Automatic Calibration of SWMM Model with Adaptive Genetic Algorithm Xi JIN
Wenyan WU,
School of Civil Engineering & Architecture, Wuhan University of Technology, Wuhan, Wuhan, China
[email protected]
Faculty of Computing, Engineering and Technology Staffordshire University , Stafford, UK
[email protected]
Ying-he JIANG,
Jian-hua JIN
School of Civil Engineering & Architecture Wuhan University of Technology Wuhan, China
[email protected]
School of Civil Engineering & Architecture Wuhan University of Technology Wuhan, China
[email protected]
Abstract—Storm Water Management Model (SWMM) is a popular simulation and management tool for sewer system or storm water management. Since it is a physically based model, the calibration process is necessary before a successful implementation. By separating the calibrated parameters into universal and special styles, the shortcoming of ignore differences among subcatchments’ width is conquered, and solution space is also reduced greatly than the way of regarding all calibrated parameters as special parameters. Using flow rate of pipes as objective values, an objective function of difference between simulated results and objective values is build as the objective function of calibration optimal model. A case sewer network is used to evaluate the proposed calibration method, and by comparison with the calibrated results of calibration optimal model using the all universal calibrated parameter selection concept, the advantages of proposed method were summarized. Keywords-Sewers network; Adaptive genetic algorithm
I.
Hydraulic
models;
II.
CALIBRATION PARAMETERS SELECTION
Theoretically, all parameters that can not be obtained or hardly be obtained by measurements should be regarded as calibrated parameters. For calibration problem, the search space will expand in a geometric series way with the increase amount of calibrated parameters, and the calibration result will be badly affected in the case of large count of calibrated parameters. So the calibration parameters selection is an important job for automatic calibration methods. Janet B. (2008) [8] has evaluated the sensitivity of SWMM calibration parameters, and the result revealed that parameters of conduit manning coefficient (N), percent of impervious area (PI), storage depth of impervious area (SI) and subcatchment width (W) have an obvious impact on simulation result, and other parameters have relatively little effect. So in this paper only these four parameters will be selected as calibrated parameters. And another aspect that makes the SWMM model calibration difficult is the large number of subcatchments and conduits in the simulated sewer network. If assign a special parameter to every subcatchment and conduit as calibrated parameter, it will also meet the high dimensional search space problem. So in the previous calibration methods (Wang 1991 [9]; Franchini 1996 [10]; Liong et al. 1995 [5]), the concept of universal calibrated parameters was used. In this concept the calibrated parameters for subcatchments or conduits were all adjusted by the same percentage of one initial parameter value during the calibration process. But this concept has a distinguish difference with real network, so the automatic calibration methods based on this concept can not achieve an ideal calibrated effect. For the aim of getting balance between calibration effect and search space dimension, both concepts of universal parameters and special parameters will be used in the parameter selecting process. In one sewer system, parameters of PI, SI, N has a relatively small difference, so these parameters will be regarded as universal calibrated parameters. Parameter of W is closely related to subcatchment shape and area, and in one sewer network, the sizes of different subcatchments may distinguish a lot, and this will make the width parameters varied widely. So in the calibration process, the parameters of W will be
Calibration;
INTRODUCTION
SWMM (Storm Water Management Model) is a dynamic rainfall-runoff model developed by The American Environmental Protection Agency, aimed at simulation of quantity and quality problems associated with rainfall runoff from urban areas. Although this model has been used widely in many areas, Calibration of SWMM is still an unsolved problem (Wang & Zhou 2009) [1]. Since manual methods are inefficient, and can not give an ideal calibration result, several automatic calibration methods which based on Markov Chain Monte Carlo (Gallagher & Doherty 2007) [2], Neural Network (Zaghloul & Abukiefa 2001) [3], Expert Systems (Baffaut & Delleur 1989) [4] and Genetic Algorithm (Liong etal. 1995, Benny & James 2002, Fang & Ball 2007) [5-7] have been developed, and have gotten certain achievements. Among these optimal methodologies, genetic algorithm calibration methods have become a active study area in calibration of catchment modeling systems, for the characteristics of simply concept, well-adapted and can deal with the objective and constraint conditions without analysis equation.
Project is supported by the National Major Science and Technology Projects of Water Body Pollution Control and Treatment (2009ZX07316-001)
___________________________________ 978-1-61284-340-7/11/$26.00 ©2011 IEEE
individuals that have larger fitness values are better than the ones with smaller fitness values, and have a higher probability to be selected to the next generation. So some disposal should be done to objective values in order to establish fitness function and make better individual getting a higher fitness value. Here difference of individual’s objective value with max objective value in the current generation is selected as the individual’s fitness value. So the fitness function can be expressed with the equation below:
regarded as special calibrated parameters. According calibration parameter selection method described above, the calibrated parameter set is illustrated in tableĉ. TABLE I.
CALIBRATED PARAMETER SET
Conduit Subcatchment parameters parameters PI (universal parameter)
C f ( x), f ( x) Cmax Fit( f ( x)) max 0, else
SI (universal parameter) N (universal Subcatchment 1
W1 (special parameter)
Subcatchment 2
W2 (special parameter)
……
Wi (special parameter)
parameter)
III.
where؟Fit(f(x)) fitness value; f(x) objective value; Cmax max objective value in the current generation. B. Coding and Decoding
OPTIMAL MODEL FOR PARAMETER CALIBRATION
Binary coding system and real coding system are the most popular coding methods in GAs, since the real coding system has some advantages than binary coding system (Janikow & Michalewicz 1991, Michalewicz 1994, Yoon & Shoemaker 1999) [11-13] and all calibrated parameters are continuous variables of positive real number, so real coding system is selected in the coding operation. For the purpose that (0, 1) random data generation method can be used in the population initialization process, a real number between 0 and 1 is defined as gene value at each gene position corresponding to each calibrated parameter. In order to make the initialization population has an appropriate gene value, and control the evaluation process in a logical search space, an upper and lower boundary was given in the decoding operation, to make sure that the parameter values which transformed from gene value, are in a reasonable zone. Equation (3) is used to transform gene values into parameter values:
Conduit flow is main simulation result of hydraulic model, and it is a parameter that can be obtained relative easily. So in many previous researches, conduit flow was selected as observed values in calibration process. In this paper, the conduit flow is also used as the parameter in calculation of objective function. The equation of objective function is showed as (1): n
m
i 1
j 1
o b˖ j m i n ( (QSij QOij ) 2 )
(1)
Where؟n count of monitored flow values of conduits; m total count of simulation steps; QijS simulated flow values of the ith monitored conduit at the jth simulation step, L/s; QijO observed flow values of the ith monitored conduit at the jth simulation step, L/s. IV.
(2)
Coeffi = LBi + (UBi - LBi) × GVi LBi = initValuei × LBcoeffi
SOLVING METHOD
UBi = initValuei × UBcoeffi
In previous research, the genetic algorithms used in calibration of catchment modeling systems are basically standard genetic algorithm (SGA). Although many studies shows that GAs are suitable for solving catchment modeling system calibration problems, but new characteristics of improved GAs have not been utilized in calibration process. For single objective optimization problem solving oriented GAs, the main improved aspect is self-adaptive of GA parameters such as crossover probability and mutation probability, which resulted in the adaptive genetic algorithm (AGA). Fang & Ball (2007) [7] discussed the relationship between SGA parameters and algorithm behavior, and gave the recommended ranges of crossover and mutation probabilities. But the recommended parameter values were obtained by a certain study case, so the representative is limited. In this paper the AGA was used in calibration optimal model solving process.
Where: Coeffi parameter value corresponding to the ith gene value in chromosome; LBi the lower boundary of the Coeffi; UBi the upper boundary of the Coeffi ; GVi the gene value of the ith position in chromosome; initValuei the initial parameter value (this value is fetched from non-calibrated SWMM model); LBcoeffi the lower coefficient boundary of the parameter value; UBcoeffi the upper coefficient boundary of the parameter value.
TABLE II.
A. Fitness Function
name
The objective function of calibration model is a minimization function, but in the genetic algorithm, the
W1
DECODING ILLUSTRATION
initValue
LBcoeff
UBcoeff
LB
UB
GV
Coeff
150.00
0.50
1.50
75.00
225.00
0.63
169.80
For different optimization problems, determining mutation and crossover probability by plenty of tests are a niggling job, and it is hard to find a fixed universal value for mutation and crossover probabilities to different problems. To solving this problem, an adaptive genetic algorithm was developed by Srinivas & Patnaik (1994) [15]. In this method, individual’s mutation and crossover probabilities will be adjusted by its fitness value, so that the best probabilities will be chosen in the whole evolution process. In this research, the mutation and crossover probabilities of each individual in each generation will be adjusted by equation 4:
A decoding example is showed by tableĊ. This example explained how to transform gene value (value in GV column) into SWMM model parameter value (value in Coeff column). In tableĊ, the values in initValue column were fetched from the initial sewer network model, LBcoeff and UBcoeff were given before evolution process, with the values of initValue, LBcoeff and UBcoeff the values of LB and UB columns can be obtained, and each parameter value can only be modified in the range limited by its LB and UB values in the whole evolution process, for example, subcatchment1’ width (W1) can only be modified between 75.00~225.00. The values of GV column were obtained by population initialization operation at the beginning of evolution or genetic operations when evolution process undergoing, and it is always a float number in the range of (0,1), and then we can get the value of Coeff by (3).
Fmax Fm Fmax F , Fm F avg , F F avg PC 0 Pm 0 PC ˗Pm Fmax Favg Fmax Favg PC 0 , Fm Favg Pm 0 , F Favg
Where: PC ؚPm adjusted value of crossover and mutation
C. Evolution Operation
probabilities; PC0 ؚPm0 initial value of crossover and
Evolution operation is composed by selection operator, crossover operator and mutation operator, Cooperated by the three operators insures that the population will undergo a useful evolution and obtain better individuals. With the development of genetic algorithms, many kinds of operators were developed by researchers, especially of crossover and mutation operators. The selection of operators will have a significant difference to evolution effect. Kusum & Manoj (2007) [14] have done some detailed research between the main crossover and mutation operators, in their research the main operators such as Laplace crossover operator (LX), Heuristic crossover operator (HX), Makinen Periaux Toivanen mutation operator (MPTM), Non-uniform mutation operator (NUM) have been tested by several representative problems and characteristics of each operator have been summarized. With the conclusions summarized by Kusum & Manoj, and the numerical tests of some sewer model examples, a genetic operation composed by tournament selection operator, HX operator and MPTM operator is used. The flow chart of genetic algorithm used in this study is showed in figure 1: Setting initial values of AGA paramters
mutation probabilities; Fmax the fitness value of best individual of current generation; Favg average fitness value of current generation; Fm the larger fitness value of two parent individuals in crossover operation; F fitness value of mutated individual. V.
Heuristic crossover
Decoding operation
Makinen Periaux Toivanen mutation
Solve corresponding hydraulic models of all individuals of current generation by calling swmm5.dll
CASE STUDY
The USER1 example of QUALITY ASSURANCE REPORT of EPASWMM (Lewis A. R. 2005) [16] is used in this case study. In this example the water flow and water level of conduit and junction were supplied. And the parameter values in this model can be regarded as true value for evaluating the final calibration results. Before calibration, the all calibrated parameters were all modified in order to act as initial parameter values of non-calibrated state. The topology of sewer system is shown in Figure 2. There are two kinds of parameter calibration optimal models were built for case study. One is built with the original concept of all universal parameter mode (AUM), the other is built with the concept proposed in this paper (universal and special parameter mode, UASM). AUM model was solved by AGA , and UASM model was solved by AGA and SGA respectively. The comparison between results of these three methods can reveal the advantage of AGA (by comparison between results of UASM model solved by AGA and SGA)
Roulette wheel selection
Population initialization
New generation
Calculate objective value of individuals
Subcatchment1 No
Calculate fitness value of individuals
Algorithm parameters adaption by adaptive strategy
(4)
Meet terminate condition
Subcatchment2 Subcatchment4
Yes Result
Subcatchment6
Conduit7
Subcatchment3 S
Subcatchment7 Conduit3
Subcatchment8
Figure 1. Flow chart of proposed genetic algorithm Figure 2. Topology of sewer system
D. Adaptive Strategy Mutation probability and crossover probability selection have a significant effect to efficiency of genetic algorithms.
and advantage of UASM parameter selection method (by comparison between results of AUM model and UASM model
solved by AGA). The algorithm parameters in three methods are same: population size 200, generations 400, initial crossover probabilities 0.6 and initial mutation probabilities 0.1. VI.
RESULTS AND DISCUSSION
Figure 3 shows the evolution curves of average objective values and best objective values of UASM model solved by SGA and AGA. In this figure it can be seen that evolution curves of AGA showed a better convergence performance than evolution curves of SGA, for example evolution curve of average objective values according to SGA result almost has no obvious evolving trend after 30 generations and shows a nearly random behavior, while the curve according to AGA result kept evolving until nearly 350 generations. The same phenomenon also can be discovered from the comparison between evolution curves of best objective values of these two methods, but is not as clear as curves of average objective values. The advantage of AGA is also evidenced by the generation of global best individual in solving procedure. The best individual in solving procedure of SGA was generated in the 237th generation and its objective value is 3.875, while the best individual in solving procedure of AGA was generated in the 392th generation and its objective value is 0.469, this revealed that with the strategy of adaptive parameter modify, the AGA algorithm search the solution space more thoroughly, and has a better ability in protecting good gene composition and in generation more excellent individuals. The comparison among initial parameter values and their true values and calibrated parameter values of UASM calibration model solved by AGA, are shown in table Ш.
Initial values
Relative error (%)
Conduit parameter
N
0.013
0.013
0.020
0.000
Subcatchment
PI (%)
70.13
70.00
50.00
0.002
parameter
SI (mm)
2.02
2.00
1.50
0.010
50.00
19.573
W2 (m)
172.30
170.00
200.00
1.352
W3 (m)
122.90
126.51
200.00
2.583
W4 (m)
79.22
77.65
100.00
2.021
W5 (m)
172.93
165.56
250.00
4.451
W6 (m)
120.64
127.45
100.00
5.343
W7 (m)
141.81
140.74
100.00
0.760
W8 (m)
104.55
106.26
80.00
1.609
Figure 5 shows the comparisons among simulation results by calibrated models and observed values. By the comparison, it can be seen that the simulation results of calibrated model in UASM have a significant improvement than simulation result of calibrated model in AUM, and the simulation result is almost superposed with the objective values, no matter in summit flow or cumulative flow, the simulation results of calibrated model are more near to the observed values.
TABLE III. COMPARISON OF CALIBRATION RESULTS BETWEEN INITIAL PARAMETER VALUES AND ITS TRUE VALUES True values
37.50
It can be seen from table Ш that the difference between calibrated parameter values and their true values are relatively small, except W1 has an error larger than 10%, other parameters’ errors are all less than 6%, and all parameter values got an improvement contrast with their initial values, especially for N, PI and SI. Since these three parameters are regarded as universal parameters in the calibration model, the modification to these parameters will affect individual fitness value significantly and be more sensitivity to the evolution operations than the special parameters, so these universal parameters will get a relatively more ideal calibration result than the special parameters. This character is shown more clearly in Figure 4. N, PI, SI, W1 and W5 of the best individual in each generation have been shown, and it can be seen that the universal parameters have a more convergence speed than special parameters.
generation in evolve process of two methods
Calibrated values
44.84
Figure 4. Evolution curves of some calibrated parameters
Figure 3. Curves of average fitness values and max fitness values of each
Parameter name
W1 (m)
[7]
[8]
[9]
[10] Figure 5. Comparison between simulate results by original model and calibrated model and objective values [11]
VII. CONCLUSIONS An application of solving calibration model of SWMM with adaptive genetic algorithm was investigated. In this adaptive genetic algorithm a composition of real coding system, tournament selection operator, Heuristic crossover operator, Makinen Periaux Toivanen mutation operator and adaptive strategy were used.
[12] [13]
[14]
By separating the calibrated parameters of SWMM into universal and special styles, the shortcoming of ignore differences among subcatchments’ characteristics is conquered, and solution space is also reduced greatly than the way of regarding all calibrated parameters as special parameters, so the solving efficiency and effect of calibration optimal model have been improved greatly.
[15]
[16]
By introduction of adaptive strategy, the shortcoming of evolution efficiency affected by setting of GA parameters is conquered. By comparing the results solved by SGA and AGA respectively, the advantage and feasibility of AGA are shown and evaluated. The result of AGA shows that AGA has a more efficient and persistent evolution effect. By self adaption of GA parameters, the good genes are more likely to be protected and passed by to next generations, so the algorithm will be prone to search for better individual in the promising space. Testing by a case sewer network, the UASM calibration model coupled with AGA has a relatively high calibration precision. REFERENES [1] [2]
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