Tank simulation for the Optimization of Water Distribution Networks L. S. Vamvakeridou-Lyroudia1, D. A. Savic2, and G. A. Walters3 Abstract In this paper an original approach to the simulation of floating-on-the-system tanks as decision variables for water distribution system design optimization is presented, aiming to bridge the gap between traditional engineering practice and mathematical considerations needed for genetic algorithms. The paper includes a systematic and detailed critical overview of various mathematical approaches in literature, as well as a novel, more “engineering oriented” approach to the simulation of tanks as decision variables for water distribution system design optimization, describing in detail assumptions and impacts to the evaluation of potential solutions. Tank simulation is based on two decision variables: capacity and minimum normal operational level, omitting risers. Shape and ratio between emergency/total capacities are taken into consideration as design parameters. Assessment of tank performance is carried out by four criteria for the normal daily operational cycle, differentiating between operational and filling capacity, as well as two further criteria for emergency flows. The original design and operational mathematical assumptions are implemented in a fuzzy multiobjective GA model, which is applied to the well-known example from literature “Anytown” water distribution network to benchmark the results.
Subject Heading List to the CEDB fuzzy sets, optimization models, multiple objective analysis, design, water distribution, pipe networks, water tanks
1
Senior research fellow, School of Engineering, University of Exeter, Exeter EX4 4QF, U.K.(corresponding author) E-mail:
[email protected] 2 Professor, School of Engineering, University of Exeter, Exeter EX4 4QF, U.K. E-mail:
[email protected] 3 Professor, School of Engineering, University of Exeter, Exeter EX4 4QF, U.K. E-mail:
[email protected]
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Introduction Within urban water distribution networks floating-on-the-system tanks are generally used in order to provide (a) energy head (pressure) regulation through their water level, and (b) storage capacity through their volume, i.e. filling with water at off peak periods, returning it to the system when needed, smoothing the peaks and fulfilling emergency demands, such as fire flows.
Large tanks, with
considerable volume, can perform both of these functions. Small tanks, with no considerable storage volume, where the rate of inflow is (almost) equal to the rate of outflow at all times, may be used solely for head regulating purposes (i.e. pressure zoning). Speaking strictly in terms of costs, tanks represent quite a small portion of the whole network costs, as opposed to the pipes’ considerable construction and/or replacement costs, or the pumps’ operational costs. Nevertheless their impact on the overall network performance is significant, disproportionate to their costs: An ill-placed tank may significantly increase pipe design costs for the network as a system, or cause exceedingly high operational costs, affecting pump design, while at the same time reducing quality performance indices, such as reliability and resilience. Mathematical simulation of tanks as components and decision variables for a water distribution design optimization model is hard and complex. Apart from (one or more) peak loadings used for meeting minimum pressure requirements, operational aspects of the network have also to be taken into account, in order to ensure tank operation for an extended time period, e.g. a daily cycle. Moreover tanks have to be filled at off peak periods using pumps, thus adding pump decision variables alongside tank variables. Up to the late 1980s, prior to the development of genetic algorithms, selection of tank location as part of network optimization models was based on engineering experience alone. A new tank would be introduced at a certain location and level by choice (manual selection - Step 1), followed by pipe optimization (mathematical model - Step 2), while the necessary volume would be decided by extended period simulation (manual check - Step 3) (Walski et al 1987). Obviously such approach permitted only a limited number of trials for tank location/level/storage selection, all classic optimization methods being unable to include tanks as independent decision variables within the algorithm. Genetic algorithms (GA), nowadays widely used for water network design optimization, enabled researchers to include tanks and storage as decision variables in the models, by assessing each solution as
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a whole, and thus relieving from the effort of developing analytic mathematical formulas and derivatives for each design variable or hydraulic parameter-the main setback of classic optimization methods (Savic and Walters, 1997). On the other hand, GA in general, require a very large number of potential network designs (solutions) to be assessed, by that meaning that hydraulic analysis of the network, a time consuming procedure, must be carried out for each solution and loading separately. If tanks are to be simulated properly within the GA process, detailed extended period simulation with a small time step should be performed, e.g. the network solved every 15 min of the daily operational cycle. However, calling repeatedly a network solver for hundreds of thousands of times for each generation, is excessively time consuming for any model. Therefore all GA models contain simplifying assumptions as to the way tank performance is simulated, calculated and graded, whereas literature concerning tank and storage mathematical simulation within water design optimization models is very limited. Murphy et al (1994) included tank location and tank storage (volume) decision variables to a singleobjective GA model. In this model tank locations (tank nodes) and the fractions of additional storage needed at each tank site are independent decision variables, determined by the GA. Simplifying assumptions include tanks filling and draining simultaneously, using a 6h time step for daily operation simulation, which in turn determines tank water levels (high and low achievable levels). These water levels are compared to predicted operational levels defined by the tank volume and elevation, through mass balance procedure. Any mismatches are treated as penalties to the objective function, while emergency volumes are added by manual checking after the GA optimization process. Another similar approach with single objective function and penalties has been introduced with the use of simulated annealing, a different heuristic method, where, tank emergency volume checks have also been included in the penalties function (Oliveira Sousa, 2005). Walters et al (1999) developed a network design optimization model including location, volume and water levels as decision variables for tanks, using structured messy GA and multiobjective optimization. Two different approaches have been used for tank variables. The first approach, is similar to Murphy et al (1994), while the second, aiming at more detailed mathematical simulation, introduces five independent decision variables for each new tank: location, volume, percentage of the maximum possible depth for each tank, bottom level of the tank and diameter of the riser (the short pipe linking the nearest network node to the tank), while tank level mismatches (tank operating level differences) are included in a
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benefits objective function, and not directly as penalties to the costs objective function. Demand balancing inflows and outflows are computed at each tank node, while volume changes and the corresponding tank water levels are computed through a 6-hours time step simulation (4 daily loadings) assuming that all tanks fill or drain simultaneously. Additionally the system total emergency volume is calculated and compared to the network emergency volume available, thus introducing another component in the benefits function. In the previously described approaches, all tanks are assumed to fill and drain simultaneously, based on tank “demand” nodes, defined through the system total inflow/outflow balance, a condition rarely occurring in real networks. Moreover emergency volume (if included in the design) is checked only for the network as a whole, assuming that all tanks fully co-operate for fire flows. Both these assumptions seldom happen in real networks, especially in water distribution network expansion and rehabilitation, where new tanks, selected by whatever method, should be integrated with previously existing, sometimes ill-fitted tanks. Additionally tank shape is not taken into consideration; the GA may produce extremely high or flat tanks, while secondary factors, such as riser diameters are introduced as independent decisions, thus increasing the number of decision variables and the search space, without any significant impact to the objective function values. However in engineering practice, the use of such models is almost non-existent (Walski 2001); the engineers still prefer the traditional 3-step procedure –if they apply network optimization at all. Obviously there exists a gap between engineering and mathematical approach, as far as water distribution network design optimization is concerned. Oddly this gap widens as GA models become more sophisticated and complex. Researchers, increasingly attracted by the undeniably huge potential of GA of “detecting” optimality in tough (mathematical) problems, often distance themselves from traditional engineering thinking: it does not matter much for a “GA expert”, if , for instance, some decision variables are quasi-constant, and could be (from the engineering point of view) omitted. This is partly due to the fact that nowadays “GA experts” who develop the (sophisticated) models are usually more experienced in mathematics and computer programming than engineering practice, so that subtle differences to the impact of various mathematical assumptions elude them. This paper aims at bridging the gap between hydraulic engineering practice and water distribution network optimization models, as far as tanks and storage simulation is concerned. It includes a systematic
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and detailed critical overview of various mathematical approaches in literature, as well as a novel, more “engineering oriented” approach to the simulation of tanks as decision variables for water distribution system design optimization, describing in detail assumptions and impacts to the evaluation of potential solutions. The original design and operational assumptions are implemented in a multiobjective GA model, using fuzzy reasoning for benefit/quality evaluation. Some of the ideas presented in this paper originate from the problems encountered during the application of an initial single objective fuzzy model for the simulation of small tanks for a real network (Vamvakeridou-Lyroudia 2003). The model presented in this paper is, however, applied to the wellknown example from literature “Anytown” water distribution network (Walski et al 1987, Murphy et al 1994, Walters et al 1999), to benchmark the results. Although the application of a model to a benchmark network may not offer the chance of facing “real” problems, it should be considered as the first step for demonstrating the efficiency of this new approach.
Multiobjective optimization of water distribution networks When genetic algorithms (GA) are applied for water network multiobjective optimization, the model in general has two objective functions: For any solution defined through the decision variable set (or string) X={x1, x2, … xn}, the first objective function C(X) stands for the sum of all costs, while the second objective function B(X) represents the benefit/quality/acceptance of the solution. The optimization algorithm takes into consideration both objective functions, aiming at minimizing costs and maximizing benefits: min C(X) and
max B(X)
(1)
No additional constraints are needed, because constraint violation is taken into account through the second objective function B(X). Indeed, any constraint violation affects (reduces) the solution’s benefit/quality function value and results in lower “acceptance” or “quality” of solution X. Multiobjective optimization using a genetic algorithm, results in a trade-off curve consisting of non-inferior cost-benefit points {C(X),B(X)}, which evolve within the GA, as generations proceed (Michalewicz 1996). It is based on elitist Pareto optimality ranking (Deb et al 2002) but adapted as to some details: Roulette wheel selection is applied, each string being selected for reproduction with probability 1/r, where r is its respective Pareto rank, while in case a string is selected twice or more, it is only retained once, so as to avoid overdominance of the population by a single string. The selected population undergoes two-point
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binary crossover (probability 0.8) and binary mutation allowing the random change of any sub-string binary digit from 0 to 1, or vice versa, with probability 0.05. All runs have been performed with population size 50. Selection of population size, genetic operators and probabilities is based on previous work by the same authors, where various alternatives for these parameters have been tried (Vamvakeridou-Lyroudia et al 2005) The quality/benefit objective function aims at evaluating the performance of the system for any solution. There can be several approaches to it: The benefit from solution X can be determined as the difference between the predicted deficiencies of the initial unimproved system and those of solution X, estimated as a scaled and weighted sum of deficiency measures. Deficiency measures usually include hydraulic parameters, such as pressure and storage shortfalls (Walters et al, 1999, Prasad and Park, 2004), although other measures of benefit, such as reliability can be taken into account (Farmani et al, 2005). The multiobjective GA applied in this paper uses fuzzy reasoning for assessing the benefit/quality objective function (Vamvakeridou-Lyroudia et al, 2005). In order to achieve this, a (theoretically unlimited) number of criteria are taken into account. Criteria refer to various network components and performance indicators (i.e. pressure, storage, pump capacity etc). For each criterion and network component involved, the performance of the system is assessed independently, by estimating the partial membership function of solution (Xi) to the set of feasible solutions for the specific criterion. The overall performance or benefit/quality of every solution X is evaluated, by aggregating (combining) the partial membership functions, with the use of different relative weights, because of the variation in accuracy and importance assigned to each criterion. The fuzzy benefit/quality function takes values in the range (0,1]. The highest value 1 refers to fully acceptable solution, representing membership value equal to 1 to the set of acceptable solutions, while lower fuzzy benefit/quality values represent solutions with smaller degrees of membership to the fuzzy set of acceptable solutions (Klir and Folger, 1988). An additional advantage is that by using fuzzy membership functions, tolerance to small constraint violations is simulated (as would any engineer do for real networks). On the other hand, by estimating benefit values through aggregators, the whole design algorithm moves away from strict mathematical functions, and resembles more a Decision Support System (DSS).
There is no need for scaling;
constraints, requirements and network properties are all being handled as a population of criteria (or
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factors), acting as decision makers. Each criterion assesses benefits separately, while in the end, all of them combine “opinions” in order to estimate the merits (benefits) of each solution. This approach offers two advantages compared to penalty functions, which are traditionally used for constraint violation in single objective GA models: (a) there is no need to scale penalties, since fuzzy membership functions are based on dimensionless deviation parameters and (b) the decision maker’s requirements are simulated through linguistic parameters. It is beyond the scope of this paper to describe the fuzzy reasoning approach in full, which can be found in previous published work by the same authors (Vamvakeridou et al, 2005). Multiobjective optimization of water distribution networks takes into account multiple loadings (scenarios), representing peaks, fire flows and normal day operation, the latter divided in multiple loadings referring to specific day periods, according to a selected time step.
Pipes and Pumps as Decision Variables Generally for water distribution network models of fixed layout, there exist three types of decision variables: pipes, pumps and storage (tanks). The coding assumptions for pipes and pumps are quasisimilar in all (or most) GA models in literature, multiobjective or not: New or duplicate pipes added to the network are simulated with the use of an integer number, referring to a set of commercial diameters. Cleaning and lining of existing old pipes are simulated using a binary (two state) variable, simulated as an integer taking the values 0 (no action) or 1 (cleaning). Pumps can only be added to predefined links and their operation curve is assumed known, so that only the number of pumps is assumed to be a decision variable, in the form of a single integer number, for each relevant loading. The representation of storage decision variables is again integer numbers, but the concept is more complex, as described in detail in the following section.
Storage Decision Variables Let us assume a complex water distribution system to be optimized using GA, involving expansion and partial rehabilitation of an existing network, including a number of existing floating-on-the-system tanks, which should be retained and exploited in the best possible way. The system also includes new tanks, the total number and location of which are open to optimization, together with their volume and level. Let us also assume, without loss of generality, that all (or most) of the new tanks will be elevated
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ones, since possible locations for new ground tanks are so heavily constrained by topography, that location optimization is practically non-existent. From the engineering point of view, in order to fully design a tank the following properties need to be defined: a)
Location.
b) Volume (capacity). c)
Shape (sphere or cylinder)
d) The ratio between diameter and height for cylindrical tanks (or the diameter). e)
Maximum operational level
f)
Minimum normal operational level
g) Overflow level (or the distance between maximum and overflow level) h) Top level (or free height/volume, or percentage of total height for free height). Free height is assumed to be the difference between the top level and maximum level of the tank. i)
Safety/emergency volume (or bottom level, or percentage of the total volume for emergency purpose)
j)
Length and diameter of the riser
Obviously, some of these properties are independently defined, while others are derived. For instance, top level can be derived through bottom level (or vice versa), directly for spherical tanks, or once the ratio between diameter and height is determined for cylindrical tanks. Theoretically, all (or most) of the above independently defined properties could be included as decision variables, as such, or transformed through equivalent numerical expressions. It is obvious that such an approach is not feasible, both for engineering and mathematical reasons. From the engineering point of view, some of the above properties are parameters, rather than decision variables. For instance the decision about spherical or cylindrical tanks, although important, is usually made beforehand, taking into account commercial, structural and technical factors, entered as data to the GA. The overflow level is set a little higher than maximum operational level (e.g. 0.10-0.20m) at most tanks, irrespective of tank volume or shape; there is no need to include both levels in the GA process as decision variables. In the same way, free height is an engineering parameter (e.g. 1.0-1.5 m) related to tank volume or height, but not subject to optimization.
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From the mathematical point of view, only independent variables should be included in objective functions, and generally, it is not recommended to include decision variables with quasi constant values, or with little or no effect on the overall objective function value. Such variables tend to make the algorithm oscillate needlessly within the decision space, hampering or, at least, delaying convergence by flattening unnecessarily the objective function. On the other hand, if too many variables are left out (e.g. ratio between diameter and height), the optimization algorithm may produce solutions resulting in extremely flat, high or wide tanks, which might be mathematically feasible, but technically and commercially unacceptable. Consequently the following assumptions have been made: 1.
Each new tank is assumed to be either spherical, or cylindrical with a predefined ratio between diameter (d) and water height from bottom to maximum operational level (h).
2.
Safety (emergency) volume is a design parameter, represented as a constant percentage of the tank water volume.
3.
Free height (hf) is a constant percentage of the total water height (h).
4.
The maximum operational level is defined through the minimum normal operational level, the volume and the ratio between diameter and height.
5.
The riser is altogether omitted during optimization, because its impact on the objective function value is considered unimportant and quasi constant. Costs for risers are added after the GA process.
Therefore within the GA each tank is assigned two independent decision variables: Tank volume and minimum (normal) operational level. All other design parameters are derived from these two variables, including shape and emergency volume. Both variables are simulated as positive integer numbers, by discretizing the decision space. Finally, although tank shape parameters are entered as data for the GA, thus avoiding excesses for tank height or diameter in the proposed solution, trimming of the final tank shape might still be necessary after the GA process. Indeed, from the engineering point of view, modifications of the diameter/height ratio might still be necessary to “fit” a tank, maintaining the total capacity defined by the GA, at a specific location, or to round up final tank dimensions. It might also be necessary to alter the emergency volume ratio, after a detailed (small time step) extended time simulation detects e.g. an early emptying of
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a specific tank. On the whole, however, these shape modifications are considered to be minor, not affecting the overall network design features and hardly (or not at all) altering costs, as shown in the application that follows.
Coding for storage variables Two different approaches have been applied for the coding of storage variables within the GA, each representing a different simulation for the water distribution system. According to the first approach (called conventionally hereafter “undefined maximum number of tanks” or 1st approach) the maximum total number of new tanks is unknown; theoretically there can be as many new tanks as the number of potential locations and all locations (nodes) potentially eligible for new tanks have to be equally taken into consideration. Therefore each possible location (node) for a new tank is simulated by the use of a pair of consecutive decision variables (xi, xi+1). Both variables refer to the same location (node) and can take positive integer values. Zero values are allowed only for the first variable xi. When the first variable xi is zero, there is no tank at the node, irrespective of the second variable value. When the first value is a positive integer, there is a tank, its volume (first variable) determined by xi, while the second integer xi+1 gives the minimum normal operational level (second variable) as a discretized integer decision. Because all possible tank nodes are assigned this pair of variables, the variable index defines the specific node, without the need for a third variable to define the location. The second approach (called conventionally hereafter “defined maximum number of tanks” or 2nd approach) assumes that the maximum total number of new tanks maxnt is known beforehand and entered as data (e.g. if maxnt =3 there can only be up to three new tanks in the system). However the possible locations for the new tanks are unknown and have to be selected out of a total number of eligible nl nodes, where nl> maxnt. In this case each new tank is simulated using three consecutive decision variables (xi, xi+1, xi+2). All three variables are positive integers, while zero values are allowed only for the first variable xi. The first variable xi is a positive integer number referring to the node identification list; the second xi+1 is the discretized volume decision, while the third xi+2 refers to the discretized minimum normal operational level. When the first variable xi is zero, there is no tank, irrespective of the second and third variable values. When the first value is a positive integer, there is a tank at the node defined by it, its volume (first variable) determined by xi+1, and its minimum normal operational level defined respectively
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by xi+2. The total number of triplets (xi, xi+1, xi+2) is equal to the maximum total number of new tanks maxnt. In the application that follows both coding assumptions have been applied and their impact discussed.
Extended period simulation loadings and assumptions Extended time period simulation loadings aim at defining network operation, during a normal day (24 hours operational cycle), under a daily use water demand pattern. They are not peaks, so that no conclusions can be drawn for pressure requirements, but they are necessary for defining: •
Whether the water volume pumped into the system is enough to meet demands
•
The operational volume required for each tank and the system as a whole
•
Whether the pumps can fill each tank separately and all of the tanks as a whole
•
The energy costs for each solution
In order to perform all these tasks with accuracy a small simulation time step is required (e.g. 530min). Given that within any GA a huge number of possible solutions have to be assessed (30-100 strings per generation for 5000-50000 generations), proper extended period simulation is not feasible for practical reasons. Previous approaches in literature include 6-hour time step (Murphy et al, 1994, Walters et al, 1999) and 1hour time step (Farmani et al, 2005), with additional computational assumptions, as follows: According to one approach (Murphy et al, 1994, Walters et al, 1999), only the volume of the tank is defined prior to the extended period simulation by the GA. Within each period of the day corresponding to the chosen time step, the total flow for the system into or out of storage is calculated, according to whether the demand for that period is greater or less that the average of the day. The flows into and out of each tank (“demand balancing flows”) are calculated for each period, by allocating the system inflows to the tanks in proportion to their volumes, thus constraining all tanks to empty and fill in a similar manner. The tanks are then considered as normal network nodes with known demands (tank demand nodes). The heads at the tank demand nodes are then determined by steady-state analysis for each period of the day. For each tank demand node, the maximum and the minimum head identified over all time periods are interpreted as maximum and minimum operating levels for each tank. Initial mismatches between the
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tank heads calculated as above and the water levels resulting from volume changes within each time period are treated as tank operating level differences (TLD) and assigned penalties within the GA. According to the second approach (Walters et al, 1999, Farmani et al, 2005), volume, maximum and minimum normal operational levels are produced as independent variables by the GA, prior to the extended period simulation. “Demand balancing flows” are calculated for each period as in the first approach. For each time period the change in volume and water levels for each tank are calculated according to the “demand balancing flows”.
The fixed water level tanks produced as above are
consequently used for steady-state analysis within each day period, from which the flows in and out of each tank are calculated and compared to the “demand balancing flows”. The accumulated difference between the two sets of flows over the whole of the operational cycle (24h) are treated as tank flow difference (TFD) mismatches and assigned penalties. Both these approaches have a disadvantage: According to the way “demand balancing flows” are estimated, it is assumed that inflows to the system are proportional to the volume of each tank and that all tanks empty or fill simultaneously, an assumption that does not necessarily hold for real networks (Walski et al, 2003, Chapter 8.5). When there is more than one tank in a pressure zone, the problem of designing and operating the system (let alone mathematically simulating it) becomes much more complicated. Even if the tanks have similar maximum operational levels (or overflow levels), the main problem is that the tank closer to the network source fills up quickly and drains slowly, while a tank at the perimeter of the system fills more slowly and drains quickly (Walski et al, 2003). This happens because the hydraulic grade line (HGL) will be higher closer to the source than the HGL at the perimeter of the network (Figure 1). One solution to the problem in practice is to use a throttle control valve to throttle the flow when the tank closer to the source is (nearly) full, enabling more water to flow to the distant tank. Also all tanks (especially the tanks closer to the source) should drain through a separate line, using a check valve, so that the draining of the tank could be easily controlled. In this way when the water in a tank is close to the minimum operational level the draining of the tank can stop. When the location of the tanks is subject to optimization within the GA, there is no way to know beforehand whether the new tanks in a solution will be close to or far away from the source, or whether they will prevent the filling or draining of other tanks (old and new) in the system. Therefore the approach applied in this paper is different from other approaches in literature and, although approximate,
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remains closer to the procedure of proper extended period simulation, from the engineering point of view. A time step ∆t=3 hours has been selected (8 loadings for the 24h operational cycle), which exceeds previous approaches, where a 6-hour time step had been applied (Walters et al, 1999). The steps involved for assessing the operational performance of tanks within the GA are as follows: 1) The volume (operational capacity), and minimum normal operational level for each new tank are defined as independent decisions from the GA. Diameter and maximum operational level are estimated according to the predetermined shape ratio for each new tank. 2) The start of the operational cycle t0 coincides with the end of the off-peak period (e.g. 6.00am) and all tanks (new and old) are considered full, i.e. the initial water level h0 is set as equal to the maximum operational for all tanks. 3) The network is solved, using steady-state analysis, at t= t0 in order to define flows in and out of each tank. 4) According to inflows/outflows calculated at step (3) the new level h for each tank (old and new) at the start of the next simulation cycle t+∆t is calculated internally within the GA, taking into account the tank operational volume and diameter, the initial conditions of the previous time period (h0), as well as operational constraints, i.e. a full tank cannot accept any additional inflow and a tank reaching its minimum operational level cannot feed the system. 5) Steps 3 and 4 are repeated until the end of the 24h operational cycle. 6) The system capacity for filling tanks within the 24h cycle, and the tank operational capacity is assessed, according to four criteria described in detail in the next section. Starting the simulation at the end of the night period (Step 2), with all tanks full, is a deliberate selection, made after careful consideration. Within the GA tanks are produced by two independent decisions: volume and water level. Filling capacity primarily checks the selection of levels and location for tanks, not volumes. Indeed a great number of solutions are examined and assessed, many of which are far from feasible: wrong location of tanks, wrong level, and wrong number of pumps. Moreover the capacity of tanks (new and old) may not be exploited in full (wrong volume), so that they would not drain completely during the day, although they might fill during the night. Therefore the assessment of filling capacity should be clearly separated from the assessment of operational volume, and the best way to assess it (for tanks and pumps), is to check whether the pumps can refill the tanks after a day peak period
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and a night off-peak period. If the proposed solution cannot return the tanks to full status after 24h, then the system filling capacity is seriously flawed and the solution has to be given a lower ranking from the benefits point of view. This assumption for checking filling capacity separately within the GA is largely ignored in literature, where the 24h cycle starts at the end of the day at 18.00h (Walski et al, 1999) or later (21.00h) at the beginning of the filling time (Oliveira Sousa et al, 2005), with the water level in all tanks assumed to be close to the minimum normal operational level. Although according to step (4) tanks do not fill or drain simultaneously as in previous approaches in literature, the use of a large time step (3h) has led to other simplifying assumptions: For each time period
∆t, each system pump is either on or off, by that meaning, that status changing in the middle of ∆t is not possible for pumps. Moreover, since the network is only solved once for each time period, the tank status from open to close, or vice versa, still cannot change within each time period. For instance, if at the beginning of a certain time period a specific tank is filling and reaches the maximum operational level in less than 3 hours, it still cannot close, as it should in normal extended time period simulation, and the network is not solved again during the same time period (to save computational time), so as to estimate the changes to all other network hydraulic parameters; next time period will simply start with this tank full. Since the scope of this approximate extended time period simulation is not to accurately determine hydraulic parameters, but to estimate pump capacity, tank levels/operational volume and energy costs, only the impact of these assumptions to the grading of possible solutions is of interest: •
The water volume pumped into the system will be slightly overestimated, because pumps cannot close in the middle of a time period, if e.g. tanks fill sooner than the 3h time step.
•
Energy costs will be accordingly overestimated. Detailed computations carried out for various solutions, have shown that they may exceed by 3-5% the actual energy costs for “good” solutions, as shown in the example application that follows.
•
The required operational volume for each tank separately is overestimated.
•
Filling the tanks is harder under these approximations. Consequently, assessment of filling capacity for each tank separately is affected, by labeling as “problematic” tanks that are not problematic at all, should computations be carried out using a smaller time step.
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Mathematically, in this model, computational inaccuracies are treated by introducing fuzzy reasoning. However, even if it were not for the fuzzy logic ability of simulating small numerical deviations, there is an additional general remark: All computational inaccuracies are on the “overestimation” side (energy costs, required storage or filling deficiencies). Therefore, a solution might appear “less good” than it is, but there is no way for a “bad” solution to be labeled as “good”, under any circumstances. A detailed numerical example of the impact of these assumptions is given in the application that follows.
Assessment of tank operational performance within the GA In this approach all tanks in the system (old and new alike) are equally assessed. In this way solutions, which make better use of the existing infrastructure, score better for the benefit function, and vice-versa. Performance assessment for the operational volume and the minimum and maximum operational level is carried out for each solution X within the GA, according to the following four criteria: 1.
Operational volume capacity to meet hourly demand variations for 24 hours under “normal day” loadings, for each tank separately (per tank). Using the network solver results from the loadings referring to extended time period simulation, the required operational volume for each tank is computed and checked against the existing one. The required operational volume is calculated, with the use of a double cumulative (volume-time) curve or, alternatively, with the use of a time-volume balance table, if the initial conditions are known, while 10% tolerance is adopted, in order to account for mathematical inaccuracies. This criterion performs better for water networks without pre-existing ill-conditioned tanks.
2.
Filling capacity for 24 hours under “normal day” loadings, for each tank separately (per tank). This criterion evaluates whether pumps can fill the tanks during the night off-peak period. Each tank is assumed to be full at the end of the night period (initial conditions) and should be full again (at the same level – no more, no less) after 24 hours, using extended time period simulation. This criterion is the hardest to satisfy, for any solution, and the one most affected by the large time step adopted for extended time period simulation. Indeed, in reality the network tanks would probably fill in turns during the night, turning to “closed” status once they fill, allowing increased inflow for the rest of the tanks (Figure 1). Due to the increased time step of the simulation, the night is divided only in two 3h periods and such detailed status changes for each tank are not possible. It is the only criterion that
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might be omitted (or assigned a small weight coefficient), in case it hampers convergence after a few initial trial runs. 3.
Operational volume capacity for the network as a whole. Similar to criterion 1, the only difference being that both required operational volume and existing operational volume are checked for the system as a whole. The sum of all tanks’ operational volume is graded, as opposed to the volume required by a double cumulative curve (volume-time), treating the network as a whole, namely summing inflows/outflows from all tanks at each time step. Again 10% tolerance is introduced to account for mathematical inaccuracies.
4.
Filling capacity for the network as a whole. It is similar to criterion 2, except for the fact that all tanks are treated as a whole, summing up inflows/outflows. The total water volume at the tanks, at the end of the 24h extended period simulation should be equal (no more, no less) to the total operational capacity of all the tanks in the network. In contrast to criterion 2, ill-conditioned preexisting tanks do not carry a significant weight on the evaluation. Indeed, even in cases where filling of each tank assessed separately is problematic, this criterion may be satisfactorily fulfilled. On the other hand, for a specific solution, failure to comply with filling all tanks indicates severe faults in the overall structure of the network and the benefit function should be affected accordingly.
Assessment of tank emergency volumes within the GA In order to check emergency volume for each tank and the system as a whole, only fire flow loadings (scenarios) should be taken into consideration. Emergency (safety) volume should be adequate for sustaining fire flows for a specified duration (e.g. 2h), the tanks starting at their lower operational level, emptying towards the end of the specified duration. Tanks would not empty simultaneously during fire flows. Some would drain faster than others, so that towards the end of the duration a number of tanks would be empty and the required flow to the system would be provided by the remaining tanks. Therefore the best way to check emergency volumes would be extended period simulation for the duration of each fire flow, with a small time step (5-15 min), taking into account the changes in the status of each tank. However this is not feasible within a GA, because of the increased computational time required. Consequently in the model applied here, as in all previous approaches in literature, the network is solved only once for each fire flow scenario, at the beginning of the duration of fire flows, with all tanks at their minimum normal operational level.
16
However assessment of tank emergency volume varies: Previous approaches in literature (Walters et al, 1999) compare the total required emergency volume (according to the tanks’ outflows at the beginning of the fire flow and the fire flow duration) to the emergency volume of all tanks within the system. Other approaches (Murphy et al, 1994) define the required emergency volume for each new tank by the outflows at the beginning of each fire flow, and sometimes even modify the emergency volume at existing tanks in this way (Oliveira Sousa et al, 2005). In this model, the emergency volume for each existing tank is considered fixed, while the emergency volume for each new tank is produced by the GA. However performance assessment for the emergency volume for each solution X within the GA is carried out both for new and existing tanks, in order to evaluate the emergency behavior of the system as a whole, according to two criteria, as follows: 1.
Total emergency volume. The first criterion is similar to previous approaches (Walters et al, 1999); the total required emergency volume is compared to the emergency volume of all tanks within the system as a whole, although 10% tolerance is included to account for computational inaccuracies, because the required emergency volume is estimated according to the tanks’ outflows at the beginning of the fire flow and the fire flow duration (and not according to precise extended period simulation with a small time step, as it should, for greater accuracy).
2.
Emergency volume per tank. The second criterion introduces an original approach: instead of checking emergency volumes directly, this criterion checks and compares “time to empty” for each tank separately, namely by dividing the emergency volume of each tank with the outflows at the beginning of each fire flow, the “time to empty” is estimated and compared to the required time the network should sustain fire flows, both for existing and new tanks. A 10% tolerance is included, to account for computational inaccuracies, as with the previous criterion.
If the time to empty is too small, then the tank emergency volume should increase. On the other hand, should it be too big, there are two possible reasons for it, from the engineering point of view: Either the tank emergency volume is too big, or the location and level of the tank is such, that it cannot contribute to fire flows as it should, i.e. it is under-exploited. Often this was proved to be the case with existing tanks within the network, which cooperated badly with the new tanks. Moreover, there is the
17
possibility of the “time to empty” being a negative number, by that meaning that the tank during fire flows receives water from the system, instead of feeding it. This indicates an ill-fitting tank, at the wrong location and/or level; the GA should eliminate or modify it in the next generations. There was no way to assess all these factors simply by summing up volumes, a procedure which smoothes effects for problematic tanks.
Application The storage optimization technique proposed in this paper has been applied to the “Anytown” water distribution network. “Anytown” was first introduced in 1987 (Walski et al 1987) as a realistic benchmark with which to compare and test network optimization methodologies. Details about data can be found in literature (Walski et al 1987, Murphy et al 1994, Walters et al 1999) and in the Exeter Benchmark Problem Library on the web (Centre for Water Systems, 2004). Briefly, “Anytown” is a hypothetical urban community, with a problematic water distribution network (Figure 2). The only source is a nearby river, from which water is fed to the treatment plant at node 10. Three identical pumps in parallel take water from the clear well and pump it into the system, the water level at the clear well being kept constant. The town has developed around an old central part. There is a surrounding residential area, and a planned industrial expansion to the north. Additional links (dashed lines) are generally treated as optional, except one. Each node should at least be connected to two pipes for redundancy. The design optimization problem is to partially renew and expand the water distribution network, by adding new pipes or pipes in parallel (duplication of existing links), by optional cleaning and lining of existing old pipes, by adding new tanks (selecting the number, size and location) and by adding new pumps, if necessary, alongside the 3 existing ones. The target is to minimize construction and energy costs, while meeting increasing water demands, pressure requirements, and urban expansion. Node demands, daily water use pattern, topographical layout, existing diameters and roughness coefficients are all given as data. There are two existing floating-on-the-system elevated tanks, at nodes 65 and 165 respectively, with 1136 m3 (250000gal) capacity each. The tanks are full at water level 77.7m (255ft), their maximum operational level is 76.2m (250ft), the minimum normal operational level 68.6m (225ft), while their bottom is at 65.5m (215ft). Tank 165 is already problematic, the existing pumps having trouble filling it. Tanks can be added at all nodes, except the nodes that are directly connected to existing
18
tanks, their costs being given as a function of tank capacity, thus leaving 17 nodes as possible tank locations. In this work “Anytown” has been optimized taking into account 12 different scenarios (loadings), as follows: •
Scenario 1 relates to peak (peak factor 1.8) demands, with 2 out of the 3 existing pumps in operation and all tanks at minimum level.
•
Scenarios 2, 3, and 4 relate to the three different critical fire flow and peak flow (peak factor 1.3) demands. Only 2 pumps are allowed, all tanks start at minimum normal operational level (conditions at the beginning of the fire flow duration) and the safety (emergency) volume of each tank must be enough for 2 hours’ (7200 sec) fire and peak flow.
•
Scenarios 5 to 12 (8 different loadings) refer to the normal day demand variations. Each scenario relates to the 3-hour demand variation pattern. The GA performs extended period simulation using the 3-hour time step, getting “snapshot” steady-state hydraulic analysis results for each loading, with tank water levels selected according to the inflow/outflow conditions of the previous scenario. The starting time of “normal day” scenarios (scenario 5) coincides with the end of the night period (6.00am), when all tanks should be full.
The evaluation of the quality/benefit objective function is carried out taking into account and aggregating, using fuzzy reasoning, the following 9 criteria: 1.
Minimum pressure at the nodes for peak and fire flow scenarios.
2.
Maximum velocity constraints in the links for peak and fire flow scenarios.
3.
Safety volume capacity to meet fire flow demands for the required time for each tank separately (“time to empty” for each tank for fire flow scenarios).
4.
Safety volume capacity to meet fire flow demands for the required time for the network as a whole (total emergency/safety volume comparison for fire flow scenarios).
5.
Pump operational capacity to meet hourly demand variations for 24 hours under “normal day” loadings.
6.
Operational volume capacity to meet hourly demand variations for 24 hours under “normal day” loadings, for each tank separately.
7.
Filling capacity for 24 hours under “normal day” loadings, for each tank separately.
19
8.
Operational volume capacity for the network as a whole, for “normal day” loadings.
9.
Filling capacity for the network as a whole, for “normal day” loadings.
Criteria (1) and (2) are only valid for peak flows. Criteria (3) and (4) aim at assessing the emergency volume of tanks (separately and as a whole) for the three fire flows. Criterion (5) rates pump efficiency, while criteria (6) to (9) rate tank levels and operational volumes.
Decision variables and coding In all, the “Anytown” optimization model contains 74 decision variables for pipes (new and rehabilitated). There is only 1 decision variable for pumps (number of new pumps). This last assumption about the pumps is only acceptable in this case study, because additional pumps are assumed to operate in parallel with the old pumps, with the same curves. When selection of location and/or type of pump is subject to optimization, a different, more complicated simulation approach should be applied. The operational pattern of pumps is not subject to optimization: The GA has been set to operate the 3 existing pumps, together with any new pumps during the night, in order to ascertain the solution capability of filling the tanks. For storage decision variables, two different coding approaches have been taken into account: (a) The “undefined maximum number of tanks” (1st approach), according to which there are 17 decisions for volume and 17 for tank levels respectively. Thus the problem consists of 109 decision variables (integers) in all for each string. (b) The “defined maximum number of tanks” (2nd approach), defining the maximum total number of new tanks maxnt=3. In this approach there are 3 decisions for tank locations, 3 for volume and 3 for tank levels, totalling 84 decision variables (integers) in all for each string. Shape parameters for new tanks have been selected using the existing tanks in “Anytown” as guidelines: •
Cylindrical tanks, with diameter (d) equal to water height (h), i.e. d=h.
•
Safety (emergency) volume is set to 40% of the total water volume. Some initial runs have been performed using 25% ratio, in accordance with the shape of the existing tanks, but they were unsuccessful.
•
Free height (hf) is set to hf=h/9.
20
The discretized search space for new pipe decision variables comprises 10 commercial diameters ranging from 152.4mm (6 in.) to 762mm (30 in.), while cleaning of existing old pipes is simulated with the use of two integer numbers {0 (no action),1 (cleaning)}. The decision variable for the number of new pumps takes values in the range {0,1,2}. The search space for tank volumes ranges from 227.2 m3 (50000gal) to 9088m3 (2000000 gal). The search step has been set to 227.2m3 (50000gal) for volumes up to 4544m3 (1000000gal) and to 454.4m3(100000gal) for larger volumes; in all it contains 30 possible values and it is the same for all potential tank locations. The discretized search space for tank level decision variables depends on topography and is presented in Table 1. It should be noted that tank level elevation steps larger than 2.0m caused the GA to oscillate without convergence at some initial runs of the model. The need for a fine search space grid is not so great for tank volumes, as the GA is less sensible to volume, than level modifications. On the other hand, care has been taken to include possible alternatives for large tank volumes, because it is impossible for the GA to put more than one tank at each location, should excessive storage be needed.
Runs and convergence “Anytown” has been solved using a fuzzy multiobjective GA (Vamvakeridou-Lyroudia et al, 2005). Initially the “undefined maximum number of tanks” concept (1st approach) was applied many times, using various (fuzzy) model parameters, GA parameters and relative weights for the 9 criteria. “Anytown” has pre-existing problematic tanks. Indeed all criteria representing storage, when each tank was assessed separately, tend to score poorly. This problem has been addressed by assigning smaller weights to Criteria 3, 6 and 7, but retaining them all (Vamvakeridou-Lyroudia et al, 2005). The resulting Pareto trade-off curve, out of a 30000 generations run, with population size 50, is presented in Figure 3. The proposed solution, described in the next section in detail, has been selected out of the Pareto points of this curve, as the point that represented the “best” trade-off between costs and benefits. It should be noticed that no run, whatever the model parameters or weights, produced a fully acceptable solution, satisfying all criteria within the GA, which is due to the fact that pre-existing conditions within the network, as well as computational approximations, made it impossible to achieve full compliance. Consequently, the “defined maximum number of tanks” concept (2nd approach) has been applied, retaining the other parameters of the model which yielded the first solution (i.e. population, as well as genetic and aggregation operators), aiming to compare convergence between the two concepts. Five times
21
the GA successfully converged in 11800 to 14200 generations (averaging 13000 generations), much faster than with the first approach (“undefined maximum number of tanks”). The Pareto trade-off curve of a 14200 generation run is presented in Figure 3. It should be noticed that trade-off curves in both approaches (undefined and defined maximum number of tanks) are similar, the only difference being the reduced number of generations. This has been expected, since the latter approach only reduces the decision space. However, it should be kept in mind that these runs were performed in hindsight, namely the parameters of the model leading to good solutions (especially the relative weights for criteria and the number of new tanks) were already known. The solution itself was not improved, i.e. no better solution was produced, while in two runs the 2nd approach model failed to converge to the same results, although it was allowed to run for 20000 generations. In conclusion, although the “defined maximum number of tanks” concept speeds up convergence for “Anytown”, it should be applied with care, because it is less generic than the “undefined maximum number of tanks” concept for real networks.
The Proposed Solution: Results, assumptions and accuracy The proposed solution is a Pareto trade-off point presenting total costs $10.743X106, with total fuzzy benefit/quality function value 0.836343. The fact that the benefit function value is smaller than 1, means there are some criteria that have been only partially fulfilled within the GA. There are two new tanks at nodes 150 and 170. According to the way storage variables have been simulated, risers have been omitted within the GA and added afterwards, slightly modifying the pipe costs. The length of each riser was assumed to be 30.5m (100ft) long (in accordance to the existing risers at tanks 65 and 165), and their diameter 304.8mm (12 in.). The costs of risers are insignificant: $2.92X103 each (0.07% of total costs), while costs of the next cheapest link (from node 55 to node 75) are $76.8X103 (2.03% of total costs). Energy costs were corrected according to detailed estimations, using EPANET (Rossman, 2000) with a much smaller time step (5 min), improving also the use of pumps by modifying their time schedule (not their total number). No new pumps are needed. The network operates with 2 pumps for 18 hours and the third pump is used only for 6 hours, as was also in the original GA produced solution. The only difference is that the third pump is shut off during the night, and operates during the peak period of the day. The GA could not automatically select this option, because pump time schedule was not a decision
22
variable: The GA was set to operate the 3 existing pumps, together with any new pumps during the night, in order to ascertain the solution capability of filling the tanks. The final energy costs have been defined using these modified pump time schedules, while risers have been added to the pipe costs, producing the “fuzzy solution” in Table 2, with total costs $10.540 X106. It should be noticed that both simplifying assumptions (pump operation and the large time step) overestimated energy costs by 3.4%. The original solution, as it was produced by the GA, added two new tanks at nodes 150 and 170 respectively, the larger being the tank at 150, selecting as decisions volumes and minimum normal operational levels. The operational level originally selected by the GA for tank 150 (70.00m) had to be slightly increased to 70.25m, to account for head losses at the riser, which was introduced afterwards. This change has been decided so as not to affect minimum pressures (Peak scenario 1) at the critical nodes. Minimum normal operational level for tank 170 (selected by the GA at 66.00m) has been retained, despite its riser head losses, because it was not critical. It should be noted, that the GA could not select levels out of the mapping configuration (Table 1). Out of all the fuzzy criteria used for benefit/quality evaluation the original solution performed the worst for Criterion 3 (emergency volume assessed separately for each tank), mainly because the existing tank 165 remains almost inactive at the beginning of the fire flow. As it can be seen in Table 3 the flow out of this tank at the beginning of each fire flow is close to zero, thus resulting in very large “time to empty” values for this tank. This feature remains, even if detailed extended period simulation is performed with a small time step (5min). Tank 165 starts being active only after another nearby tank (Tank 65) empties (scenario 3), while it remains practically inactive at another case (scenario 4). Results are shown in Figure 4, for the most critical fire flow (scenario 3), and in Figure 5, for the worst loading (scenario 4) for pre-existing tank 165. It should be noticed that criterion 4 (system “time to empty” according to total emergency volume and outflows) could not detect this problem, as shown in Table 3. This led to some necessary shape trimming for the new tanks (especially tank 150), which needed increased emergency volume (initially set to 40%) and elevated bottom to account for minimum pressure requirements at the end of the fire flows. The need for the latter was detected after performing extended period simulation with a small time step (5 min); the tank had to be shallower and wider. Finally a ratio of 1.6 has been adopted (diameter/height =1.6) and the emergency/total ratio was increased close to 50%. Similar adjustment (but on a smaller scale), increasing the diameter/water height ratio to 1.1 was also
23
needed for tank 170. It should be noticed, that the necessary shape adjustments did not affect the two decisions (tank volume and tank minimum normal operational level) of the GA model, or the tank costs of the solution; they only had to do with ratios (diameter/height and emergency/total volume). Final tank properties are presented in Table 4. Another novelty, improving accuracy in storage simulation is separating active water volume from overflow volume (free height), taking into account both as shape factors, but assigning tank costs, within the GA, to total tank capacity and not to “storage” in general, as happened in previous approaches (Walters et al, 1999, Oliveira-Sousa, 2005). All tanks fill during the night and empty gradually during the day, remaining at their minimum normal operational level in the afternoon and early evening (Figure 6). It should be noted that the tanks do not fill or empty simultaneously. Tank 165 (pre-existing) is the only one to fill partially also during the day. It also does not fully exploit the existing operational volume, accounting again for the operational volume criterion scoring poorly within the GA. Detailed hydraulic results can be found in Vamvakeridou-Lyroudia et al (2004). There exist several previous published solutions for the same problem (Walski et al 1987, Murphy et al 1994, Walters et al 1999). The solution presented here is the cheapest of all: $10.5x103, compared to $10.9x103 respectively, for the cheapest of the previous approaches in Walters et al (1999).
Conclusions In this paper a new approach to the simulation of floating-on-the-system tanks as decision variables for water distribution system design optimization is presented, aiming to bridge the gap between traditional engineering practice and mathematical considerations needed for any GA model. The paper includes a systematic and detailed critical overview of various mathematical approaches in literature, as well as a novel, more “engineering oriented” approach to the simulation of tanks as decision variables for water distribution system design optimization, describing in detail assumptions and impacts to the evaluation of potential solutions. According to this approach, tank simulation is based on two decision variables: capacity and minimum operational volume, omitting risers. Shape and ratio between emergency/total capacities are taken into consideration as design parameters. Assessment of tank performance is carried out by four criteria for the 24 hour (daily) operational cycle, differentiating between operational and filling capacity, as well as two further criteria for emergency flows. The original
24
design and operational mathematical assumptions are implemented in a fuzzy multiobjective GA model, which is applied to the well-known example from literature “Anytown” water distribution network.
Acknowledgments The authors would like to thank Prof. Roger King (CWS) for reviewing the paper and the ASCE anonymous reviewers for their valuable comments.
Notation The following symbols are used in this paper: B
benefit/quality objective function
C
costs objective function
d
tank diameter
h
water height in a tank
hf
free height of a tank
h0
initial condition for the water height in a tank
maxnt
maximum total number of new tanks
nl
network nodes eligible for the location of new tanks
t
time variable
t0
start of the operational cycle for extended period simulation
X
string of decision variables (solution) for the genetic algorithm
x
decision variable
∆t
extended period simulation time step
References 1.
Centre for Water Systems, University of Exeter (2004). “Anytown Water Distribution Network Benchmark Data”, (June 20, 2006).
2.
Farmani, R., Walters, G.A. and Savic, D.A. (2005). “Trade-off between total cost and reliability for Anytown Water Distribution Network”, Jour. Wat. Res. Plan. Man. ASCE, 131 (3), 161-171.
3.
Deb, K., Pratap A., Agarwal S. and Meyarivan, T. (2002). “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II”, IEEE Transactions on Evolutionary Computation, 6 (2), 182-197.
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4.
Klir, G.J. and Folger T.A. (1988). “Fuzzy Sets, Uncertainty and Information”, Prentice-Hall, Englewood Cliffs, New Jersey.
5.
Michalewicz, Z. (1996). “Genetic Algorithms + Data Structures=Evolution Programs”, Springer Verlag, Berlin, Germany.
6.
Murphy, L.J., Dandy, G.C. and Simpson, A.R. (1994). “Optimum design and operation of pumped water distribution systems”, Proc. Conf. on Hydraulics in Civil Engineering, Feb. 1994, Institution of Engineers, Brisbane, Australia, 149-155.
7.
Oliveira Sousa, J.J., Conceicao Cunha, M. and Almeida Sa Marques, J.A. (2005). “Simulated Annealing reaches Anytown”, in Water Management for the 21st Century, Proc. 8th Int. Conf. CCWI05, University of Exeter, Exeter, U.K., Vol. 2, pp. 69-74.
8.
Prasad, T.D. and Park, N.-S. (2004). “Multiobjective genetic algorithms for design of water distribution networks”, Jour. Wat. Res. Plan. Man. ASCE, 130(1), 73-82.
9.
Rossman L.A. (2000). “EPANET user’s manual”, U.S. Environmental Protection Agency, Cincinnati, Ohio.
10. Savic, D.A. and Walters, G.A. (1997). “Genetic algorithms for least-cost design of water distribution networks”, Jour. Wat. Res. Plan. Man. ASCE, 123(2) 67-77. 11. Vamvakeridou-Lyroudia, L.S. (2003). “Optimal extension and partial renewal of an urban water supply network, using fuzzy reasoning and genetic algorithms”, Proc. XXX IAHR Congress, Aug. 2003, Theme B, 329-336. 12. Vamvakeridou-Lyroudia, L.S., Walters, G.A. and Savic D.A. (2004). “Fuzzy multiobjective design optimization of water distribution networks”, Report No.2004/01, Centre for Water Systems, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, U.K., 98p. 13. Vamvakeridou-Lyroudia, L.S., Walters, G.A. and Savic D.A. (2005). “Fuzzy multiobjective design optimization of water distribution networks”, Jour. Wat. Res. Plan. Man. ASCE, 131(6), pp. 467476. 14. Walski, T.M. (2000). “Hydraulic design of water distribution storage tanks”, in Water Distribution Systems Handbook (Mays L.W. ed), Mc Graw-Hill, U.S.A., pp. 10.1-10.20 15. Walski, T.M. (2001). “The wrong paradigm-Why water distribution optimization doesn’t work”, Jour. Wat. Res. Plan. Man. ASCE, 127(4), 203-205.
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16. Walski, T.M., Brill, D.Jr, Gessler, J., Goulter, I.C., Jeppson, R.M., Lansey, K., Lee, H-L., Liebman, J., Mays, L., Morgan, D.R. and Ormsbee L. (1987). “Battle of the network models: Epilogue”, Jour. Wat. Res. Plan. Man. ASCE, 113(2),191-203. 17. Walski, T.M., Chase, D.V., Savic, D.A., Grayman, W., Beckwith S. and Koelle E. (2003). “Advanced water distribution modelling and management”, Haestad Methods, First Edition, Haestad Press, U.S.A. 18. Walters, G.A., Halhal, D., Savic, D. and Ouazar, D. (1999). “Improved design of “Anytown” distribution network using structured messy genetic algorithms”, Jour. Urban Water, 1(1), 23-38.
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Tables: List of Captions TABLE 1: Discretized search space for potential tank levels TABLE 2: Costs and pumping station operation for the proposed solution TABLE 3: Emergency loadings-Conditions at minimum normal operational level TABLE 4: Tank properties
28
TABLE 1: Discretized search space for potential tank levels Network Nodes (potential tank locations) Genetic algorithm 30, 40, 50, 70, 120, 130, 140, 55, 20 decision variable 80, 90, 100, 110 150, 170 75, 115 code number Tank potential min operational level (m) 0 53.0 65.0 53.0 70.0 1 55.0 66.0 54.5 71.0 2 57.0 67.0 56.0 72.0 3 59.0 68.0 57.5 73.0 4 61.0 69.0 59.0 74.0 5 63.0 70.0 60.5 75.0 6 65.0 71.0 62.0 76.0 7 67.0 72.0 63.5 77.0 8 69.0 73.0 65.0 78.0 9 71.0 74.0 66.5 79.0 10 73.0 75.0 68.0 80.0 11 75.0 76.0 69.5 81.0 12 77.0 77.0 71.0 82.0 13 79.0 78.0 72.5 83.0 14 81.0 79.0 74.0 84.0 15 83.0 80.0 75.5 85.0
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TABLE 2: Costs and pumping station operation for the proposed solution Costs
Initial solution Fuzzy solution (Genetic Algorithm) ($x103) ($x103) Pipes 3776.00a 3781.84 Tanks 610.00 610.00 Energy 6357.34 6148.42b Total 10743.34 10540.26 a: Costs of the risers to the tanks are not included in the Genetic Algorithm b: Energy costs have been modified by refining the time step of the 24h extended period simulation and by adjusting the daily time schedule of the pumps
30
TABLE 3: Emergency loadings-Conditions at minimum normal operational level Tank T150 T170 T165 T65 System
Status new new existing existing
Scenario 2 Outflow Time to empty (l/s) (h) 108.28 2.41 22.12 3.73 999.00 91.69 0.87 222.19 2.26
Outflow (l/s) 124.78 46.16 999.00 0.80 1.86
Scenario 4 Outflow Time to empty (l/s) (h) 71.65 3.64 35.27 2.34 999.00 57.77 1.38 164.79 3.05
TABLE 4: Tank properties Properties Tank bottom Minimum operating level Maximum operating level Top (full tank) Diameter Total tank height Diameter/tank height Emergency /capacity Effective volume Emergency volume Free height volume Total volume (capacity) Costs
Units m m m m m m Ratio Ratio m3 m3 m3 m3 $x103
32
New tanks Existing tanks T150 T170 T65 and T165 65.55 62.10 65.53 70.25 66.00 68.60 74.35 70.05 76.20 75.80 71.10 77.70 15.94 9.85 10.90 10.25 8.95 12.17 1.60 1.10 0.90 0.46 0.43 0.25 818 309 710 938 297 284 289 80 142 2045 686 1136 405 205 -
Figures: List of Captions FIGURE 1: Tanks close and away from the pumping station at the same pressure zone FIGURE 2: “Anytown” water distribution network FIGURE 3: Pareto trade-off curve FIGURE 3: Emergency water volumes in the tanks for the duration of fire flow at nodes 75, 55, 115 – scenario 3 FIGURE 4: Emergency water volumes in the tanks for the duration of fire flow at nodes 120 and 160 – scenario 4 FIGURE 5: Tank operating levels over a cycle of 24 hours
33
HGL filling
HGL draining Check valve
Throttle valve
Source
Pump
FIGURE 1: Tanks close and away from the pumping station at the same pressure zone
34
Central city pipes Residential area pipes New pipes 6’ Pipe diameter (in) 115 D New duplicate pipe
55
6’
75
12’
6’
50
12’
10’ 140
New Tank 150
18’D
150
New Tank 170
80
8’D
12’
12’ Tank 165
170
40
Tank 65
60
90
14’D
100
8’D
70 160 130
30
20’D
Clean
10’D
6’
30’D
110 120
Pumps
20
10 Source
FIGURE 2: “Anytown” water distribution network
35
1
0.9
Selected pareto point
0.8
Benefits/Quality
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1st approach
2nd approach
0 4000
6000
8000
10000
12000
14000
Costs ($X1000)
FIGURE 3: Pareto trade-off curve
36
16000
18000
Safety volume in the tanks (%)
100.00 80.00 60.00 40.00 20.00 0.00 0
15
30
45
60
75
90
105
120
Fire flow duration (min) Tank 65
Tank 165
Tank 150
Tank 170
Total system
FIGURE 4: Emergency water volumes in the tanks for the duration of fire flow at nodes 75, 55, 115 – scenario 3
37
Safety volume in the tanks (%)
100.00 80.00 60.00 40.00 20.00 0.00 0
15
30
45
60
75
90
105
120
Fire flow duration (min) Tank 65
Tank 165
Tank 150
Tank 170
Total system
FIGURE 5: Emergency water volumes in the tanks for the duration of fire flow at nodes 120 and 160 – scenario 4
38
Tank water level (m)
78 76 74 72 70 68 66 64 6.00
9.00
12.00
15.00
18.00
21.00
24.00
Time of the day (h) Tank 65
Tank 165
FIGURE 6: Tank operating levels over a cycle of 24 hours
39
Tank 150
Tank 170
3.00
6.00
