DK201470104A - System and method to optimize operation of a water network - Google Patents

Description

Title: System and method to optimize operation of a water network

The invention relates to a system and a method to optimize operation of a water transmission and/or distribution network, in the following also called water network. The system comprises at least one data storage unit for storing a hydraulic model as well as at least one operational constraint of the water network, where the hydraulic model represents the dependence of heads and flows in the water network on an operational state of at least one actuating unit of the water network and on an expected water demand, at least one processing unit containing an optimizing unit, where the optimizing unit is adapted to generate at least one operational configuration information for the at least one actuating 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 providing the at least one operational configuration information to the at least one actuating unit. The steps performed by the above named system elements define the corresponding method.

As energy prices are constantly rising, more and more water utilities are reviewing the operation strategy of their energy intensive water transmission and/or distribution networks in order to reduce energy and operational 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 hundred kilometers, for example from a seawater desalination plant towards one or several water distribution networks. The operation of these water networks offers great improvement in operational efficiency. Replacing or overhauling energy intensive actuators in the water network such as pumps is one possibility that is most commonly done already today. The other possibility is to review and optimize the interaction of the different actuators and the overall operation strategy of the complete network.

The upper diagram in Fig. 1a shows an operation procedure for a water network as it is done today: water utilities try to adjust the network planning such that the water levels 1 in the water storage facilities, also called water storage heads, are kept in between upper bounds 2 and lower bounds 3. The upper and lower bounds are marked as dotted lines in Fig. 1a. Changing the water level 1 in between these bounds is controlled in dependence on the water consumption in the network by the configuration of the pump system and by a direct link of the water storage level to the pump control system: if the water level is close to the lower bound, pumps are automatically switched on in order to increase the water storage level. If the water level is high enough, pumps are switched off. The number of pumps which are switched on simultaneously are shown in the lower diagram in Fig. 1a.

Thus, the current operation procedure focuses on maintaining high water levels in the water storages to fulfill customer demands at every moment in time, where the customer demands are subject to uncertainty. To handle this uncertainty in demand, the water utilities set artificially high lower limits for the storage head bounds, up to 80% of filling capacity, to maintain high water levels in the storage facilities. In this way, enough water is in the storage facilities to cover the demand even in the case of unexpected changes in demand. However, this approach leads to an increase in operational costs due to the required higher pumping capacity.

Further, this approach neglects possible benefits of timely shifting the filling operations, e.g. when energy tariffs are low, or of running pumps for transporting the water through the network at reduced speed or in their efficiency point, which may be achieved by storing the water when it is not needed in storage facilities, such as storage tanks.

In general, optimization of water network operation is the determination of operational information to be taken into account when subsequently generating the control signals for at least one of the actuating units of the network, where the actuation units usually are pumps, valves inside the network, as well as actuating units controlling the inflow and/or outflow of water storages and controlling one or multiple source flows of the water network. The operational information for the actuating units is determined so as to minimize the network operation costs, such as energy consumption costs, while satisfying the water demand, the minimum and/or maximum pressure or flow levels in the water pipes of the network as well as other operational constraints of the water network.

Methods to optimize the operation of water distribution networks are described in WO2011/092012 A2, as well as in "Combined Energy and Pressure Management in Water Distribution Systems", by P. Skworcow et al., Proc. 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 Informationstechik, Berlin (2004). The optimization methods are based on deterministic hydraulic or specially adapted deterministic models representing the water distribution network physical properties. Based on these models, optimization algorithms are used to generate optimized actuators operating schedules and parameters over a certain time horizon, e.g. 24 hours, taking into account changing energy tariffs, time varying demands and possible storage capabilities over this time horizon. Taking the complete optimization horizon into account enables to take full benefit of storage capabilities in the water network, to shifting pumping or energy extensive operations into times with 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 demands over time as well as changing parameters such as pump efficiencies or differences between the model of the water network and the real process. The uncertainties in the demands make a prediction that is needed for the optimal actuator schedules for e.g. pumps, valves, storage tanks, sources, difficult and affect the quality of the optimization result. In fact the uncertainties can even lead to infeasibilities or unwanted system behavior.

The left hand diagram in Fig. 1b shows an example for a predicted water demand trend 5, predicted at OhOO for the next 24 hours. The predicted water de- mand trend 5 is the darker one of the two solid line curves. The right hand diagram in Fig. 1b shows the resulting optimized water level trend 6 in a water storage facility, in particular a water tank, as it is expected for the predicted water demand 5 when applying optimized actuator schedules calculated at OhOO for the next 24 hours. Here, it is again the darker one of the two solid line curves. The optimization of the actuator schedules is, in the example of Fig. 1b, based on known deterministic optimization algorithms. In case that the real demand varies and differs from the forecasted one - which will almost certainly occur at some point in time, e.g. due to increased outdoor temperatures or due to a water consuming sport event - the lower or upper bound of the water storage tank might be under- or overrun. The lower and upper bound of the water level in the water tank are shown as dotted lines in the right hand diagram of Fig. 1b. In the example of Fig. 1b, optimized actuator schedules are calculated at OhOO for the predicted demand curve 5 and are applied for the next 24 hours. The brighter one of the two curves in the left hand diagram of Fig. 1b visualizes the demand as real-world demand realization 6. As can be seen, the real-world demand 6 is higher than the expected demand 5 for about 6 hours, between approximately 6h00 and noon. As a result, the real-world water level 8 in the water storage facility undercuts the accepted lower bound considerably. When the undercutting goes too far, the pressure may become too low at least in parts of the water network. A mechanism to overcome the crossing of the lower bound, e.g. by automatically turning on a pump in the water storage facility, can be integrated in the optimization algorithm but would potentially result in reduced energy savings. In the worst case, it could even lead to higher energy consumption than without any optimization at all. In general, undercutting the lower bound has to be allowed in the optimization algorithm as to feasibly solve the optimization problem. In order to take this into account, it is known to implement a relaxation of lower level bounds. This is done by introducing slack variables. In return, they are penalized in the objective function to be optimized in order to avoid their use. However, it occurs quite often that the optimal deterministic solution requires undercutting the lower water level bound while accounting for an arbitrary demand. Mostly, this is not controllable in advance, since it cannot be predicted how far the undercutting will go using a deterministic model.

To overcome this situation, existing solutions that interact with a control system or a SCADA (Supervisory Control and Data Acquisition) system collect real-time network operation data and recalculate optimized actuator schedules or alarm limit settings to adapt to the new situation in the network. The recalculation is performed based on a periodic time schedule, e.g. every one hour, or based on events, such as the crossing of upper or lower bounds. In the solution formulation of the optimization algorithms, high security margins are included for the parameters to be optimized in order to overcome these problems and to ensure delivery according to customer demands at every moment in time even under changing conditions. This again results in a more conservative approach on how to run and operate the water network. In the case of event-based recalculations, the recalculations may become necessary to be performed comparatively often, resulting in frequent corrections of the actuator schedules and therefore in considerable actuator activities. In the end, the energy savings may be less than compared to the above described optimization without any adaption to the network situation.

Deterministic modeling strategies have the drawback that they can only take modeling uncertainties and operational uncertainties into account in a very limited way. This can result in unfeasible or unwanted optimization solutions when applied to the real world.

It is an object of the present invention to present a system and method to optimize operation of a water network, with which the conservative approaches described above are overcome in order to avoid the artificially high bounds for the water levels in the storage facilities, i.e. in order to reduce in particular the lower level bound, 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 for generating a finite set of scenarios in the form of possible realizations over time of the expected water demand and/or of the operational state of the at least one actuating unit and/or of at least one parameter of the hydraulic model, where the finite set of scenarios is based on probabilistic information on the uncertainty of the expected water demand, the operational state or the at least one parameter, respectively, and the optimizing unit is adapted to minimize an objective function of the optimization problem by performing stochastic optimization, where the optimization problem takes into account the objective function for representing at least one undesired side effect of operating the at least one actuating unit, the at least one operational constraint and the finite set of scenarios in the form of scenario based optimization models.

According to the invention, the optimization problem is formulated and solved directly taking describable model and operational uncertainties into account, thereby ensuring that defined limits and demands are met. This also allows for an overall better cost-effective solution than when using a deterministic model as a base. In particular, since it can be ensured that the lower level bound of a water storage facility is not heavily undercut, the lower level bound can be safely set to a lower level.

The uncertainties are described for any of the following: • measured signals, in particular inputs or disturbances to the hydraulic model, such as the water demand, • operational state and/or characteristics of the actuating units, such as efficiency curves, e.g. pump efficiency curves, and • modeling parameters, e.g. pipe roughness coefficients.

Further uncertainties may be defined. For example, in case where additional input information to the optimization problem, such as electricity tariffs or water source costs, is not known beforehand for the next optimization time horizon, which in the example described below is the next 24 hours, such information may also be included in the optimization problem in the form of uncertainties instead of additional deterministic input.

In a first embodiment of the invention, the scenario generating unit is adapted to apply a successive scenario reduction technique to bundle similar scenarios in order to reduce the computational effort for the optimizing unit during processing the resulting scenario-based optimization models.

In another embodiment, the scenario generating unit is adapted to generate the finite set of scenarios by constructing an upper scenario and a lower scenario, which define the region of possible realizations over time for at least one of the above named values or parameters which are subject to uncertainty. This is a considerably simple approach for the cases where only few information on the uncertainties and probabilities exist.

In a specific embodiment, the scenario generating unit is adapted to represent the uncertainties by a stochastic process, each stochastic process being defined on an underlying continuous probability space over a time horizon with limited duration, thereby generating an explicit representation of the uncertainties.

In a further embodiment, the scenario generating unit is adapted to generate the finite set of scenarios by deriving the uncertainties from measurement data of the water network. In the alternative, uncertainty data could be read from a storage unit or could be input by an operator to the system and thereby to the scenario generating unit.

The scenario generating unit could further be adapted to provide the finite set of scenarios in the form of a scenario tree for graphical visualization. The scenario tree could then be further adapted by an operator, before it is used to generate the optimization problem for the optimizing unit.

In a particular embodiment, the objective function reflects resulting energy consumption and/or water inflow from a source when operating the at least one actuating unit.

In this particular embodiment, the optimizing unit could further be adapted to take into account changing tariffs for energy and for water from the source.

In an even further embodiment, the optimization problem is formulated as a finite discretization of a scenario based two stage stochastic programming model.

The hydraulic model may 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 that case, the hydraulic model is then formed by state equations representing the dependence of the time dependent state of the water network on the at least one operational configuration information of the at least one actuating unit and on the expected water demand and where the time dependent state consists of heads in the nodes of the node set and flows along the edges of the edge set.

In order to take into account the changing of a lower level bound of a water storage facility explicitly, the at least one processing unit may further comprise a level adjusting unit which is adapted to estimate and to provide to the optimizing unit as further input to the optimization problem a dependency between a decreasing in the lower level bound and an extent to which the at least one undesired side effect is reduced while simultaneously abiding to the at least one operational constraint.

Even further, the level configuration unit may be adapted to estimate and to provide to the optimizing unit as further input to the optimization problem a dependency between the reduction in the undesired side effect and a risk for the water storage facility to reach a critical water level, where the critical water level may indicate that the water storage facility is about to run dry or that it reaches a water level which no longer is sufficient to provide enough pressure for transporting the water through the water pipes.

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

The invention and its embodiments will also become apparent from the examples described below in connection with the appended drawings which illustrate: Fig. 1a, b operational diagrams of a water storage facilities as known from the art,

Fig. 2 a scenario tree for a parameter P,

Fig. 3 uncertainties of a parameter M modeled by three scenarios,

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

Fig. 5 a comparison between optimization without and with modeled uncertainties,

Fig. 6 examples for outputs generated by a level adjusting unit.

As was explained above, the main idea of the invention is the following: Instead of formulating only one optimization problem as done with deterministic modeling approaches, the uncertainties in the different 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 certain point in time in the future a certain value for the respective parameter P is assumed with a specific probability ωι... ωΓ leading to a so called scenario tree, as is shown in Fig. 2.

Each path from the root to one of its leaves accords with one scenario. The nodes correspond to particular points in time t {0, ..., T}, where decisions c(t) are taken.

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

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

The computational effort for solving scenario-based optimization models strongly depends on the number of modeled scenarios. Thus, a compromise between the precision of the approximation and the model size, which increases exponentially with the number of scenarios, has to be taken into account. Applying special scenario reduction techniques helps to reduce calculation time. In an embodiment of the invention, the idea is therefore to reduce the number of scenario tree nodes by bundling similar scenarios. In the extreme, the finite set of possible parameter realizations, i.e. the finite set of scenarios, could for example be reduced to just two scenarios, a lower and an upper scenario, where each of the two scenarios or curves represents the one curve from the lower or upper halve of all curves, respectively, with the highest probability.

Taking uncertainties into account in the optimization problem formulation leads to more robust operational configuration information for the at least one actuating unit as solution infeasibilities can be more easily avoided. On a long-term basis it also results in higher cost savings as the security margins in the upper and lower bounds of water levels, necessary today due to deterministic problem formulations, can be reduced.

The operational configuration information for the at least one actuating unit can be one of • an actuator schedule in the form of a time series of operational states of the at least one actuating unit, or • a look-up table of operational instructions for the generation of a control signal for the at least one actuating unit, where the operational instructions are defined in dependence on the current point in time and/or the current state of the water network, or • a set of parameters, including for example minimum and/or maximum values for the control signals to be generated for the at least one actuating unit.

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

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

The task of water supply systems is to transport and distribute the required amount of water in the needed quality at the right time to designated consumers, or storage or water treatment facilities. Raw water is fed into the network at sources, such as groundwater or surface water, like rivers or lakes. If required, the raw water gets a chemical treatment. Afterwards the clean water is either stored in small storage facilities which are directly situated at the sources, or pumped into the network. Pumps convey the water through pipes by boosting the head at particular locations to overcome elevation differences and to compensate head decreases in pipes, which are caused by friction losses. The number of individual pumps being switched on in a pump station is described by a discrete decision variable. It can take any integer values between its lower bound of zero and its upper bound which is defined by the maximum number of pumps in that pump station. Valves are network elements to control flows and heads. They can be throttled to different extents to control the movement of water through pipes. Special attention has to be paid to the storage facilities, 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 for different possible operation strategies.

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

The node set N = Nrn u Ncn u Ndn u Nsn consists of storage nodes Nrn, connection nodes Ncn, water demand nodes Ndn and source nodes Nsn, and the edge set E = EPi u Epu u Eva encompasses or represents water pipes (Epi), pumps (Epu) and valves (Eva).

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

The differential-algebraic equations are affected by the control and decision variables c(t), some of which are integral, representing the network operation configuration for the pumps, valves and source flow, i.e. the operational configuration information for the actuating units, in particular the actuator schedules. Moreover, the water demand ξ (t) determines the outflow of the system.

Formally, the state equations F, i.e. the system of DAEs, can be written as F(c(t), y(t), y(t), ξ (t)) = 0, vt e [0,T] (1), with y(t) being the first time derivative of the network state y(t).

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

The objective function Φ = (φρυ+φ8η) which is to be minimized consists of operating costs, such as pump energy costs Φρυ and source flow costs Φδη being incurred during the time horizon T.

The constraints G restrict the state of the system to its practical operating range, such as the requirement that the pumps work in the physical range of positive head increase.

The operation planning problem and thereby the optimization problem is then

(2), subject to F(c(t), y(t), y(t), ξ (t)) = 0, vt e [0,T] state equation (3) and G(c(t), y(t)) < 0, vt e [0,T] constraints (4).

The optimum solution of the decision variables c(t) is the answer the user, i.e. the water utility, has to implement in the real technical system.

In general, the DAE system is discretized on a fixed temporal grid, and the control and decision variables c(t) are chosen to be piecewise constant, which transforms 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 nonlinear part comprises the objective function and the head-flow relationships. Depending on the numerical solver and its underlying algorithm, smooth functions are strongly desired, especially when applying derivative-based optimization methods.

Current practice in water network operation planning is to solve the optimization problem for a fixed demand ξ (t), usually the expected demand, which makes the optimization deterministic.

According to the invention, stochastic optimization is applied instead which makes it possible to take modelling und demand uncertainties into account. In general, a common practice to correct random disturbances in dynamical processes relies on the application of a so called moving horizon. Provided that probabilistic information on the uncertain data is available, stochastic models take a step forward by including explicitly this stochastic information on future events. The decision process becomes predictive, instead of just reacting to the past water demand realizations. Therefore, a stochastic process is required, which describes the random water demand events to appear within their likelihood. The stochastic process for the random water demand ξ = with te[0,T] is defined on some underlying continuous probability space, which among others is 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.

Evidently the explicit representation of uncertainty by a stochastic process leads to a gain of information over time. This 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 the approximation by a discrete time stochastic process, with ξι taking values in Rs (for t e {0, ...T}), where s :=|Ndn| denotes the number of demand nodes. The stochastic variables ^ get observed directly before time t and ξ0 is assumed to be known at t = 0. So the discrete time stochastic process is a collection of random variables ξ = (ξο, ...,ξτ ) in RsT .

Moreover, to get 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 among others is a function of particular scenarios cjOj with j=1,...,r. Each scenario represents a possible water demand realization of the future and appears with probability ττω; > 0. Altogether, the modeled scenarios hold

, assuming that one of the scenarios will occur in reality.

All modeled scenarios Ω = {ωι, ..., ωΓ} can be represented in the form of a scenario tree, as is shown in Fig. 2. Each path from the root to one of its leaves accords with one scenario. The nodes correspond to particular points in time t e {0, ..., T}, where decisions c(t) are taken, i.e. where certain actuator actions are performed. The edges delineate the uncertain water demand variables. The scenario tree branches off for every modeled demand realization in each t e {0, ..., T}. To conform to the notation of the stochastic process ξ, each scenario u)j e Ω labels a possible water demand realization ξω, = (ξω,ι, ...,ξω]τ ), where ξω,ι< denotes the j-th scenario for time period k.

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

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

The computational effort for solving scenario-based optimization models strongly depends on the number of modeled scenarios. Thus, a compromise between the precision of the approximation and the model size, which increases exponentially with the number of scenarios, has to be taken into account. It is a common practice to apply special scenario reduction techniques while still representing suitable approximations. 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 by a certain distance of probability metric. New probabilities are then assigned to the reduced tree such that the new probability measure is the closest to the initial distribution in terms of a natural distance metric.

Another important fact relies on the robustness of the optimal objective value and optimal solution concerning the employed scenarios. Robustness is desired, i.e., small disturbances of the scenarios and the corresponding probabilities should at most result in small changes of the optimal objective value and optimal solution.

As described above, it is provided that probabilistic information on the uncertainty of the 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 Φ, which in particular considers expected operational costs, wherein the minimization is subject to the requirement that the obtained operational configuration information for the actuating units is feasible 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 in order to get a numerically tractable optimization problem.

Besides accounting for the uncertain demand structure, as described above, two stage stochastic programming (2SSP) exploits the fact that only part of the decisions, which are the here-and-now decisions x, so called first stage deci- sions, have to be made before the demand can be observed. This is known under the term nonanticipativity.

The remaining control variables, which are recourse actions z, so called second stage decisions, have to be fixed only later in time when part of the demand has already been observed.

In general this leads to multi-stage stochastic programming. Here we restrict the attention to the more optimistic two stage approach where the choice of recourse actions z is based on the complete knowledge of the demand realization ξ·

Subsequently, we will denote the 1st and 2nd stage variables with x(t) and z(t, ξι(ω)), respectively. A scenario based optimization problem in the form of a 2SSP-model formulation, where the recourse action is defined from time point t~ e (0,T) on, reads:

(5), subject to F(x(t), z(t, ξ( (ω,))) = 0, vt e [0,T], vujj e Ω model equalities (6), G(x(t), z(t, ξ( (ω,))) < 0, vt e [0,T] , Vujj e Ω model inequalities (7), a<(x(t), z(t, ξ( (ω,))) < A, vt e [0,T], νω, e Ω variable bounds (8), with <t>2ssp being the objective function to be minimized, Φρυ being energy consumption cost in pump station pu and Φδη being source flow cost.

The energy consumption cost Φρυ takes into account, among others, the pump efficiency. Pumps transform electrical energy into mechanical energy of water. The pump efficiency, called wire-to-water efficiency, describes the efficiency of this transformation. It increases with the flow rate through the pump up to a certain point, called peak efficiency, and then declines with further rising flow rate.

The source flow cost <t>sn reflect the costs for the provision of water from the water sources, such as water treatment facilities, desalination facilities or water reservoirs.

Further costs which may be taken into account in the objective function (t>2ssp can be so called penalty costs for violations of boundary conditions in the water reservoirs.

The term cost is to be understood not only in the monetary sense, but is used to represent in general the level of undesired side effects which are to be minimized. These undesired side effects are in general connected to the operation of the actuating units, since any actuation action may result in at least one of the production of too much noise or carbon dioxide, the usage of too much electrical energy, the increase of the amount of costly water pumped into the network from a reservoir, or the reaching of an undesired frequency and/or amplitude of the actuator operation itself. All these and more undesired side effects may be modelled 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 the finite discretization of the underlying stochastic process, the equations (5) to (8) have the form of a deterministic optimization model. Hence, it is also called the corresponding deterministic equivalent of the 2SSP problem. The benefit of the deterministic equivalent consists of its better numerical manageability 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 providing security of water supply. The water network of Fig. 4 consists of water pipes interconnecting the following operational elements: sources S, from where water flows into the network, consuming units C, where the water leaves the network, pumps 20 for boosting the head at particular locations to overcome elevation differences and to compensate head decreases in the pipes, water storage facilities, such as water reservoirs or tanks 21, for providing a buffer between network inflow and outflow and for decoupling different water network sections, and various types of valves, as depicted by all the remaining elements in the network 17, for controlling the movement of water through the pipes and thereby controlling flows and heads, by throttling the valves to different extents.

To determine the potential benefit from solving a stochastic problem over solving a corresponding deterministic problem, in which the expected demand replaces the uncertain demand, the value of the stochastic solution (VSS) is employed. The VSS is an adequate feature, because it indicates how well the optimal solution of a 2SSP-model performs relative to the deterministic optimal solution being embedded into an uncertain demand framework.

The structure of 2SSP-models ensures that 1st stage decisions are only feasible, if there exists a feasible recourse action for every mapped demand scenario. This means that the 1st stage variables are calculated in order to make the model feasible in the long term, so that the optimal water storage level trajectories lie in between the predefined bounds. Flence, the water levels are kept under control in 2SSP.

This can be seen from the right hand side of Fig. 5 which shows the expected water storage level 14 and the resulting real water storage level 15 for the stochastic optimization of the 2SSP-model of above described water network . Both, the expected and the resulting real water storage levels 12 and 13 lie in between the upper and the lower level bounds, depicted as dotted line. Further experiments for arbitrary demand realizations affirmed feasibility of 2SSP, thus enabling the decrease of the lower bounds on the water storage levels in the storage facilities. As a consequence, less pumping is required resulting in additional cost savings.

On the other hand, the corresponding model which is based on an expected and not an uncertain demand value, is infeasible, illustrating that an optimal deterministic schedule for an expected demand is not necessarily optimal or even feasible for the real demand. To feasibly operate the water networks, slack variables had to be introduced and in reaction to a sudden increase in water demand, even an undercutting of the lower level bound had to be accepted, as can be seen from the expected water storage level 12 and the resulting real water storage level 13 in the left hand side of Fig. 5.

In general, it was determined that the optimal objective values hold 02ssp « Φε-ev, further illustrating the significance 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 quadratic growth with respect to the number of scenarios. But even a rough approximation of the underlying probability distribution comprising few scenarios provides good results compared to the deterministic optimization at a moderate increase in computational complexity. Additionally small perturbations of the modelled demand scenarios only lead to marginal difference in the optimal objective values or in the optimal solution of the stochastic programs.

Fig. 4 shows an example for a system 16 for the stochastic optimization of operation of a water network. The system 16 contains an optimizing unit OP to perform the above described minimization of the objective function for generating at least one operational configuration information, in particular at least one actuator schedule, which is then output to a control system CS in order to be applied to the intended at least one actuating unit of water network 17 via an output interface 18. The actuating units of water network 17 are the pumps 20 for boosting the head at particular locations inside the network, the pumps and valves at the water storage facilities 21, and the various types of valves inside the network.

System 16 contains further a scenario generating unit SG. The scenario generating unit SG is arranged to generate scenarios ooj for the future water demand from uncertainties for the water demand derived from real-world data of the water network, i.e. from historical measurement data MD which include measurement data of the water demand. The measurement data MD are stored in a storage unit 23 which belongs to system 16. Storage unit 23 can be either a volatile data memory, e.g. a RAM, or a permanent data memory.

The scenarios are assumptions for the water demand of the future, where the expected water demand varies for example depending on the time of the day, the week day or the time of the year. Accordingly, day-night variations, higher water demands during hot and sunny months of the year or due to mass events can be taken into account by scenario generating unit SG. In addition, the scenario generating unit SG may take into account additional information which may implicitely influence the water demand, such as information on electricity tariffs ET and/or on water source costs WSC, which may be stored in storage unit 23 or which may be input directly to the system by an operator. Further such additional information may be outdoor temperatures, wheather forcasts and information on public mass events. Besides the modeling of scenarios for the water demand, the scenario generating unit SG may for example generate scenarios for the variation in the efficiency of the water pumps in the water network. For this purpose, pump efficiency characteristics may be stored in storage unit 23 and may be used by the scenario generation unit together with measurement data on the operation of the water pumps to model the future uncertainty in the water pumps efficiency.

The finite set of scenarios for the future water demand, generated by the scenario generating unit SG, is then provided to the optimizing 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 operational constraint G (see equations (4) and (7)) and at least one objective function Φ for representing an undesired side effect U of operating the at least one actuating unit, modeled for example as energy consumption cost Φρυ and source flow cost Φ8η. The optimization problem may even further take into account variations in parameters from which the at least undesired side effect depends on, such as variations in the electricity tariffs ET which influence the energy consumption cost Φρυ and variations in tariffs for water source costs WSC which influence the source flow cost Φ8η. Accordingly, optimizing unit OP is adapted to obtain the respective variations of the parameters from storage unit 23.

The scenarios may also be presented via a human-machine interface to an operator, which may amend them according to his experience.

In particular 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 adapted to provide the water demand scenarios to a graphic display, for visualization for example in form of a tree as in Fig. 2 or in 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 graphical form.

The system 16 contains further a level adjusting unit LA which is adapted to determine a dependency between a decreasing in a lower level bound of a water storage facility and an extent to which the at least one undesired side effect U is reduced while simultaneously abiding to the at least one operational constraint G and to provide to the optimizing unit OP as one updated operational constraint G to the optimization problem a new value for the lower level bound. An example for such a dependency is depicted in Fig. 6, where energy savings are shown over a varying lower bound. The energy savings are in fact an inversed undesired effect U in the sense the undesired effect are the monetary energy costs. The higher the energy costs, the lower the energy savings. The hatched area indicates an area where none of the at least one operational constraints G is violated.

The level adjusting unit LA may further be adapted to determine the new value for the lower level bound by further taking into account a dependency between the reduction in the undesired side effect U and a risk for the water storage facility to reach a critical water level. An example for such a dependency is shown in Fig. 7 as a risk level over energy savings. The higher the energy savings, the higher the risk to reach the critical water level. Based on such a relationship, an optimum can be found at a risk level which the operator of the water network is willing to take.

The above named dependencies may be determined in the form of functional relationships, or tables or rule based relationships.

Another element which may contained in system 16 is a decision support unit DS. The decision support unit DS receives from the scenario generating unit SG the finite set of scenarios and uses it as input for evaluating for an identified potential cause of a fault in the water network 17 a corresponding impact on the water network 17. A method performed by the decision support unit may contain the steps of a) receiving a fault notification indicative of a problem in the water network 17; b) determining from the fault notification a series of potential causes of the notified fault; c) determining, for each potential cause, an estimated impact; d) aggregating the estimated impacts for each potential cause to derive an importance indication for the notified fault.

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

The output of the decision support unit DS, i.e. the importance indication, may then be used by the control system CS to determine how quickly the notified fault requires a response and/or a priority or order in which a series of notified faults should be reacted to.