NO20140362A1 - System and procedure for optimizing operation of a water network - Google Patents

Abstract

A system for optimizing the operation of a water network comprises at least one data storage unit (23) for storing a hydraulic model as well as at least one operational limitation in the water network (17), wherein the hydraulic model represents the dependence on pressure and flow in the water network at an operating state of at least one activation unit (20) in the water network and at an expected water need, at least one processing unit containing an optimization unit (OP) wherein the optimization unit is arranged to generate at least one operational configuration information for the at least one activating unit by minimizing an objective function of an optimization problem, wherein the optimization problem is based on the hydraulic model, at least one output is cleaned (18) to supply the at least one operating configuration information to the at least one actuator. In order to be able to operate the water network with a reduced lower level limit for a water storage facility, the at least one processing unit further includes a scenario generating unit (SG) to generate a final set of scenarios in the form of possible realizations over time of the anticipated water demand and / or of the operating condition and / or characteristics of the at least one actuator and / or of at least one parameter of the hydraulic model where the final set of scenarios is based on probabilistic information about the uncertainty of these variables. The optimizer is designed to minimize the objective function of the optimization problem by performing stochastic optimization, wherein the optimization problem further takes into account the objective function of representing at least one undesirable side effect in the operation of the at least one activator, at least one operational limitation and the final set. of scenarios.

Description

The invention relates to a system and a method for optimizing the operation of a water transmission and/or distribution network, hereinafter also referred to as a water network. The system comprises at least one data storage unit for storing a hydraulic model as well as an operating constraint for the water network, where the hydraulic model represents how pressure heights (heads) and flows in the water network depend on an operating state of at least one activation unit in the water network and an expected water demand, at least one processing unit containing an optimization unit, where the optimization unit is arranged to generate at least one operating configuration information for the at least one activation unit by minimizing an objective function of an optimization problem, where the optimization problem is based on the hydraulic model, and at least one output interface for delivering the at least one operating configuration information to the at least one activation unit. The steps performed by the above-mentioned system elements define the corresponding method.

As energy prices constantly rise, more and more water utilities are reviewing the operational strategy of their energy-intensive water transmission and/or distribution networks to reduce energy and operating costs. While a water distribution network delivers water from a water source to various industrial and/or private water consumers, a water transmission network transports water over long distances of several hundreds of kilometers, for example from a seawater desalination plant to one or more water distribution networks. The operation of these water networks offers major improvements in operational efficiency. Replacing or overhauling energy-intensive activators in the water network, such as pumps, is an option that is already most common today. The other possibility is to look over and optimize the interaction of the various activators and the overall operating strategy for the entire network.

The upper diagram in fig. la shows an operating procedure for a water network like that

is done today: water utilities try to adjust the network planning so that the water levels 1 in water storage facilities, also called water storage reservoirs, are kept between upper limits 2 and lower limits 3. The upper and lower limits are marked as dotted lines in fig. let. The change of the water level 1 between these limits is controlled depending on

the water consumption in the network by configuration of the pumping system and by a direct connection of the water storage level to the pump control system: if the water level is close to the lower limit, pumps are automatically switched on to increase the water storage level. If the water level is high enough, pumps are switched off. The number of pumps that are switched on at the same time is shown in the lower diagram in fig. let.

Current operating procedures thereby focus on maintaining high water levels in the water storage to meet customer demand at any time, where customer demand is uncertain. To manage this uncertainty in demand, the water utilities set artificially high lower limits for the storage reservoirs, up to 80% of the filling capacity, to maintain high water levels in the storage facilities. In this way, there is enough water in the storage facilities to cover the need even in cases of unexpected changes in need. However, this method leads to an increase in operating costs due to the required higher pump capacity.

Furthermore, this approach neglects the possible benefits of shifting the filling operations in time, for example when energy tariffs are low, or of running pumps to transport the water through the network at reduced speed or at their efficiency point, which can be achieved by storing the water when it is not need for it in storage facilities, such as storage tanks.

In general, the optimization of van network operation is to determine operating information to be taken into account in the subsequent generation of control signals where at least one of the actuation units in the network, where the actuation units are usually pumps, valves inside the network, as well as actuation units that control the flow in and/or the flow out of the water storage and which controls one or more source flows in the water network. The operating information of the activation units is determined so that it minimizes the operating costs of the network, such as costs of energy consumption, while satisfying the water demand, minimum and/or maximum pressure or flow levels in the water pipes of the network, as well as other operating constraints of the water network.

Methods for optimizing the operation of water distribution networks are described in WO 2011/092012 A2, as well as in "Combined Energy and Pressure Management in Water Distribution Systems", by P. Skworcow et al„ Proe. World Environmental and Water Resources Congress 2009, Great Rivers, pp. 709-718, and in "Optimization Models for Operative Planning in Drinking Water Networks" by J. Burgschweiger et al. ZIB-Report 04-48, Konrad-Zuse-Zentrum fur Informationstechnik, Berlin (2004). The optimization methods are based on deterministic hydraulic or specially designed deterministic models that represent the physical properties of the water distribution network. Based on these models, optimization algorithms are used to generate optimized operating plans for activators and parameters over a specific time horizon, for example 24 hours, which takes into account changes in energy tariffs, time-varying needs and possible storage capabilities over this time horizon. By taking into account the entire optimization horizon, it is possible to take full advantage of the storage capabilities of the water network, to shift pumping or energy-intensive operations to times of low energy tariffs and to run pumps at energy-efficient speed even if the demand is higher than the amount of pumped water.

Unfortunately, the operation of water networks is very complex and affected by many uncertainties. Uncertainties are, for example, varying water demand over time as well as changed parameters, such as pump efficiencies or differences between the model of the water network and the real process. The uncertainties in the needs make it difficult to make a prediction that is necessary for optimal activator plans such as pumps, valves, storage tanks, sources, and affect the quality of the optimization result. Indeed, the uncertainties may even lead to infeasibility or undesirable system behavior.

The diagram on the left side in Figure 1b shows an example of an edited trend 5 in water demand, predicted at OhOO for the next 24 hours. The predicted trend 5 in water demand is the darker of the two curves with solid lines. The diagram on the right side in fig. 1b shows the resulting optimized trend in water level 6 in a water storage facility, in particular a water tank, as expected for the predicted water demand 5 using optimized activator plans calculated by OhOO for the next 24 hours. Here again it is the darker of the two curves with a solid line. The optimization of the actuator plans is in the example in fig. lb based on known deterministic optimization algorithms. In the event that the actual demand varies and deviates from the calculated one - which will almost certainly happen at some point, for example due to increased outdoor temperatures or due to a water-consuming sporting event - the lower or upper limit of the water storage tank can be under or exceeded. The lower and upper limits of the water level in the water tank are shown as dotted lines in the diagram on the right side of fig. lb. In the example in fig. lb optimized actuator plans are calculated at OhOO for the predicted demand curve 5 and used for the next 24 hours. The lighter of the two curves in the diagram on the left side in fig. lb visualizes the need as the realization of true need. As can be seen, the true demand 6 is higher than the expected demand 5 for about 6 hours, between about 6:00 a.m. and noon. As a result, the actual water level 8 in the water storage facility will be considerably below the accepted lower limit. When the undershoot goes too far, the pressure can become too low, at least in parts of the water network. A mechanism to overcome the crossing of the lower bound, for example by automatically turning on a pump in the water storage facility, can be integrated into the optimization algorithm, but could potentially result in reduced energy savings. In the worst case, it may even lead to higher energy consumption than without any optimization at all. In general, falling below the lower limit must be allowed i

the optimization algorithm to feasibly solve

the optimization problem. To take this into account, it is known to implement a relaxation of lower level limits. This is done by introducing wobble variables. In return, they are penalized in the objective function to be optimized to avoid their use. However, it happens quite often that the optimal deterministic solution requires undercutting the lower water level limit, while taking into account a random need. In most cases, this is not controllable in advance since it cannot be predicted how far the undershoot will go when using a deterministic model.

To overcome this situation, existing solutions that interact with a management system or a SCADA (Supervisory Control and Data Acquisition) system will collect real-time network operation data and recalculate optimized activator plans or alarm limit settings to adapt to the new situation in the network. The restocking is performed based on a periodic schedule, such as every hour, or based on events, such as crossing upper or lower limits. In the solution formulation of the optimization algorithms, high safety margins are included for the parameters to be optimized to overcome these problems and to ensure delivery according to customer needs at any time even under changing conditions. This in turn results in a more conservative approach to how the water network should be run and operated. In the case of event-based recalculations, it may be necessary to perform the recalculations relatively often, resulting in frequent corrections of the activator plans and therefore in considerable activator activity. Finally, the energy savings may be less than compared to the optimization described above without any adaptation to the network situation.

Deterministic modeling strategies have the disadvantage that they can only handle modeling uncertainties and operational uncertainties in a very limited way. This can result in infeasible or undesirable optimization solutions when applied to the real world.

It is a purpose of the present invention to present a system and method for optimizing the operation of a water network, where the conservative approaches described above are overcome in order to avoid the artificially high limits for the water levels in the storage reservoirs, i.e. to reduce in particular the lower level limit, while at the same time reducing the risk of running out of water.

According to the invention, the at least one processing unit further contains a scenario generating unit to generate a final set of scenarios in the form of possible realizations over time of the expected water demand or to the operational state of the at least one activating unit and/or at least one parameter in the hydraulic model, where the final set of scenarios is based on probabilistic information about the uncertainty in the respective expected water demand and the operational state of the at least one parameter, and the optimization unit is designed to minimize an objective function of the optimization problem by performing stochastic optimization, where the optimization problem takes into account the objective function to represent at least one undesirable side effect of operating the at least one activation device, the at least one operating constraint, and the final set of scenario-based optimization models.

According to the invention, the optimization problem is formulated and solved directly by taking into account the model that can be described and operational uncertainties, thereby ensuring that defined limits and needs are met. This also allows an overall better cost-effective solution than using a deterministic model as a basis. Since it can be ensured that the lower level limit of a water storage facility is not massively undershot, the lower level limit in particular can be safely set at a lower level.

The uncertainties described are one of the following:

• Measured signals, especially inputs or disturbances to the hydraulic model, such as the water demand. • Operating condition and/or characteristics of the activation units, such as efficiency curves, for example curves of pump efficiency, and

• Modeling parameters, for example pipe roughness coefficients.

Additional uncertainties can be defined. For example, in the case where additional input information to the optimization problem, such as efficiency tariffs or water source costs, is not known in advance for the next optimization time horizon, which in examples described below is the next 24 hours, such information can also be included in the optimization problem in the form of uncertainties in the site of further deterministic input.

In a first embodiment of the invention, the scenario generating unit is arranged to use a successive scenario reduction technique to bundle similar scenarios to reduce the computational effort of the optimization unit during processing of the resulting scenario-based optimization models.

In another embodiment, the scenario generating unit is arranged to generate the final set of scenarios by constructing an upper scenario and a lower scenario that define the range of possible realizations over time for at least one of the above-mentioned values or parameters that cause uncertainty . This is a considerably simple procedure for those cases where only little information about the uncertainty and probabilities exists. In a specific embodiment, the scenario-generating unit is designed to represent the uncertainties of a stochastic process, each stochastic process being defined on an underlying continuous probability space over a time horizon of limited duration, thereby generating an explicit representation of the uncertainty.

In a further embodiment, the scenario generating unit is arranged to generate a final set of scenarios by deriving the uncertainties from measurement data in the water network. Alternatively, uncertain accuracy data can be read from a storage unit or entered by an operator into the system and thereby into the scenario generating unit.

The scenario generating unit can further be arranged to provide the final set of scenarios in the form of a scenario tree for graphical visualization. The scenario tree can then be further aligned by an operator before it is used to generate the optimization problem for the optimizer. In a particular embodiment, the objective function reflects resulting energy consumption and/or water inflow from a source upon operation of the at least one activation unit.

In this particular embodiment, the optimization unit can further be arranged to take into account changed tariffs for energy and for water from the source.

In yet another embodiment, the optimization problem is formulated as a finite discretization of a scenario-based two-stage stochastic programming model.

The hydraulic model can be based on a description of the topology of the water network comprising a node set and an edge set, where the node set consists of storage nodes, connection nodes, water demand nodes and source nodes, and where the edge set represents water pipes, pumps and valves.

In this case, the hydraulic model is then formed by equations of state that represent the dependence of the time-dependent state of the water network on the at least one operating configuration information of the at least one activation unit and on the expected water demand and where the time-dependent state consists of pressure heads at the nodes in the node set and flows along the edges of the edge set.

In order to explicitly take into account the change of a lower level limit in a water storage facility, the at least one processing unit may further comprise a level adjustment unit which is arranged to estimate and provide to the optimization unit, as a further input to the optimization problem, a dependence between a decrease in the lower level limit and an extent to which the at least one undesirable side effect is reduced while at the same time complying with the at least one operational limitation.

The level configuration unit may further be arranged to estimate and provide to the optimization unit as further input to the optimization problem a dependence between the reduction in the undesirable side effect and a risk of the water storage facility reaching a critical water level, where the critical water level may indicate that the water storage facility is about to empty or that it reaches a water level that is no longer sufficient to provide enough pressure to transport the water through the water pipes.

In another specific embodiment, the scenario generating unit is arranged to provide the final set of scenarios to a decision support unit, where the decision support unit is arranged to evaluate for an identified potential cause of a failure in the water network a corresponding impact on the water network, while taking into account the final set of scenarios.

The invention and its embodiments will also be apparent from the examples described below in connection with the attached drawings which illustrate: Fig. la, b operational diagrams for a water storage facility, as previously known,

Fig. 2 a scenario 3 for a parameter P,

Fig. 3 uncertainties for a parameter M moderated by three scenarios,

Fig. 4 a system for optimized operation of a water network,

Fig. 5 a comparison between optimization without and with modeled

uncertainties,

Fig. 6. examples of outputs generated by a level adjustment unit.

As explained above, the main idea of the invention is the following: instead of formulating only one optimization problem as is done with deterministic moderation approaches, the uncertainties in the various parameters of the hydraulic model of the water network are modeled as a finite set of possible realizations of a respective parameter P over time, leading to scenario-based optimization models. Each scenario represents a possible parameter realization over time, i.e. where at a specific time in the future a specific value for the respective parameter P is assumed to lead to a so-called scenario 3 with a specific probability (Oi...cor, as shown in fig .2.

Each path from the root to one of the leaves follows a scenario. The nodes correspond to specific times t{0,...,T}, where decisions c(t) are made.

The optimization problem can, for example, be solved by using two-stage stochastic programming techniques (2SSP).

In many situations, only incomplete information about the underlying probability distribution is available. The scenario generation is then mostly based on the use of historical data to be used on estimation, simulation and sampling techniques, or instead to include the knowledge of an experienced network operator.

The computational effort to solve scenario-based optimization models strongly depends on the number of modeled scenarios. Therefore, a trade-off must be taken into account between the precision of the approximation and the model size, which increases exponentially with the number of scenarios. Application of special scenario reduction techniques helps to reduce the calculation time. In one embodiment of the invention, the idea is therefore to reduce the number of scenario tree nodes by bundling similar scenarios. In the extreme case, the final set of possible parameter realizations, i.e. the final set of scenarios, can for example be reduced to only two scenarios, a lower and an upper scenario, where each of the two scenarios or curves represents one curve respectively from the lower or upper half of all curves, with the highest probability.

Taking uncertainties into account in the formulation of the optimization problem leads to more robust operational configuration information for the at least one actuation unit as infeasible solutions can be more easily avoided. In the longer term, it also results in higher cost savings as the safety margins in the upper and lower limits of the water levels, which are necessary today due to deterministic problem formulations, can be reduced.

The operational configuration information for the at least one activation unit can be one of: • An activator schedule in the form of a time series of operating states for the at least one activation unit, or • A lookup table of operating instructions for the generation of a control signal for the at least one activation unit where the operating instructions are defined depending of the current time and/or current state of the water network, or • A set of parameters, which for example include minimum and/or maximum values for the control signals to be generated for the at least one activation unit.

In what follows, an example of a stochastic approach for optimizing a water network under uncertainties is described in more detail for the example of modeling the uncertainty for water demand.

The first step is to model the structure and dynamics of the van network.

The task of water supply systems is to transport and distribute the required amount of water in the required quality at the same time to specified consumers or storage or water treatment facilities. Raw water is fed into the network of sources, such as groundwater or surface water, such as rivers or lakes. If required, the raw water receives a chemical treatment. Afterwards, the clean water is either stored in small storage facilities located directly at the sources, or pumped into the network. Pumps move the water through pipes by increasing the pressure head in specific locations to overcome differences in height and to compensate for pressure head losses in pipes caused by friction losses. The number of individual pumps that are switched on in a pumping station is described by a discrete decision variable. It can take any integer value between its low limit of zero and its upper limit defined by the maximum number of pumps in this pump station. Valves are network elements to control flows and pressure levels. They can be throttled to different extents to control the movement of water through pipes. The storage facilities must be given special follow-up, because they are the only network elements that provide a buffer between network inflow and outflow. They decouple different water network sections and make the system flexible, thus allowing different possible operating strategies.

The water network topology is described by a graph G=(N,E), where N stands for a node set and E stands for an edge set.

The node set N= U NcnU NdnU Nsn consists of storage nodes Nrn, connection nodes Ncn, water demand nodes Nd and source nodes Nsn, and the edge set E=EpiU EpuU Eva includes or represents water pipes (Epi), pumps (Epu) and valves (Eva).

The physical time t dependent state y(t) of the network consists of pressure heights in the nodes and flows along the edges. State y(t) satisfies a system of differential-algebraic equations (DAEs) that include Kirschhoff-type conservation laws, nonlinear head-flow relationships of pumps, valves, and pipes, and the temporal rate of change of water storage levels in storage facilities due to flows.

The differential-algebraic equations are affected by the control and decision variables c(t). where some are integrals representing the network operational configuration for the pumps, valves and source flow, i.e. the operational configuration information for the actuation units, particularly the actuation plans. Furthermore, the water demand Jj(t) will determine the outflow from the system.

Formally, the state equations F, i.e. the system of DAEs, can be written as

where y(t) is the first derivative of the network state y(t).

The formulation of the optimization problem includes the objective function cd, the state equations F and operational constraints G, some of which are simple limits.

The objective function <D=(øpU+øsn) to be minimized consists of operating costs, such as pump energy costs Opu and source flow costs c<p>zn which are applied during the time horizon T.

The constraints G limit the state of the system to its practical operating range, such as the requirement that the pumps operate in the physical range of positive head rise.

The operational planning problem and thereby the optimization problem is then

presumed that:

The optimal solution of the decision variables c(t) is the response the user, i.e. the water utility, must implement in the real technical system.

In general, the DAE system is discretized on a fixed temporal network (grid), and the control and decision variables c(t) are chosen to be piecewise constant, transforming the optimization problem into a dynamic mixed integer nonlinear programming (MINLP) problem.

The mixed integer component is due to the coexistence of continuous and discrete pump variables. The non-linear part includes the objective function and the pressure-flow conditions. Depending on the numerical solver and its underlying algorithm, smooth functions are strongly desired, especially when using derivative-based optimization methods.

Current practice in operational planning of waterworks is to solve

the optimization problem for a fixed demand ^ (t), usually the expected demand, making the optimization deterministic.

According to the invention, stochastic optimization is used instead, which makes it possible to take modeling and demand uncertainties into account. In general, a common practice for correcting random disturbances in dynamic processes is based on the use of a so-called moving horizon. Assuming that probabilistic information about the uncertainty data is available, stochastic models go a step further by explicitly including this stochastic information on future events. The decision-making process becomes predictive, instead of simply reacting to previous realizations of water needs. Therefore, a stochastic process is required, which describes that the random water demand events occur within their probability. The stochastic process for the random water demand with te[0,T] is defined on an underlying continuous probability space, which is, among other things, a function of the above-described set of state equations F and of a probability distribution P, and where T is the duration of the time horizon.

Obviously, the explicit representation of uncertainty in a stochastic process will lead to an increase of information overtime. The additional modeled information increases the complexity of the optimization model, resulting in large-scale stochastic programs. Most solution procedures for such large-scale programs are based on approximation by a discrete-time stochastic process, where Jjttar values in Rs (for t e {0, ...T», where s : = |Ndn| denotes the number of consumption nodes. The stochastic variables ij is observed directly before time t and J is assumed to be known at t=0. The discrete time stochastic process is a collection of random variables $<=>(So, ...,St) in R<ST>.

Furthermore, to obtain a numerically tractable optimization problem, uncertainty in water demand is modeled as a finite set of possible demand realizations over time leading to scenario-based optimization models. This means that a discrete approximation of the underlying probability distribution is considered, given by a finite probability space, which is, among other things, a function of certain scenariosC0jwith j=l,...,r. Each scenario represents a possible realization of water demand in the future and appears with probability nwj> 0. Together, the modeled scenarios, jr^ =i assume that one of the scenarios will occur in reality.

All modeled scenarios £2 = {oøi, ..., oor} can be represented in the form of a scenario tree, as shown in fig. 2. Each path from the root to one of its leaves corresponds to one scenario. The nodes correspond to specific points in time t e {0, ..., T}, where decisions c(t) are made, i.e. where specific actuator interventions are performed. The edges delimit the uncertain water demand variables. The scenario tree is branched for each modeled need realization in each t e {0, ..., T}. To match the notation of the stochastic process Jj marks each scenario (Oj e £2 d a possible water demand realization ^ = {^ n, ..., ^j), where ^ denotes the j-th scenario for time period k.

Since the results of the real underlying problem are of strong interest, the generalization of such a scenario tree is an important task. The main goal consists in a suitable representation of the underlying probability distribution of {j, because the optimal value and the optimal solution depend on the selected scenarios. In many applications, only incomplete information is available about the underlying available probability distribution. The scenario generalization is then mostly on the use of historical data for use on estimation, simulation and sampling techniques, or instead to include the knowledge of an experienced network operator.

In general, four different types of approximation of the true probability distribution P of a finite set of scenarios can be distinguished with regard to the degree of available information: full knowledge of P, known parametric family, sampled information and low level of information. If little information exists that is based on observed data, the construction of upper and lower scenarios are typical samples, as shown in Fig. 3. The upper scenario 10 and the lower scenario 11 then define the range of possible need realizations and provide limits for the optimal value of the optimization problem. In addition, an expected value scenario can be generated, denoted by 9, which may be the scenario with the currently highest probability.

The computational effort to solve scenario-based optimization models strongly depends on the number of modeled scenarios. A compromise must therefore be made between the precision of the approximation and the model size, which increases exponentially with the number of scenarios. It is common practice to use special scenario reduction techniques which still represent proprietary approaches. These methodologies are based on the application of successive scenario reduction.

The idea is to reduce the number of scenario tree nodes by bundling similar scenarios. The reduced trees and the original tree will then be compared at a certain distance of the probability metric. New probabilities are then assigned to the reduced tree so that the new probability measure is the one closest to the initial distribution in terms of natural distance metrics.

Another important fact is based on the robustness of the optimal objective value and the optimal solution regarding the scenarios used. Robustness is desirable, i.e. small disturbances of the scenarios and the corresponding probabilities should at most result in small changes in the optimal objective value and the optimal solution.

As described above, it is provided that probabilistic information about the uncertainty in water demand is available and stochastic optimization methods can be used to exploit it. The stochastic optimization problem is then to minimize an objective function O, which particularly considers expected operating costs, where the minimization is subject to the requirement that the provided operational configuration information for the activation units is suitable for all water demand realizations. As was also explained above, uncertainty in water demand is modeled as a finite set of possible demand realizations leading to so-called scenario-based optimization models to obtain a numerically tractable optimization problem.

In addition to taking into account the uncertain demand structure, as described above, two-stage stochastic programming (2SSP) exploits the fact that only parts of the decisions, which are here-and-now decisions x, so-called first-stage decisions, have to be carried out before the demand can be observed. This is known as unpredictability.

The remaining control variables, which are resort actions z, so-called second stage decisions, only have to be determined at a later time when parts of the need have already been observed.

In general, this leads to multistage stochastic programming. Here we limit our attention to the more optimistic two-step procedure, where the choice of solution actions z is based on the complete knowledge of the need realization

In what follows, we will denote the first and second stage variables by x(t) and bzw respectively. z(t, ^t(æ)). A scenario-based optimization problem in the form of a 2SSP model formulation, where the exit action is defined from time t~e (0,T) becomes:

Where ø2ssp is the objective function to be minimized,

cDpuer energy consumption costs in pumping station pu and

cDsner source flow costs.

The energy consumption cost takes into account pump efficiency, among other things. Pumps convert electrical energy into mechanical energy of water. The pumping efficiency, called line-to-water efficiency, describes the efficiency of this transformation. It increases with flow rate through the pump up to a certain point, called peak efficiency, and then decreases with further increasing flow rate.

The source flow cost cDsnreflects the cost of providing water from the water sources, such as water treatment plants, desalination plants or water reservoirs.

Additional costs that can be taken into account in the objective function cd2ssp can be so-called penalty costs for exceeding boundary conditions in water reservoirs.

The term cost is to be understood not only in a monetary sense, but is used to generally represent the level of unwanted side effects to be minimised. These undesirable side effects are generally related to the operation of the activation units, since any activation action can result in at least one of the production of too much noise or carbon dioxide, the use of too much electrical energy, an increase in the amount of costly water pumped into the network from a reservoir, or that an unwanted frequency and/or amplitude is reached by the activation operation itself. All these and more unwanted side effects can be modeled into the objective function and thereby into the optimization problem and are then avoided as much as possible due to the minimization of this objective function.

Due to a finite discretization of the underlying stochastic process, equations (5) to (8) take the form of a deterministic optimization model. Therefore, it is also called the corresponding deterministic equivalent of the 2SSP problem. The advantage of the deterministic equivalent consists of better numerical tractability by avoiding probability integrals.

Using the example of a water network 17, as shown in fig. 4, it can be understood that 2SSP leads to more cost-effective and robust water network operations while adding security to the water supply. The water network in fig. 4 consists of water pipes that connect the following operating elements: sources S from which the water flows into the network, consumption units C, where the water leaves the network, pumps 20 to increase the pressure head in specific locations to overcome differences in height and compensate for pressure drop in the pipes, water storage facilities , such as water reservoirs or tanks 21 to provide a buffer between inflow and outflow of the network and to separate different sections of the water network, and different types of valves, as depicted by all the remaining elements of the network 17, to control the movement of water through the pipes and thereby control flows and pressure, by throttling the valves to different extents.

To determine the potential benefit of solving a stochastic problem in relation to solving a corresponding deterministic problem, where the expected need replaces the uncertain need, the value of the stochastic solution (VSS) is used. VSS is an adequate feature because it indicates how well the optimal solution of a 2SSP model works in relation to the deterministic optimal solution built into an uncertain needs framework.

The structure of 2SSP models ensures that first-stage decisions are only feasible if a feasible solution exists for each mapped needs scenario. This means that the first stage variables are calculated to make the model feasible in the long term, so that the curves for the optimal water storage levels lie between the predetermined limits. Therefore, the water levels are kept under control in 2SSP.

This can be seen from the right side of fig. 5 showing the expected water storage level 14 and the resulting actual water storage level 15 for the stochastic optimization of the 2SSP model of the water network described above.

Both the expected and resulting actual water storage levels 12 and 13 lie between the upper and lower level limits, shown as dashed lines. Further experiments for arbitrary demand realizations confirmed the suitability of the 2SSP, therefore allowing the reduction of the lower limits of the water storage levels in the storage facilities. As a consequence, less pumping is required resulting in further cost savings.

On the other hand, the corresponding model based on an expected and not an uncertain demand value is unsuitable, which illustrates that an optimal deterministic scheme for an expected demand is not necessarily optimal or even suitable for the real demand. To feasibly operate the water network, approximate (slack) variables had to be introduced and in response to a sudden increase in water demand even an undershoot of the lower level limit had to be accepted, as can be seen from the expected water storage level 12 and the resulting actual water storage level 13 on the left page in fig. 5.

In general, it was determined that the optimal objective values hold <t>2ssp « Oeev, which further illustrates the importance of stochastic optimization for water network operations. Another important aspect of the stochastic approach is the computational effort. The model size of the 2SSP models increases linearly with the number of scenarios and the calculation time indicates a quadratic increase with respect to the number of scenarios. But even a rough approximation of the underlying probability distribution that includes few scenarios gives good results compared to the deterministic optimization at a moderate increase in computational complexity. Further small perturbations of the modeled demand scenarios only lead to marginal differences in the optimal objective values or in the optimal solution of the stochastic programs.

Fig. 4 shows an example of a system 16 for the stochastic optimization of the operation of a water network. The system 16 contains an optimization unit OP to perform the above-described minimization of the objective function to generate at least one operating configuration information, more specifically at least one actuator layout, which is then transmitted to a control system CS to be used on the intended at least one activation unit in the water network 17 via an output interface 18. The activation units in the water network 17 are the pumps 20 to increase the pressure at specific locations within the network, the pumps and valves in the water storage facilities 21, and the various types of valves within the network.

System 16 also contains a scenario generating unit SG. The scenario generating unit SG is designed to generate scenarios for the future water demand from uncertainties for the water demand derived from real data in the water network, i.e. from historical measurement data MD which includes measurement data of the water demand. The measurement data MD is stored in a storage unit 23 belonging to system 16. Storage unit 23 can either be a volatile data memory, for example a RAM, or a permanent data memory.

The scenarios are assumptions for the water demand in the future, where the expected water demand varies e.g. depending on the time of day, day of the week or time of year. Consequently, day-night variations, higher water needs during hot or sunny months of the year, or due to large public events at the scenario-generating unit SG can be taken into account. In addition, the scenario generating unit SG can take into account additional information that can implicitly influence the water demand, such as information about electricity tariffs ET and/or water source costs WSC, which can be stored in storage unit 23 or which can be entered directly into the system by an operator. Furthermore, such additional information can be outdoor temperatures, weather reports or information about public events. Besides modeling scenarios for the water demand, the scenario-generating SG can e.g. generate scenarios for the variation of the efficiency of the water pumps in the water network. For this purpose, pump efficiency characteristics can be stored in the storage unit 23 and can be used by the scenario generating unit together with measurement data on the operation of the water pumps to model the future uncertainty in the efficiency of the water pumps.

The final set of scenarios for future water needs, generated by the scenario generating unit SG, is then delivered to the optimization unit OP, where it is input to the optimization problem. The optimization problem further takes into account a hydraulic model F of the water network 17 (see equations (3) and (6)), at least one operating constraint G (see equations (4) and (7)) and at least one objective function CD to represent an unwanted side effect U of the operation of at least one activation unit, modeled for example as energy consumption cost Opu and source flow cost øsn. The optimization problem can even take into account variations in parameters on which the least undesirable side effect depends, such as variations in the electricity tariffs ET which affect the energy consumption cost øpu and variations in tariffs for water source costs WSC which affect the source flow cost cDsn. Accordingly, the optimization unit OP is arranged to provide the respective variations of the parameters from the storage unit 23.

The scenarios can also be presented via a human-machine interface to an operator, who can modify them according to his experience.

In certain embodiments, the scenario generating unit SG is arranged to derive upper and lower scenarios, and - if required - one or more additional expected value scenarios. The scenario generating unit SG may be arranged to deliver the water demand scenarios to a graphical display unit for visualization, e.g. in the form of a tree as in figure 2 or in the form of a time-dependent graph showing the upper and lower scenarios as well as the expected value scenario as in fig. 3 or in any other suitable graphic form.

System 16 further contains a level adjustment unit LA which is arranged to determine a dependency between a reduction in a lower level limit of a water storage facility and an extent to which the at least one undesirable side effect U is reduced while simultaneously adhering to the at least one time limitation G and to providing to the optimization unit OP, as an updated time constraint G to the optimization problem, a new value for the lower level limit. An example of such a dependence is depicted in fig. 6, where energy saving is shown over a varying lower limit. The energy saving is in reality an inverted unwanted effect U in the sense that the unwanted effect is the monetary energy costs. The higher the energy costs, the lower the energy savings. The shaded area indicates an area where none of the at least one operational limitation G is exceeded.

The level adjusting unit LA can further be arranged to determine the new value for the lower level limit by also taking into account a dependence between the reduction in the unwanted side effect U and a risk of the water storage facility reaching a critical water level. An example of such a dependence is shown in fig. 7 as a risk level above energy savings. The higher the energy savings, the higher the risk of reaching the critical water level. Based on such a relationship, an optimum can be found at a risk level that the operator of the water network is willing to take.

The above-mentioned dependencies can be determined in the form of functional relationships, or tables, or rule-based relationships.

Another element that can be contained in system 16 is a decision support unit DS. The decision support unit DS receives the final set of scenarios from the scenario generating unit SG and uses it as input to evaluate for an identified potential cause of a failure in the water network 17 a corresponding impact on the water network 17.

A method performed by the decision support unit may include the steps to

a) receive an error message indicating a problem in the water network 17,

b) determine from the error message a series of potential causes for the reported error,

c) determine, for each potential cause, an estimated impact,

d) aggregate the estimated impacts for each potential cause to derive one

importance indication for the reported error.

In this approach, the step of determining an estimated impact on each potential cause includes a risk scenario with initial network conditions and assumed future network conditions in the water network, where the final set of scenarios is used to represent the future network conditions.

The output of the decision support unit DS, i.e. the importance indication, can then be used by the management system CS to determine how quickly the reported error requires a response and/or a priority, or order in which a series of reported errors should be responded to.

Claims (15)

1. System for optimizing the operation of a water network comprising • at least one data storage unit (23) for storing a hydraulic model (F) as well as at least one operational limitation (G) of the water network where the hydraulic model (F) represents the dependence of pressure and flows in the water network (17) on an operational state of at least one activation unit (20) in the water network (17) and on an expected water demand (£(t)), • at least one processing unit containing an optimization unit (OP) where the optimization unit (OP) is arranged to generate at least one operating configuration information (c(t)) for the at least one actuation unit (20) by minimizing an objective function (CD) of an optimization problem where the optimization problem is based on the hydraulic model (F), • at least one output interface ( 18) to deliver the at least one operating configuration information (c(t)) to the at least one activation unit (20), characterized in that • the at least one processing unit further contains a scenario generating unit (SG) to generate a final set of scenarios (£2) in the form of possible realizations over time of the expected water demand ($(t)) and/or of the operational condition and/or characteristics of the at least one activation unit and/or at least one parameter of the hydraulic model (F), where the final set of scenarios (£2) is based on probabilistic information about the uncertainty of the expected water demand, the operational condition and /or characteristics of the at least one activation unit or the at least one parameter of the hydraulic model, respectively, • the optimization unit (OP) is designed to optimize the objective function (<t>) of the optimization problem by performing stochastic optimization, where the optimization problem further takes consideration of the objective function (CD) to represent at least one undesirable side effect (U) of operation of the least e ne activation unit, the at least one operational constraint (G) and the final set of scenarios (£2). 2. System according to claim 1, where the scenario generating unit (SG) is arranged to apply a successive scenario reduction technique to bundle similar scenarios (coj). 3. System according to one of claims 1 or 2, where the scenario generating unit (SG) is arranged to generate the final set of scenarios (Q) by constructing an upper scenario (10) and a lower scenario (11), which define the range of possible realizations over time. 4. System according to one of the preceding claims, where the scenario generating unit (SG) is designed to represent the uncertainties of a stochastic process (Jjt), each stochastic process being defined on an underlying, continuous probability space over a time horizon of limited duration. 5. System according to one of the preceding claims, where the scenario generating unit (SG) is arranged to generate the final set of scenarios (Q) by deriving the uncertainties from measurement data (MD) in the water network (17). 6. System according to one of the preceding claims, where the scenario generating unit (SG) is arranged to provide the final set of scenarios (Q) in the form of a scenario tree (22) for graphical visualization. 7. System according to one of the preceding claims, where the objective function (CD) reflects resulting energy consumption (cDpu) and/or water inflow from a source (cDsn) when operating the at least one activation unit (20). 8. System according to claim 7, where the optimization unit (OP) is arranged to take into account changing tariffs for energy (ET) and for water from the source (WSC). 9. System according to one of the preceding claims, where the objective function (^ssp) is formulated as a final discretization of a scenario-based two-stage stochastic programming model. 10. System according to one of the preceding claims, where the hydraulic model (F) is based on a description of the topology of the water network (17) comprising a node set (N) and an edge set (E), where the node set consists of storage nodes (Nrn) , connection nodes (Ncn), water demand nodes (Ndn) and source nodes (Nsn) and where the edge set represents water pipes (Epi), pumps (Epu) and valves (Eva). 11. System according to claim 10, where the hydraulic model (F) is formed by state equations that represent the dependence of the time-dependent state (y(t)) of the water network (17) on the operational state of the at least one activation unit (20) and on the expected water demand (£(t)) and where the time-dependent condition consists of pressure head in nodes in the node set (N) and flows along the edges of the edge set (E). 12. System according to one of the preceding claims, wherein the at least one processing unit further comprises a level adjusting unit (LA) which is arranged to determine a dependence between a reduction in a lower level limit (3) of a water storage facility (21) and an extent in which the at least one undesirable side effect (U) is reduced while simultaneously maintaining the at least one operational constraint (G) and supplying to the optimization unit (OP) a new value for the lower level limit (3) as an updated operational constraint to the optimization problem. 13. System according to claim 12, where the level adjusting unit (LA) is arranged to determine the new value for the lower level limit (3) by further taking into account a dependence between the reduction in the unwanted side effect (U) and a risk of that the water storage facility (21) reaches a critical water level. 14. System according to one of the preceding claims, wherein the scenario generating unit (SG) is arranged to deliver the final set of scenarios (£2) to a decision support unit (DSU), wherein the decision support unit (DSU) is arranged to evaluate for a identified, potential cause of a fault in the water network (17) a corresponding effect on the water network, while taking into account the final set of scans (£2). 15. Procedure for optimizing operation of a water network comprising the steps storage of a hydraulic model (F) as well as at least one operational model limitation (G) of the water network (17), where the hydraulic model (F) represents the dependence of pressures and flows in the water network (17) on an operational condition and/or characteristics of at least one activation unit (20) in the water network and on an expected water demand (£(t)), • generating at least one operating configuration information (c(t)) for the at least one actuation unit (20) by minimizing an objective function (O) of an optimization problem, where the optimization problem is based on the hydraulic model (F ), • provision of the at least one operational configuration information (c(t)) to the at least one activation unit (20), characterized by the steps • generation of a final set of scenarios (Q) in the form of possible realizations over time of the expected water demand (£(t)) and/or of the operational condition and/or the characteristics of at least one activation unit and/or of at least one parameter in the hydraulic model (F), where the final set of scenarios (Q) is based on probabilistic information on the uncertainty of the expected water demand, the operational condition and/or characteristics of the at least one activation unit and/or the at least one parameter in the hydraulic model, respectively, • minimization of the objective function (cp) by performing stochastic optimization, where the optimization problem further takes into account the objective function (CD) to represent at least one undesirable side effect (U) of operation of the least one enabling unit (20), the at least one operational constraint (G) and the final set of scenarios (£2).