Parameter estimation of two innovative Muskingum models
Majid Niazkar
1and Seied Hosein Afzali
21- PhD. Candidate, Department of Civil and Environmental Engineering, School of Engineering, Shiraz University, Shiraz, Iran.
2- Assistant Professor, Department of Civil and Environmental Engineering, School of Engineering, Shiraz University, Shiraz, Iran.
[email protected] Abstract
Nonlinear Muskingum models improve the accuracy of the lumped hydrologic flood routing methods. In this study, two new improved versions of nonlinear Muskingum model with seven and nine parameters are proposed. The parameters of these models are calibrated using Modified Honey Bee Mating Optimization algorithm. The performances of the new models were compared with fourteen models available in the literature for the most widely used case study in this field. The obtained results show that the new models achieve the best results among the ones considered in the comparison and reduced the best SSQ value of three-parameter Muskingum model by 85%.
Keywords: Flood routing, nonlinear Muskingum models, Storage function, Parameter estimation, Modified Honey Bee Optimization algorithm.
1. INTRODUCTION
Muskingum flood routing model is the most widespread lumped flood routing method. This model utilizes the continuity equation and a hypothetical relation for channel storage. The former equation considers water balance in the channel while the latter one speculates a relation between inflow and outflow magnitudes with the amount of the water in the channel. Although the inflow and outflow in a typical flood varies with time, the storage relation still applies for all inflow and outflow magnitudes in all time intervals of flood period.
This relation comprises some parameters, namely Muskingum parameters, which are basically calibrated based on the historical records. In essence, Muskingum parameters are presumed to capture the characteristics of the channel by exclusive usage of previous flood records. Consequently, the more the storage relation has potential to capture channel characteristics, the more accurate flood routing results can be achieved using Muskingum model. In this regard, various versions of Muskingum models have been proposed which their focus was particularly to improve the storage relation.
The first version of Muskingum model was the linear model. In this model, the channel storage is assumed to be proportional to the weighted inflow and outflow values of the same time interval (Eq. 1). Eq. 1 with the continuity equation (Eq. 2) is recognized as the simplest version of Muskingum model. Since the linear model was appeared to be inadequate for some channels, nonlinear Muskingum models were recommended. Unlike the parameters in the linear Muskingum model, the parameters in the nonlinear models lose their physical interpretation. For instance, the exponent parameter in Gill’s version has no physical meaning. Although the calibration process of Muskingum parameters in the nonlinear versions is more complex than the one in the linear ones, the nonlinear Muskingum models achieved much more precise results in comparison to the linear ones. Therefore, the nonlinear Muskingum models and their modifications were placed in the center of the focus of recent studies.
] ) 1 (
[ t t
t K xI x O
S (1)
t t
t I O
dt
dS (2) where St is channel storage at time t, It and Ot are inflow and outflow magnitudes at time t, respectively, K is storage parameter, x is weighting parameter, and m is exponent parameter.
Various versions of nonlinear Muskingum models were presented, which are summarized in Table 1.
In this table, K1, and K2 are weighting parameters, m1, m2 and m3 are exponent parameters, and β is lateral-
constant value throughout the whole flood period for a specific channel. On the contrary, Muskingum models with variable parameters provide the chance of considering variable parameters during the flood period. The scheme for considering variable parameters in these modified versions are done by considering either a function for Muskingum parameters or different discrete values for different sub-periods. Although the available Muskingum versions have improved the accuracy of the flood routing results, there is still an active field of research in light of searching for more reliable versions in the literature.
Table 1 – Different versions of nonlinear Muskingum model
No. Muskingum Version Year Storage relation
1 Gill [1] 1959 St K[xItm (1x)Otm] 2 Gill [1] 1978 St K[xIt (1x)Ot]m 3 Gavilan and Houck [2] 1985 St K[xItm1 (1x)Otm2] 4 Easa [3] 2013 St K[xItm1 (1x)Otm1]m2 5 Easa et al. [4] 2014 St K[xItm1 (1x)Otm2]m3 6 Vatankhah [5] 2014 St K[x(1)It (1x)Ot]m 7 Karahan et al. [6] 2015 St [xK1Itm1 (1x)K2Otm2]m3 8 Haddad et al. [7] 2015 St [x(K1Itm1)(1x)(K2Otm2)]m3
In this paper, two new Muskingum models with seven and nine constant parameters are proposed.
Since Modified Honey Bee Mating Optimization (MHBMO) algorithm achieved the best results in calibrating the three-parameter version, it was utilized to estimate the parameters of the new Muskingum models for a common case study selected from the literature. Based on the achieved results, these new Muskingum models acceptably reduced the sum of the square of the deviations between the observed and routed outflows (SSQ).
2. N
EWV
ERSIONS OFM
USKINGUM MODELIn this section, two new versions of Muskingum model are proposed:
The first model: The first new Muskingum model uses seven constant parameters in order to relate the storage of the reach to the inflow and outflow values:
3 2 2
1 1 ]
[ tm tm m
t K x I x O
S (3) This model considers not only different exponent parameters but also different weighted parameters for inflow and outflow in the storage relation. In this new version, the storage relation has six constant parameters while the seventh parameter, i.e., θ, is used in modifying inflow in the simulation process. This modification is illustrated in the following equation:
1 mod It (1 )It
I (4) where Imod is the modified inflow at time t.
The second model: The second proposed model utilizes a storage relation with eight constant parameters. The ninth parameter of the second model is also the inflow-modification parameter (θ). In this new model, the weighted inflow parameter is a function of inflow values. Hence, this new Muskingum model has variable inflow-weighted parameter.
4 3 3
2 2
1 1(1 ) ]
[ tm tm tm m
t K x I x I x O
S (5) in which xi for i=1, 2, 3 is the weighted parameter, and mi for i=1, 2, 3, 4 is the exponent parameter.
It should be noted that it was found that the constant-parameter version of Muskingum model with more complex storage function performs better than the variable-parameter versions in previous studies (Karahan 2014). Not only this statement can be tested once again here but also the performance of the proposed versions of Muskingum models can be investigated in comparison with the available versions of Muskingum model listed in Table 1.
3. S
IMULATION PROCEDURE OF THE PROPOSED MODELSIn the routing procedure for the proposed Muskingum models, Euler’s method is employed. The simulation procedure for the introduced models is presented separately as below:
The first model: In order to estimate the parameters of the first proposed Muskingum model, the routing procedure can be conducted as follows:
First, by rearranging Eq. 3, the magnitude of the outflow Ot can be expressed as:
2 1 1 mod 2 3 1 1
2
] )
1 (
[ t m m m
t I
x x K
S
O x (6) Secondly, by combining the corresponding storage relation (Eq. 3) and continuity equation (Eq. 2), the time rate of change of the storage volume can be declared as:
2 1 1 mod 2 3 1 1
2
] )
1 (
[ t m m m
t I
x x K
S I x
t
S
(7) where Δt = time interval.
Finally, the simulation procedure particularly includes the following steps:
Step 1: Determine the seven hydrologic parameters (K, x1, m1, x2, m2, m3, and θ) by implementing the MHBMO algorithm. The MHBMO algorithm randomly choses initial values of the parameters in the defined range.
Step 2: Calculate the initial storage volume, in which the initial outflow is assumed to be the same as the initial inflow. Therefore, the initial storage volume can be obtained by:
3 2 1 2 1 1 1
2 K[x Im x Om ]m
S (8) Step 3: Check the applicability of the selected values for the new proposed Muskingum model, i.e., (K, x1, m1, x2, m2, and m3). This check is particularly conducted in order to avoid producing complex values for the outflow using Eq. 6. According to this check, Eq. 9 should be satisfied to obtain acceptable and feasible outflow values.
) log(
) log(
1 1 m t t
I x
K S
m (9) Step 4: Compute the time rate of change of the channel storage by Eq. 7.
Step 5: Estimate the next accumulated storage using Euler’s method by
t t
t S S
S1 (10) It should be noted that negative values for reach storage is not acceptable and should be penalized in the routing procedure.
Step 6: Calculate the magnitude of the outflow at the next time by Eq. 6.
The 3th step to the 6th step should be repeated for all time intervals.
The second model: In order to estimate the parameters of the second proposed Muskingum model, the routing procedure can be conducted as follows:
First, by rearranging Eq. 4, the magnitude of the outflow Ot can be expressed as:
3 1 2 2 mod 1
mod 3 4 2 1
3
)]
1 ( )
1 (
[ t m m m m
t I x I
x x K
S
O x (11) Secondly, by combining the related storage relation (Eq. 5) and continuity equation (Eq. 2), the time rate of change of the storage volume can be stated as:
3 2 1
1
mod 2 mod 3 2 1
3
)]
1 ( )
1 (
[ I x I
x x K S I x
t
St t m
(12)
Finally, the simulation procedure particularly includes the following steps:
Step 1: Determine the nine hydrologic parameters (K, x1, m1, x2, m2, x3, m3, m4, and θ) by implementing the MHBMO algorithm. The MHBMO algorithm randomly choses initial values of the parameters in the defined bounds.
Step 2: Calculate the initial storage volume, in which the initial outflow is assumed to be the same as the initial inflow. Therefore, the initial storage volume can be obtained by:
3 4 2
1(1 ) ]
[x Im x Im x Om m K
S (13)
Step 3: Check the applicability of the selected values for the second proposed Muskingum model, i.e., (K, x1, m1, x2, m2, x3, m3, and m4). This check is particularly conducted in order to avoid producing complex values for the outflow using Eq. 11. According to this check, the following should be satisfied to obtain acceptable and feasible outflow values.
)) 1
( log(
) log(
2 2 1 1 1 1
m m
t
I x I x
K S
m (13) Step 4: Compute the time rate of change of the channel storage by Eq. 12.
Step 5: Estimate the next accumulated storage using the Euler’s method (Eq. 10).
It should be noted that negative values for reach storage is not acceptable and should be penalized in the routing procedure.
Step 6: Calculate the magnitude of the outflow at the next time by Eq. 11.
The 3th step to the 6th step should be repeated for all time intervals.
3. MHBMO ALGORITHM
The Modified Honey Bee Mating Optimization (MHBMO) algorithm method is a search-based optimization technique which was has been already used for parameter estimation of different versions of Muskingum models [8-10]. The reason why this technique was selected to estimate the parameters of the new proposed Muskingum models is that this algorithm achieved the best SSQ values for Gill’s version in comparison with seventeen techniques available in the literature [8]. The MHBMO algorithm, which simulates the mating process of honey bees, has the following five main stages [8]:
• The algorithm starts with the mating flight, where a queen (best solution) selects drones probabilistically to form the spermatheca. One drone is randomly selected for the creation of broods,
• Producing of new broods (trial solutions),
• Workers conduct local search on new broods (trial solutions),
• Adjustment of workers fitness based on the amount of improvement achieved on broods, and
• Compering between the queen and the best brood and Replacement of weaker queens by fitter broods.
The flowchart of this algorithm is depicted in Figure 1. Furthermore, the comprehensive details of this algorithm and its application in Muskingum parameter estimation are fully presented in Niazkar and Afzali [8-10] for interested readers.
4. PERFORMANCE EVALUATION CRITERIA
The utilized performance evaluation criteria which were adopted for verifying the efficiency of the hybrid MHBMO-GRG algorithm are as followings [8]:
1. Accuracy of Procedure Consideration: Accuracy of the estimation procedure can be measured by both SSQ and SAD which are given respectively as:
N
i
t routed t
observed O O
SSQ
1
)2
( (14)
N
i
t routed t
observed O O
SAD
1
|
| (15) where
O
observedt = observed outflow at time t;O
routedt = routed outflow at time t; and N = total number of time intervals.2. Magnitude of Peak Consideration
The absolute value of the deviations of peak of observed and routed outflows (DPO) is considered as a measure of the accuracy of the amount of peak outflow:
|
|Oobservedpeak Oroutedpeakt
DPO (16) where
O
observedpeak = peak of observed outflow;O
routedpeakt = peak of routed outflow.Figure 1. The flowchart of the MHBMO algorithm
5. APPLICATION AND RESULTS
The performances of the new proposed versions of Muskingum model were demonstrated using an experimental data set [8]. Since this example has already been applied in lots of studies on parameter estimation of the Muskingum models, the applicability of the new model Muskingum model potentially could be compared with most of previous models in literature.
In this research, the utilized objective function is to minimize the residual SSQ between the observed and routed outflows
N
i
t routed t
observed O O
SSQ
1
}2
{
min (17) This example is conventionally used for Muskingum parameter estimation, especially for Gill’s nonlinear model, and is traditionally accepted as a standard problem for this purpose. The optimal values for the first and second proposed versions of Muskingum model are shown in Table 2 and Table 3, respectively.
Furthermore, the inflow and observed and routed outflows for this case study for the first and second Input data
The MHBMO algorithm
Generating a random initial population and sorting based on RMSE
Queen selection and generate Queen Spermatheca matrix
Mating process and generate new broods
Feeding generated broods and queen with royal jelly by workers
Calculate objective functions for the new colony
K < Maximum No. of first step
Use the final queen values as an initial guess for parameters in the second step
Implement the problem in the Excel spreadsheet
Is the best solution better
than the queen?
K=1
K=K+1
Import the train data in the Excel spreadsheet
Is the termination criterion satisfied?
Results Yes No
proposed models are illustrated in Figure 2 and Figure 3. As these two figures depict, the new proposed versions obtain close routed results to the observed ones.
Table 2 – Optimal values of the parameters of the first proposed Muskingum model
K X1 m1 X2 m2 m3 θ
0.0102 0.1111 0.7347 2.6496 0.3631 4.3938 0.0661
Table 3 – Optimal values of the parameters of the second proposed Muskingum model
K X1 m1 X2 m2 X3 m3 m4 θ
2.3584 0.0271 0.7300 0.2359 0.0010 0.7626 0.3661 4.3657 0.0651
Figure 3. Inflow, observed and routed outflows of the case study using the first proposed model
Figure 3. Inflow, observed and routed outflows of the case study using the second
proposed model
In Table 4, the SSQ, SAD and DPO of the proposed models are compared with the available models in the literature. As it is shown in this table, both proposed models achieved the best SSQ and SAD values among all the considered models in this table. In particular, both the first and second models reduced the SSQ value of the best SSQ and SAD of the three-parameter version by 85%. Moreover, these two new models achieve better SSQ and SAD results than the present Muskingum model with both constant and variable parameters. This also indicates that even though the proposed models have more complex storage relations, they perform much better than the ones with simple storage relation with variable parameter in the flood period. The significant reduction of SSQ values for this standard case study demonstrates the new proposed models can increase the accuracy of the flood routing results using Muskingum model.
Table 4 – Comparison of the new proposed Muskingum models for the case study
Method No. of
parameters SSQ SAD DPO
LSM [8] 3 143.6 46.4 1.8
GA [8] 3 38.23 23 0.7
PSO [8] 3 36.89 24.1 0.6
DE [8] 3 36.77 23.46 0.9
BFGS [8] 3 36.768 23.46 0.9
MHBMO [8] 3 36.242 21.88 0.699
Premium solver [3] 4 7.67 10.32 0.31
SFLA-NMS [7] 4 7.54 10.2 0.3
GRG [6] 5 26.78 21.8 0.7
SFLA-NMS [7] 5 5.4 6.6 0.1
Premium solver [4] 5 5.35 - -
MHBMO (this study) 7 4.947 6.33 0.120
SFLA-NMS [7] 7 6.348 7.61 0.05
AGA [11] 8 5.730 8.58 0.10
MHBMO (this study) 9 4.944 6.36 0.128
Premium solver [3] 11 24.904 20.72 0.57
6. CONCLUSIONS
Muskingum model is one of the most utilized hydrological flood routing methods. In favor of achieving more accurate flood routing results using this model, various versions of Muskingum model were presented.
In this research, two new innovative versions of Muskingum model with seven and nine parameters are proposed. The parameters in these models were calibrated using Modified Honey Bee Mating Optimization technique for a case study selected from the literature. The obtained results of these models were compared with fourteen models having three to eleven parameters. The comparison shows that the new proposed models achieve the best SSQ values among all these Muskingum versions. In particular, these models reduce the best SSQ value of the three-parameter Muskingum version by 85%.
11. REFERENCES
1. Gill M.A. (1978),“Flood routing by the Muskingum method,” Journal of Hydrologic Engineering, 36(3), pp. 353–363.
2. Gavilan, G. and Houck M.H. (1985),“Optimal Muskingum river routing”, In: Computer applications in water resources, ASCE, pp. 1294–1302.
3. Easa, S.M. (2013),“New and improved four-parameter non-linear Muskingum model”, Proceeding of ICE-
4. Easa S.M., Barati R., Shahheydari E.J.N. and Barati T. (2014),“Discussion: New and improved four- parameter nonlinear Muskingum model”, Proceeding of ICE-Water Management, 167(10), pp. 612–615.
5. Vatankhah, A.R. (2014),“Discussion of parameter estimation of the nonlinear Muskingum flood-routing model using a hybrid Harmony Search algorithm by Halil Karahan, Gurhan Gurarslan, and Zong Woo Geem”, Journal of Hydrologic Engineering, 19(4), pp. 839–842.
6. Karahan H., Gurarslan G. and Geem Z.W. (2015),“A new nonlinear Muskingum flood routing model incorporating lateral flow”, Engineering Optimization, 47(6), pp. 737–749.
7. Haddad O.B., Hamedi F., Orouji H., Pazoki M. and Lo´aiciga H.A. (2015),“A re-parameterized and improved nonlinear Muskingum model for flood routing,” Water Resources Management, 29(9), pp.
3419–3440.
8. Niazkar M. and Afzali, S.H. (2015),“Assessment of Modified Honey Bee Mating Optimization for parameter estimation of nonlinear Muskingum models,” Journal of Hydrologic Engineering, 20(4), 04014,055.
9. Niazkar M. and Afzali, S.H. (2015),“Optimum design of lined channel sections,”. Water Resources Management, 29(6), pp. 1921–1932.
10. Afzali, S. (2016),“Variable-parameter Muskingum model,” Iranian Journal of Science and Technology, Transactions in Civil Engineeinrg, 40(1), pp. 59–68
11. Zhang, S., Kang, L., Zhou, L., and Guo, X. (2016),“A new modified nonlinear Muskingum model and its parameter estimation using the adaptive genetic algorithm,” Hydrology Research, DOI:
10.2166/nh.2016.185.
