.

Based on the data in Table 2 and Table 3, it is evident that the ARO-VEP-NMM model outperforms other methods significantly in both SSQ and SAD aspects. Furthermore, the simulation results of ARO-VEP-NMM exhibit the highest accuracy, with the smallest simulation error and the best fit to the observed inflow curve. This outcome further validates the outstanding performance and efficient computational capabilities of the ARO algorithm in parameter calibration.

Additionally, it is noteworthy that the VEP-NMM model’s performance in flood simulation surpasses that of the traditional NMM model. This indicates that the chosen variable exponent parameter model is feasible and effective in enhancing flood simulation accuracy. Therefore, the ARO-VEP-NMM model demonstrates significant advantages in flood simulation and proves to be a viable approach for flood forecasting.

3.2. Case 2—Site Floods in the Zishui River Basin

The Zishui River Basin is located within the territory of Shaoyang City, Hunan Province, China. Due to the complex topography within the river channel and the interference of tributaries, the basin exhibits intricate channel morphology and irregular flood propagation characteristics. To further validate the practical simulation performance of the ARO-VEP-NMM model, the Luo Jiamiao hydrological station is chosen as the outlet section of the basin. Six sets of observed flood data from Tangdukou and Longhui hydrological stations from 2014 to 2015 were selected for flood simulation.

After analyzing the data from the six flood events, they are divided into two different periods. The first four flood events are used for the calibration period, while the last two are reserved for the validation period. The ARO algorithm is used to calibrate the parameters, resulting in the optimal parameter values: K = 1.1039, X = 0.5000, a = 1.2618, b = 0.3487, and c = 0.8914. The simulation results for each flood event are presented in

Table 4. Flood simulation results for Tangdu, Longhui, and Luo Jiamiao River sections.

	Flood Site	Measured Peak Value /(m3/s)	Simulated Peak/(m3/s)	QE /(%)	Peak Time Difference /(h)	NSE	Qualified or Not
Regular rate	20 June 2014	5180	4715.05	8.98	0	0.96	Yes
	27 June 2014	1378.7	1290.26	6.42	−4	0.39	No
	5 July 2014	1860	1761.21	5.31	−2	0.74	Yes
	8 June 2015	3170	3468.67	9.42	−3	0.82	Yes
Average				7.53		0.73
Validation period	10 June 2015	1701	1726.87	1.52	0	0.93	Yes
	2 July 2015	3338	3883.24	16.33	−2	0.87	Yes
Average				8.93		0.9

.

From Table 4, it is evident that for the ARO-VEP-NMM flood simulations at the Luo Jiamiao station, the relative errors in the flood peaks during the calibration period are within an acceptable range, with an average of 7.53%. Additionally, the Nash coefficient averages 0.73 for the calibration period. Despite one flood event (“27 June 2014”) during the calibration period having a peak time difference exceeding two time intervals, failing that particular flood simulation, the overall pass rate for the floods during the calibration period reaches 75%. Furthermore, both flood events during the validation period passed the criteria, with an average relative error of 8.93% and a Nash coefficient average of 0.90. Overall, the flood simulation results are reasonably accurate.

Although only six flood events were selected in this study, these events can reflect the overall characteristics of the basin’s floods, and the simulation results can serve as important references for downstream flood control. The simulation hydrographs for the validation period are shown in Figure 3 and Figure 4.

From Figure 3 and Figure 4, it is evident that both floods during the calibration period are of the single-peak type, and their simulation results are excellent, closely fitting the actual flow curves. The overall trend in flood simulation closely resembles the original Luo Jiamiao flow curve, indicating good simulation performance of the ARO-VEP-NMM model for these two floods. Additionally, the small differences between the simulated and actual peak values further demonstrate the accuracy and adaptability of the model’s flood simulation.

3.3. Parameter Sensitivity Analysis

In order to comprehensively evaluate the performance of the ARO algorithm in the VEP-NMM model, this section describes a parameter sensitivity analysis of the ARO algorithm for each case. Due to the fact that the ARO algorithm itself does not involve additional parameters, the population size N of the algorithm was considered the main parameter for sensitivity analysis when solving the case.

Multiple sets of experiments were conducted on N, and each set of experiments was run independently 10 times.

Table 5. Case 1 parameter sensitivity analysis.

N	Best	Average	Worst	Standard Deviation	Time(s)
30	20.4657	21.2139	25.2026	1.6332	3.17
50	20.4657	20.5513	21.3215	0.2706	4.28
100	20.4657	20.4657	20.4657	6.86 × 10−14	6.35
200	20.4657	20.4657	20.4657	2.23 × 10−14	13.35
300	20.4657	20.4657	20.4657	5.32 × 10−14	15.67

shows the SSQ of Case 1 (Wilson River section), recording the average, best, worst, standard deviation, and solution time of the output results. Meanwhile,

Table 6. Case 2 parameter sensitivity analysis.

	Regular Rate		Validation Period
N	QE	NSE	QE	NSE	Time(s)
30	7.73	0.71	9.12	0.89	8.27
50	7.67	0.71	9.01	0.89	9.98
100	7.53	0.73	8.93	0.90	11.13
200	7.53	0.73	8.93	0.90	14.54
300	7.53	0.73	8.93	0.90	16.63

shows the Q_E, NSE, and solution time of flood events in the Zishui Basin during the regular and validation periods.

From the data in the tables, it can be observed that as N gradually increases, all evaluation indicators improve, indicating that the increase in population size increases the range of the ARO algorithm in finding the optimal parameters. Especially at N = 100, all results reach the optimal level. Further increasing the number of groups at this point does not improve the indicators, but rather leads to a continued increase in the solving time and the consumption of additional computing resources. Therefore, in this article, it is found that selecting the ARO algorithm with a population size of N of 100 can achieve beneficial results. It is worth noting that since the ARO algorithm does not have additional parameters, it does not need to consider the rationality of the proportional parameter values during its own evolution process. This is also one of the important reasons why this article chooses ARO as the tool for finding the optimal parameters.

4. Conclusions

This study aimed to improve the accuracy of river flood simulations by introducing the Variable Exponent Nonlinear Muskingum Model (VEP-NMM). To address the parameter optimization challenge of the VEP-NMM model, this study used the Artificial Rabbit Optimization algorithm (ARO) to construct a variable exponent Nonlinear Muskingum Model based on ARO (ARO-VEP-NMM). The feasibility and adaptability of the ARO-VEP-NMM model were tested using two specific case studies, and the conclusions are as follows:

• Using the VEP-NMM model can enhance flood simulation accuracy, with parameter optimization being a crucial aspect of the model’s flood simulation. By utilizing the ARO algorithm for parameter optimization and testing with the Wilson (1974) River segment data for flood simulation, a comparison of seven different optimization methods revealed that the ARO algorithm provides higher optimization accuracy and better robustness. This offers a more effective method for Muskingum model parameter optimization.
• The ARO-VEP-NMM model, applied to simulate measured flood events in the Zishui River basin, accurately reproduces the movement and characteristics of floods. This further validates the excellent applicability of the ARO-VEP-NMM model in flood simulation.

Therefore, the proposed ARO-VEP-NMM flood simulation method combines the advantages of the VEP-NMM model and the ARO algorithm. The research results indicate that their integration can effectively conduct flood simulation, offering a new approach to flood prediction.

However, the model studied in this article may not have fully taken into account the backwater effect caused by hydraulic structures such as bridges and culverts. The impact of these structures on flood routing may be an important factor, and future research will further explore this aspect. In addition, it was noted in this study that the ARO algorithm may have higher solving time and may consume more computing resources compared with other algorithms. Therefore, efforts will be made to improve the ARO algorithm or consider combining optimization with other techniques (such as machine learning) to optimize its convergence speed while improving the accuracy of the results. This is also a new direction for future research.