An Overview of the Applications of Particle Swarm in Water Resources Optimization Rosemary Cyriac1 , A. K. Rastogi2 1
M.Tech Student, Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India Professor, Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India {
[email protected],
[email protected]} 2
Abstract Optimization methods have evolved over the years to solve many water resources engineering problems of varying complexity. Today researchers are working on soft computing based meta heuristics for optimization as these are able to overcome several limitations of conventional optimization methods. Particle Swarm is one such swarm intelligence based optimization algorithm which has shown a great potential to solve practical water resources management problems. This paper examines the basic concepts of Particle Swarm Optimization (PSO) and its successful application in the different areas of water resources optimization. Keywords : Water Resources Engineering, Particle Swarm Optimization, Swarm Intelligence
1 Introduction Planning, development and management of water resources falls within the domain of water resources engineering. Freshwater demand for domestic, irrigational, industrial and recreational purposes already exceeds supply in many parts of the world and continues to rise due to rapid urbanisation and population growth. Proper management of the available ground water and surface water resources in all user sectors is of utmost importance for any nation for the best utilization of the available sources of water. One of the areas where this is more important than others is irrigation sector since over 80% of water in India is diverted towards agriculture. An entire spectrum of activities involving reservoir releases, groundwater withdrawals, use of new irrigation techniques call for optimal solutions to obtain maximum benefits from the available water while also meeting all the demands timely. Similarly, to minimize floods and droughts, to ensure water quality considerations and for well field installations water resources management is necessary to meet the competing demands. In this context, the importance of optimization in certain specific areas of water resources is considered in this paper.
2 Optimization Optimization tools are utilized to facilitate optimal decision making in the planning, design and operation of especially large water resources systems. The entire gamut of operations involved with large water resources projects are complex and directly influence the people. The application of optimization techniques is therefore necessary and also challenging in water projects, due to the large number of decision variables involved. This is further demanded by the stochastic nature of the inputs and multiple objectives such as irrigation, hydropower generation, flood control, industrial and drinking water demands which a project has to meet simultaneously. Presently certain specific cases where optimisation practices have been used successfully are considered as follows:
Reservoir planning, design and operation
River water pollution control using optimal operation policy
Regional scale groundwater pollution and utilization management
Identification of unknown groundwater pollution sources
Estimation of unknown aquifer parameters in groundwater flow through inverse modelling
Optimal design of water distribution and waste water systems
Fundamentally, optimization involves systematically choosing solutions from an allowed set of decision variables for maximizing the benefits and minimizing the losses. The conventional numerical optimization methods (viz. linear, non linear and dynamic programming) which were used in the past have limited scope in problems of water resources management where objective functions are often non convex, non linear, not continuous and non differentiable with respect to the decision variables. Nonlinear programming methods have rather slow rate of convergence and often result in local optimal solutions since they depend upon initial estimations of variables, whereas the dynamic programming approach suffers from the curse of dimensionality [20]. Thus the conventional methods which utilize gradients or higher order derivatives of objective functions are not suitable for many real world problems in water resources management. For the last two decades non conventional, metaheuristic techniques have been used successfully for obtaining optimal solutions. Although metaheuristic techniques do not have a rigorous mathematical proof like the conventional numerical methods, they follow a certain logical procedure that allows them to deliver a near global optimum solution.
3 Particle Swarm Optimization Evolutionary Computation is the general term for several computational techniques which are based to some degree on the evolution of biological life in the natural world. Particle swarm optimization (PSO) is an evolutionary computation technique based upon the behaviour of a flock of birds or a school of fish [38]. When a swarm looks for food, the individuals will spread in the environment and move around independently. Each individual has a degree of freedom and randomness in its movements which enables it to find food deposits. Sooner or later, one of them will find something digestible and being social, announce this to its neighbours. These can then approach the source of food too. Like the other evolutionary computation techniques, PSO is a population-based search algorithm and is initialized with a population of random solutions, called particles. Unlike in the other evolutionary computation techniques, each particle in PSO is also associated with a velocity. This velocity connotes an improvement in the solution which gets added to the initially assumed solution to make it move towards the optimum solution. Particles fly through the search space with velocities which are dynamically adjusted according to their historical behaviours. Therefore, the particles have a tendency to fly towards the better and better search area over the course of search process. Since its introduction by Eberhart and Kennedy [4], PSO has attracted considerable attention from the researchers around the world and seen gradual improvements with the passage of time.
3.1 Original PSO Algorithm The basic concept of the PSO can be technically summarized in the following steps: 1. Initialize a population of random solutions on D dimensions in the search space. In the D dimensional search space the ith individual (assumed solution or a particle having a position equal to the assumed solution) of the population can be represented by a D dimensional vector Xi = (xi1,xi2......xid)T (1) 2. Each of the above elements of the assumed solution set is modified in each iteration in a probabilistic manner. The improvement made to each of them in each iteration is referred to as velocity. Thus the velocity (position change or change in solution) of the particle can be represented by another D dimensional vector which is also initialized with some random values. Vi = (vi1,vi2....vid)
(2)
3. For each particle (position or assumed solution) evaluate the desired optimization fitness function in D variables. 4. The best previously obtained fitness value of each particle and the corresponding value of the particle is noted. They are stored in a D dimensional vector pid= (pi1,pi2....pid)T
(3)
5. The best fitness value obtained so far by any particle in the population space is noted and the value of the particle is stored as pgd 6. Each of the initially assumed solutions (particles) is improved upon in each iteration through the following equation. The improvement in solution is denoted by vid (velocity). (4) (5) Where c1 and c2 are positive constants, and rand1and rand2 are two random functions in the range [0, 1], m is the number of iterations; 7. Loop to step 2) until a criterion is met, which is either a sufficiently good fitness or depends upon maximum number of iterations. At the end of n iterations the modified xid for which the best fitness value has been obtained in all these iterations is denoted by pgd (global best) and in the nth iteration the value of xid in the solution set which gives the best fitness value is denoted by pid. Thus in each iteration initially assumed solution is updated with respect to the best fitness value obtained among all the other members of the population set and its own previous best. Like other evolutionary algorithms, PSO algorithms is a population based search algorithm with random initialization, and interactions among population members. However, unlike the other evolutionary algorithms, in PSO, each particle flies through the solution space, and has the ability to remember its previous best position, and survives from generation to generation [10]. A general flowchart for PSO application is shown in fig. 1. Randomly initialize the positions and velocities of each member(particle) of the population
Determine the fitness of each particle If the fitness of each particle is greater than its own previous best fitness upto the current iteration then update particle best pid with current fitness value If the particle fitness is greater than the fitness of all other particle in the current iteration then update global best pgd with the current fitness value
Obtain Optimum Solution
Yes Update particle velocity according to the velocity update equation
Update particle position according to the position update equation
Termination condition satisfied No
Fig. 1 General Flowchart of PSO Algorithm
3.2 Parameters of PSO The first new parameter added into the original PSO algorithm is the inertia weight (Eberhart and Shi 1998a, 1998b).They modified dynamic equation (4) of PSO as: (6) where a new parameter, inertia weight ω is introduced. Equation (5), however remains unchanged. The inertia weight is introduced to balance between the global and local search abilities. The large inertia weight facilitates global search while the small inertia weight facilitates local search. A value of 0.1 – 0.9 is recommended in many of the research papers. The introduction of the inertia weight also eliminates the requirement of carefully setting the maximum velocity Vmax each time the PSO algorithm is used. The Vmax can be simply set to the value of the dynamic range of each variable and the PSO algorithm still performs satisfactorily. Another parameter - constriction coefficient was introduced to accelerate PSO convergence [1][2]. A simplified method of incorporating it appears in Equation (7), where k is a function of c1 and c2 as seen in Equation (8). (7)
k=
where
(8)
= c1 + c2 ,
>4
Mathematically, Equation (6) and (7) are equivalent by setting inertia weight ω to be k, and c1 and c2 meet the condition = c1 + c2 , > 4. The PSO algorithm with the constriction factor can be considered as a special case of the PSO algorithm with inertia weight while the three parameters are connected through Equation (8). As a rule of thumb a better approach is to utilize the PSO with constriction factor while limiting Vmax to Xmax, the dynamic range of each variable on each dimension, or utilize the PSO with inertia weight while selecting ω, c1 and c2 according to Equation (8)[8]. When Clerc’s constriction method is used, is commonly set equal to 4.1 and the constant multiplier k is approximately 0.729. This is equivalent to the PSO with inertia weight when ω ≈ 0.729 and c1 = c2 = 1.49445. Since the search process of a PSO algorithm is nonlinear and complicated, a PSO with well-selected parameter set can have good performance, but much better performance could be obtained if a dynamically changing parameter is well designed. Intuitively, the PSO should favour global search ability at the beginning of PSO while it should favour local search ability at the end of PSO. Shi and Eberhart [5] first introduced a linearly decreasing inertia weight to the PSO over the course of PSO, then they further designed fuzzy systems to nonlinearly change the inertia weight [9][10]. The fuzzy systems have some measurements of the PSO performance as the input and the new inertia weight as the output of the fuzzy systems. In a more recent study, an inertia weight with a random component [0.5 + (rand/2.0)] rather than time decreasing is utilized. This produces a randomly varying number between 0.5 and 1.0, with a mean of 0.75 which is similar to Clerc's constriction factor described above [10].
3.3PSO Variants 3.3.1 Elitist Mutated PSO To accelerate the convergence of PSO, an elitist mutation strategy is adopted. This elitist process replaces the worst particle solutions by the best solution among the swarm, after performing mutation mechanism on the best particle.
This process of random perturbation tries to improve the solution, by maintaining diversity in the population, and explores the new regions in the whole search space. This strategy replaces the position vectors of a predefined number of least ranked particles in the swarm with mutated positions of the global best particle in each iteration. The fitness function of particles is sorted in ascending order and the index number for the respective particles is obtained, then elitist-mutation is performed on the particle having the worst fitness in the swarm. If the mutated value exceeds the bounds, then it is limited to the upper or lower bound. During this elitist-mutation step, the velocity vector of the particle remains unchanged.
3.3.2 Binary PSO The original PSO is designed for the real-value problems. The algorithm is further extended to tackle binary/discrete problems. Kennedy and Eberhart [17] used velocity as a probability to determine whether xid (a bit) will be in one state or another (zero or one). They squashed vid using a logistic function s(v) = 1/(1+exp(-v)) while the velocity is calculated using the same equation as the Equation (3.4a). If a randomly generated number within [0,1] is less than s(vid )then xid is set to be 1, otherwise it is set to be 0. Kennedy and Spears [19] found that this version of binary PSO outperforms several versions of GAs in all tested problems except one
3.3.3 Other Variants Recently some more variants of PSO are developed to enable the application of PSO in certain complex optimization problems. These include Multi Objective PSO implemented with the help of Pareto Front Concept, Hybrid Particle Swarms which include a combination of PSO with other techniques (like Genetic Algorithm, Differential Evolution etc), Bare bones PSO which is proposed to move particles according to a probability distribution rather than through the addition of velocity.
4 Applications of PSO in Water Resources Engineering Researchers have attempted a wide range of problems in water resources engineering using PSO. Certain problems where particle swarm techniques have been successfully applied in water resources are examined as follows:
4.1 Reservoir Planning Design and Operation Reservoir Operation optimization involves determining the optimum amount of water that should be released for flood control, irrigation, hydropower generation, navigation and municipal water supply. Being a complex problem it involves many decision variables, multiple objectives as well as considerable risk and uncertainty [21]. Traditionally, reservoir operation is based on heuristic procedures, embracing rule curves and subjective judgments by the operator. This provides general operation strategies for reservoir releases according to the current reservoir level, hydrological conditions, water demands and the time of the year. Particle Swarm Optimization is successfully applied in the optimal assessment of the relevant decision variables. Kumar and Reddy [20] discussed the implementation of Particle Swarm Optimization in multipurpose reservoir operation. They considered Bhadra reservoir system in India which serves irrigation and hydropower generation. It was required to obtain the optimum releases to the left and right bank canals (utilized for irrigation and hydropower generation) and to the river bed turbine (for hydropower generation). To handle multiple objectives of the problem, a weighted approach was adopted. The objective function dealt with minimizing the annual irrigation deficits and maximizing the annual hydropower generation with greater weightage for minimizing irrigation deficits. The decision variables were the monthly releases that should be made to the left and right bank canal and the river bed turbine in a year. The optimization was carried out under a set of constraints which included mass balance, storage, canal capacity, power production and water quality requirements.
The performance of the standard PSO algorithm was improved by incorporating an Elitist Mutated PSO (EMPSO) in which a certain number of the best performing solutions (elites) were retained with mutation during each successive iteration to increase population diversity and enhance the quality of the population. The results obtained demonstrated that EMPSO consistently performed better than the standard PSO and genetic algorithm techniques. They concluded that EMPSO is yielding better quality solutions with less number of function evaluations.
4.2 Groundwater utilization Groundwater resources form important sources of water supply in places where surface water resources are meager or polluted. Proper groundwater management involves optimal pumping of the available groundwater at minimum cost to meet the demand without exhausting the available groundwater reserves. Gaur et.al. [11] used Analytic Element Method and Particle Swarm Optimization based simulation optimization model for the solution of a groundwater management problem. The AEM-PSO model developed was applied to the Dore river basin, France to solve two groundwater hydraulic management problems: (1) maximum pumping from an aquifer, and (2) minimize the cost to develop the new pumping well system. Discharge as well as location of the pumping wells were taken as the decision variables. The influence of the piping length was examined in the total development cost for new wells. The optimal number of wells was also calculated by applying the model to different sets of wells. The constraints of the problem were identified with the help of water authority, stakeholders and officials which included maximum and minimum discharge limits for the well pumping, minimum allowable groundwater drawdown and water demand. The AEM flow model was developed to facilitate the management model in particular, as in each iteration optimization model calls a simulation model to calculate the values of groundwater heads. The AEM-PSO model was found to be efficient in identifying the optimal location and discharge of the pumping wells. A penalty function approach was used to penalize constraint violations and this was found to be valuable in PSO and also acceptable for groundwater hydraulic management problems.
4.3 Groundwater Pollution Control In many parts of our country and in the world ground water is excessively contaminated due to various anthropogenic and industry related activities. Pollution of groundwater happens due to the leachate from animal and human waste dumped on the land, fertilizer application, industrial effluents and municipal waste dumped into surface water bodies. Pump and Treat (PAT) is an often followed technique for the remediation of contaminated ground water where dissolved organic and inorganic contaminants are pumped out of the aquifer and treated on the surface of the ground. The treated groundwater is either returned to the (pumped) aquifer, or used directly, or discharged elsewhere. The treatment method depends on the physical-chemical properties of the contaminants to be removed. Mategaonkar and Eldho [22] presented a simulation optimization (SO) model for the remediation of contaminated groundwater using a PAT system. They developed a simulation model using Mesh Free Point Collocation Method (PCM) for unconfined groundwater flow and contaminant transport and an optimization model based upon PSO. These models are coupled to get an effective SO model for the groundwater remediation design using pump and treat mechanism. In groundwater pollution remediation using PAT, optimization is aimed at identification of costeffective remediation designs, while satisfying the constraints on total dissolved solids concentration and hydraulic head values at all nodal points. Also, pumping rates at the pumping wells should not be more than a given specified rate. Only minimization of the remediation cost is considered as the objective function in this remediation design. The decision variables were the pumping or injection rates for the wells considered and the purpose of the design process is to identify the best combination of those decision variables. The cost function includes both the capital and operational costs of extraction and treatment. The PCM PSO model is tested for a field unconfined aquifer near Vadodara, Gujarat, India.
4.4 Estimation of unknown aquifer parameters in groundwater flow through inverse modeling In real aquifers it may not be possible to directly determine aquifer properties like transmissivity and storage coefficient through physical observations. The optimal values of these aquifer properties are determined through an iterative procedure by minimizing the difference between the observed and simulated groundwater head values. The simulated head values are obtained using a suitable groundwater flow model employing assumed values of transmissivity and storage coefficient which are modified in each consecutive iteration until the calculated groundwater head matches the observed head value. This procedure is known as Inverse Modeling Jianqing and Hongfei [14] applied the PSO algorithm to the function optimization problem of analyzing pumping test data to estimate aquifer parameters of transmissivity and storage coefficient. The objective function was to minimize the difference between simulated and observed groundwater head values with transmissivity and storage coefficient as the decision variables. The results showed that 1) PSO algorithm may be effectively applied to solve the function optimization problem of analyzing pumping test data in aquifer to estimate transmissivity and storage coefficient, 2) the convergence of PSO algorithm and the computation time are influenced by the number of particles, and that fewer iterations are needed in computation with the larger number of particles and 3) the ranges of initial guessed values of transmissivity may also bring some effect on the convergence of PSO algorithm and the computation time. They found that larger the ranges are, the more number of iterations and longer computation time are needed for a guaranteed convergence of PSO algorithm. Some important contributions in water resources optimization by particle swarm techniques are further summarized in Table 1
Table 1 .Certain Applications of PSO in Water Resources Engineering Author/s
Mattot et.al [24]
Zhou et.al. [40]
Gill et.al [12]
Kumar & Reddy [20]
Location
USA
China
USA
India
Year
Application
Decision variable
Empirical constants Chosen
2006
Pump and treat technique is an extensively used groundwater contamination remediation technique and PSO is used for the cost minimization of a pump and treat optimization problem.
Extraction and Injection Rates of the wells and Number of wells required
2006
Training of Artificial Networks by PSO to classify and predict water quality
Weights of the input and hidden layers of ANN
1) ω = .9-.4 2) No of particles - 80 3) c1 = c2 = 2 4) k is not used5) Termination - 1000 iterations
2006
A Multi Objective PSO (MOPSO) was developed and applied to calibrate the (i) Sacramento soil moisture accounting model which is a a well-known conceptual rainfallrunoff model and (ii) a support vector machine model for soil moisture prediction
Parameters of both the models (16+3)
1) ω(linearly varying) - 0.9 0.01 2) No of particles - (i) 100 (ii) 50 3) c1 = c2 = 0.5
2007
Multipurpose Reservoir Optimization (Determine Optimum allowable Discharge to left, right and river bed turbine in a year) (1) to minimise irrigation Deficits (2) Maximize Hydropower Generation
Allowable discharge for the left bank, right bank and river bed turbine respectively.
1) ω = 1 2) No of particles 200 3) c1 = 1 and c2 = 0.5 4) k = 0.9 5)Termination - no change in fitness value for 50 iterations 6) No of particles on which elitist mutation was done - 20 7) Iteration at which elitist mutation started - 100
Real or Synthetic
Remarks
Synthetic
PSO is used for the first time to solve a pump and treat optimization problem and has been found to give better results than the other optimization techniques used(like GA and SA )
Real
PSO based ANN performed better than BP or GA based ANNs for water quality classification and better than time series modeling in water quality prediction
Real
It was noted that for the models tested MOPSO as much or more better than other calibration approaches reported in literature
Real
Elitist Mutated PSO was used to derive operating policies for the reservoir. It was shown to outperform results obtained by ordinary PSO and GA
Izquierdo et.al [15]
Mathur et al [23]
Jianqing and Hongfei [14]
Gaur et.al. [11]
Mategaonkar and Eldho [22]
Spain
India
China
France
India
The performance of a PSO variant adapted to consider mixed integer continuous optimization is investigated by applying the model to the (i) design of 2 water distribution networks, namely the Hanoi new water distribution network and the New York tunnel water supply system (ii) the design of a waste water network and (iii) the calibration and identification of leaks in a water distribution network Binary particle swarm optimisation (BPSO) algorithm for generating the optimal schedule of irrigation from lateral canals to supply the optimum quantity of water to the crop in the field to meet crop water requirement as per guidelines of FAO
pipe diameters and slopes
1). ω= 0.5 + 1/(2(ln(k) +1)) ;k -iteration no 2). No of particles - (i) 100 (ii) 100 (iii) 300 3). c1 = 3 ; c2 = 2 4). Termination after no of iterations - (i) 200 (ii) 800 (iii) 200
No of decision variables - no of minor canals(21) x no of days (120)
1) ω (linearly varying) - 0.9 0.4 2) No of particles - 200 3) c1 = c2 = 1.5 4) Termination - BPSO model run for 200 iterative steps
2010
Application of PSO for the estimation of aquifer parameters (Transmissivity and Storativity)through inverse modeling from the data of pumping test
Transmissivity and Storativity
1) ωmax = 0.9 , ωmin = 0.1 2) No of particles - 200 -500 3) 3. c1 = c2 = 2 4) Termination - value of objective function ≤ 2×10-4
2011
To Design Optimum Well Pumping Location and Operation (1) To maximize Discharge (2) To minimize Well Pumping Cost
Location(x,y ) and Discharge of pumping wells)
1) ω (linearly varying) = 0.8 -0.4 2) No of particles - 25 3) c1 and c2 - btw 2.0 -1.5 4) k = 0.95 5) Termination - no change in fitness value for 50 iterations
2012
To design a cost effective pump and treat remediation system
Pumping Rate of Wells
1) ω = 1.2 2) No of particles - 100 3) c1 = c2 = 0.5 4) k is not used5) Termination - till end of remediation time
2008
2010
Synthetic
The design of both Water Distribution and wastewater systems, the calibration of a network and the identification and detection of leaks may be adequately addressed by means of the considered PSO variant. The results obtained were compared to those obtained by other methods
Real
BPSO has been used for generating optimal schedule for lateral canals such that (1) Soil Moisture Content is maintained between Field Capacity and Wilting Point (2) all minor canals receive adequate water as per crop water requirement
Real
The value of Storativity and Transmissibility (31.831ft2/min and .0621 respectively) by PSO was found to match well with the results obtained by other conventional optimization methods
Real
AEM -PSO flow Optimization model was successfully used to determine the optimal no, location and discharge of pumping wells. The dependence of various parameters of PSO and the cost function were also examined in detail
Real
Mesh Free Point collocation Method was coupled with PSO to develop a cost effective design for pump and treat remediation system
5 Conclusions and further Scope Particle Swarm Optimization has been successfully used in various complex water resources engineering problems to decide water management policies. Some of the advantages of PSO are as follows: 1. In comparison to other evolutionary algorithms PSO is simpler to understand and implement. 2. The method does not depend on the nature of the function it maximizes or minimizes. Thus approximations made in conventional techniques are avoided. 3. It uses objective function information to guide the search in problem space. Therefore it can easily deal with non differentiable and non convex objective functions. 4. Non Linear Programming solutions are dependent upon the initial estimation of solutions. Therefore different initial estimates of parameters give different suboptimal solutions. PSO method is not affected by the initial searching points, thus ensuring a quality solution with high probability of obtaining the global optimum for any initial solution. 5. In PSO particle movement uses randomness in its search. Hence, it is a kind of stochastic optimization algorithm that can search a complicated and uncertain area. Thus it is more flexible and robust than conventional methods. 6. The convergence is not affected by the inclusion of more constraints. 7. It also has the flexibility to control the balance between the global and local exploration in search space. This property enhances the search capabilities of the PSO technique and yields better quality solutions with fewer function evaluations. 8. The algorithm of PSO, demands fewer adjusted key parameters of the algorithm and its arithmetic process is convenient and programmable. It can be easily implemented, and is computationally inexpensive, since memory and CPU speed requirements are low. PSO has been highly successful and within little more than a decade hundreds of papers have reported successful applications of PSO. As it is a technique of recent origin, the number of applications of PSO in water resources engineering is relatively less and there is still a lot of scope for a wider application of PSO to solve water related problems. Therefore there is a possibility that it may emerge as a powerful optimization tool in water resources research. Some of the possible areas in water resources where further research may be done is as follows
Ground water – utilization management, detection of unknown groundwater pollution sources, contaminant remediation, estimation of unknown aquifer parameters, estimation of water table by geo physical methods, optimization for design of multi layered sorptive systems, management of salt water intrusion in coastal aquifers. The decision variables are specific to the problem under study. It can include the location, number and discharge of pumping wells, unknown aquifer parameters, depth of water table etc Reservoir – planning, design and operation. The decision variables may include the optimum discharge values for each time period such that the all the demands are met. Hydrology – Calibration of hydrological and ecological models , Time Series Modelling, stream flow forecasting, The calibration of various models involve the estimation of the various parameters associated with them. It may not be possible to obtain them from physical observations. Hence optimization methods have a definite advantage. Irrigation – scheduling of irrigation canals, Canal design River Stage forecasting , River Water Quality Control and Prediction
Design of Water Distribution Networks, Calibration and improvement of urban drainage systems, Detection of leaks and its rectification Climate Variability and Change, Calibration of climate models There are efforts by many researchers to develop better variations of PSO to increase population diversity and ensure global convergence of the algorithm. These researches may make it more suitable for large scale complex combinatorial optimization problems.
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