JP3425160B2 - Water supply operation planning method - Google Patents

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for preparing an optimum operation plan for a water supply pipe network for waterworks.

[0002]

2. Description of the Related Art With the increase in water demand due to the concentration of population in urban areas, water facilities have become large-scale and complex, and it has become extremely difficult to control the entire system efficiently and smoothly. On the other hand, the development of a new water source requires a great deal of time and money, so there is a growing need to make effective use of limited water resources, and computer systems are being actively introduced into the water supply business. Has been. The operation plan is to perform demand forecasting, water intake planning, distribution planning, etc. for the purpose of comprehensive management and control of all systems from water intake to water distribution.

In the conventional operation plan, the plan is divided for each region and each time because of the constraints of the storage capacity of the computer, the calculation speed, and the like. First, the water system is divided into zones, each of which is a regional unit, and the distribution calculation that satisfies the forecast demand for each zone is performed. At this time, the amount of water stored in the zone is planned in advance, and the amount of water transfer and water transmission between zones is calculated at each time. When the allocation between zones is determined, the allocation within the zone is calculated next. This method is based on Matsumoto: "Large-scale water and space water supply space-time operation planning system"
Japan Automatic Control Association Multipurpose System Research Subcommittee No.79-4,11
Have been introduced to.

The problem of planning water intake and flow according to demand is to add the transportation cost and the purification cost to each arc on the network of water conduits to evaluate the total cost of water transfer from the intake source to the reservoir. It can be formulated as a minimum cost flow problem that minimizes it as a function. Various mathematical solutions such as the simplex method have been devised as the solution, and are introduced in Chapter 4 of "Network Theory" by Iri and Furubayashi (Nikka Giren), for example. further,
In the case of a network with holders, it is possible to treat as a model a multi-layered extended network in which individual networks corresponding to each time are connected by arcs corresponding to the amount of stored water. Chapter 5 of "System Optimization Theory" (Corona Publishing) p392-p3
It is written in 97. In addition, as a solution of the dynamic optimization problem of the system, T.Nishiya: CONTROL AND DYNAMIC SYST
EMS, Academic Press, inc. Vol.42 p341-p370.

[0005]

In the conventional operation plan,
Since the problem was divided temporally and spatially, there were cases where it was not possible to allocate even if a solution existed when the conditions were severe. In addition, it was not possible to calculate a suboptimal solution that satisfies all possible conditions when it cannot be allocated. Collective calculation by dynamic optimization is desirable in order to reliably obtain the planned value that effectively uses the water storage capacity of the reservoir and realizes stable supply. However, when dynamic optimization is performed, the scale of the problem becomes large, and it is difficult to find a solution in a limited calculation time with an ordinary algorithm, which is a situation that cannot be realized.

Furthermore, in actual operation, it is desirable that the amount of purified water does not fluctuate with time so that the purification facility is not burdened, and a constant flow rate is better in terms of easiness of valve operation. Further, when water is supplied by a pump, it is more efficient to always switch the number of units at the maximum discharge amount rather than continuously adjusting the flow rate by the number of rotations. For this purpose, it is necessary to set a discrete flow rate according to the capacity of the pump. In ordinary mathematical programming, all conditions required for operation are expressed as equalities and inequalities in advance and formulated. As conditions for a consistent flow that satisfies the demand, the upper and lower limits of the flow rate of each arc on the network are easily expressed as inequalities, and the material balance on each node is easily expressed as an equality condition. However, if all the conditions of such various properties are expressed as mathematical expressions, the problem becomes complicated. In addition, there is a limit in accurately describing the conditions, and there is a problem that the desired planned value is not always obtained in actual operation. An object of the present invention is to provide a method for rapidly obtaining a solution that satisfies various conditions and can withstand actual operation.

[0007]

[Means for Solving the Problems] In order to achieve the above-mentioned object, the intake source, the water treatment plant, the reservoir, the branch point are regarded as the nodes, the water pipes and the water pipes are regarded as arcs, and the water supply pipe network of the water supply is
Express as a spatial network. These are constructed and arranged by the number of unit times for each time when operation is planned, and the nodes corresponding to the reservoirs of each spatial network are connected by an arc representing the amount of stored water. In addition, each virtual intake point is set as a source node and each virtual consumption point is set as a sink node.
The source node and each intake source and reservoir of the spatial network at the initial time are connected by an arc that represents the initial storage volume, and the sink node and each intake source and distribution reservoir of the spatial network at the final time are connected by an arc that represents the final storage volume. tie. With a multi-layered expansion network configured in this way,
The model is used to express the temporal change of the entire water system. The distribution plan is constrained by the material balance at each node on this model and the upper and lower limits of the amount of water transferred, the amount of water transferred, and the amount of stored water at each arc, and the amount of water transferred at each arc, the amount of water transferred, and the unit amount of stored water. Considering the minimum cost flow problem on the extended network that minimizes the total cost obtained from the cost coefficient that represents the cost per hit,
This is obtained by mathematical programming. Next, by dynamically improving the minimum cost flow obtained by the mathematical programming so as to satisfy other operational constraints, a dynamic planned value for the amount of water transfer, water transfer, and water storage is planned. To do. The above is the outline of the solution.

Since the above minimum cost flow problem is a kind of linear programming problem, the simplex method can be applied. With the variables of the amount of water transfer, the amount of water transfer, and the amount of water storage of each arc that composes the expansion network, the equation of the material balance that should be established at each node and The inequality for the upper and lower bounds is the constraint,
The evaluation cost is the total cost determined by the flow rate at each arc and the cost coefficient. Since the time direction is also taken into consideration, the above minimum cost flow problem becomes a multistage linear programming problem, and the division solution method is applied to this. First, the whole problem is divided into small problems for each stage corresponding to each planned time. Then, the variables included in each stage are divided into non-basic variables that take upper or lower limits and non-basic variables that take values determined from the non-basic variable values and the equality condition, and the equality condition of each stage Using the inverse matrix of the base submatrix composed of the coefficients of the base variables, the relative cost coefficient that represents the change amount of the total cost when the unit amount of each non-base variable is changed is obtained. Repeat the procedure of changing the combination of the basic variable and the non-basic variable to reduce the total cost by changing the non-basic variable with the largest proportion to reduce the total cost until the other base variables reach the upper and lower limit values. .

When setting the cost in the minimum cost flow calculation, a negative cost coefficient with a large absolute value is used when the amount of water transfer or water transmission is below the lower limit, and a positive cost with a large absolute value is exceeded when it exceeds the upper limit. Set the coefficient. In addition, if the water storage amount is below the facility lower limit, a negative cost coefficient with a sufficiently large absolute value is set, and if the water storage amount is below the operating lower limit, a negative cost coefficient is set that is smaller but sufficiently larger than when the absolute value is below the facility lower limit. However, if the water storage capacity exceeds the facility upper limit, a positive cost coefficient with a sufficiently large absolute value is set, and if the water storage capacity exceeds the operation upper limit, a positive cost coefficient with a smaller but sufficiently larger absolute value than the facility upper limit is set. To do. In this way, the cost function in which the cost coefficient changes stepwise according to the value is used so that the solution can be reliably calculated even when the allocation is impossible.

Further, when it is necessary to fix the amount of water to be introduced, the amount of water to be sent, and the amount of stored water to specified values, a negative cost coefficient whose absolute value is sufficiently large below the specified value and an absolute value is sufficiently large above the specified value. Set a positive cost factor.

The improvement of the minimum cost flow is to search for a path connecting between two reservoirs including a pipeline whose flow rate is to be corrected, and to find the same path on two different layers on the extension network.
Between the layers, a loop consisting of a path that passes through an arc corresponding to the amount of water stored in the two reservoirs is constructed, and correction is performed on the loop.
Furthermore, a path with a zero cost is searched for among the paths connecting the two reservoirs, including the pipeline whose flow rate is to be corrected, and the path between the reservoirs on the layer corresponding to the time to be corrected on the extended network, A loop is constructed by connecting a path reaching the sink node through an arc corresponding to the amount of water storage from the time to be corrected to the final time of the two reservoirs, and the correction is performed on this loop. The target flow rate is approached by repeating these corrections.

[0012]

[Operation] By expressing the water supply pipe network as a multi-layered expansion network in which the networks corresponding to each time are connected by the arc corresponding to the water storage amount, the reservoir water storage amount is It can be regarded as the flow rate of the connecting pipeline, and the processing can be performed without having to distinguish it from the normal pipeline flow rate. By using such a model, the dynamic allocation plan can be reduced to the problem of finding the minimum cost flow on the extended network, and the dynamic plan can be easily calculated. However, if all conditions of various properties required for operation are expressed as mathematical expressions, the problem becomes complicated and the variables are doubled for dynamic optimization. It is difficult to find a solution in. In addition, there are limits to how accurately the conditions can be described, and they are not always desirable planned values in actual operation. Therefore, first, the minimum cost flow that satisfies the minimum necessary conditions for a consistent flow is obtained by limiting only the material balance at each node of the expansion network and the upper and lower limits of the flow rate of each arc. This problem is easy to formulate, and it is possible to find a solution at high speed by mathematical programming. Next, by improving the minimum cost flow while minimizing the increase in cost so as to satisfy other constraints required for operation, the amount of water transmission and transmission that can withstand actual operation satisfying various conditions. It is possible to quickly obtain dynamic planned values such as water volume and water storage volume. The above is the main principle of problem solving.

The above minimum cost flow calculation can be formulated as a multi-stage linear programming problem, and the coefficient matrix of the overall problem has a shape in which small matrices are arranged in a stepwise manner in the diagonal direction. By expressing the basis matrix as a product of a matrix in which a small diagonal matrix is a square matrix and a correction matrix, the whole problem can be divided into small problems in each stage. Next, the relative cost coefficient is obtained from the inverse matrix of the base matrix and the correction matrix of each stage, and the minimum cost flow is obtained by repeating the same procedure as the ordinary simplex method. By converting the base inverse matrix calculation into the repetition of the inverse matrix calculation of the small matrix, the calculation amount is greatly reduced. Also, the coefficient matrix consisting only of the material balance equation is
Since the elements are only 1, -1, 0 and the basis inverse matrix, the correction matrix, etc. all represent the exchange of substances,
Its elements are only 1, -1, 0. By utilizing this property, all matrix calculations can be expressed by addition and subtraction. This not only leads to great labor saving on a computer, but also has the advantage that no calculation error occurs.

When the formulation as a multistage linear programming problem is performed as described above, the minimum cost flow can be obtained at high speed.
If it cannot be allocated, no feasible solution exists and no plan value can be obtained. In the actual operation plan, there is a demand to calculate an incomplete plan value that divides the lower limit of some reservoirs and use it as a guideline for operation. Therefore, the solution can be calculated without fail if the staircase cost is obtained by adding a high cost outside the upper and lower limits of the reservoir water storage volume and pipeline flow rate. At that time, if the cost outside the upper and lower limits is set so that the cost of the flow is sufficiently larger than the cost of the stored water, the incomplete planned value will not exceed the upper limit of the flow or flow backward. You can In addition, it may be necessary to secure as much water storage as possible from the viewpoint of safety by drawing as much water as possible until the limit is reached. Under these conditions, by adding a positive cost to the storage volume of the intake source and a negative cost to the reservoir storage volume, the storage volume of the intake source becomes
Reservoir water storage will be as low as possible and as high as possible, which will be reflected in the planned values. Further, when it is desired to specify the flow rate, a negative cost coefficient whose absolute value is sufficiently large is set below the specified value, and a positive cost coefficient whose absolute value is sufficiently large is set above the specified value. Other than the specified value, the cost is set to be extremely high, so that the specified value is planned.

In this way, the minimum cost flow is obtained, and the solution is heuristically modified so as to satisfy other conditions, and is set to a planned value that can withstand actual operation. For example,
A planned value with little fluctuation in flow rate is desirable for the pipeline that adjusts the flow rate by valve operation, and the average flow rate is for a certain period of time, and for the pipeline that conducts water supply by the pump, it is a discrete value according to the pump discharge rate. These are calculated as target flow rates based on the solution obtained by the minimum cost flow calculation. The improvement method is
Search for a path that connects two reservoirs, including the pipeline whose flow rate is to be modified, and find the same path on two different layers on the extended network, and the two layers are equivalent to the reservoirs of two reservoirs at both ends. It is improved by constructing a loop through the arcs that perform and adding a correction flow rate on this loop. this is,
The flow rate of the pipeline between the two reservoirs will be increased or decreased by the corrected flow rate at different times, and the overall cost will not change. Alternatively, a path with a zero cost is searched for among paths connecting two reservoirs that include a pipeline whose flow rate is to be corrected, and the path with a zero cost is found on the layer corresponding to the time to be corrected on the extended network. Improve by constructing a loop that connects the path reaching the sink node through the arc corresponding to the reservoir volume from the desired time to the final time of the two reservoirs and adding the corrected flow rate on this loop To do. This will modify the flow rate of the pipeline without transportation costs and change the final reservoir volume of the two reservoirs at both ends accordingly. The target flow rate is approached by repeating these modifications.

[0016]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 1 is a diagram showing a general water system of water supply and an operation mode by application of the present invention. Water intake source 11
Raw water taken from -1, 2 reaches the water treatment plants 12-1, 2 through the water conduits 15-1 to 3 and the purified water purified there is distributed by the water pipes 16-1 to 16 in each region. Water pond 13-
Water is distributed to 1 to 5, and purified water is distributed to consumers from each distribution reservoir. It is necessary to operate the water system so as to always satisfy different demands in each region and each time, and to plan the intake amount, the flow rate of the pipeline, the reservoir storage amount, etc. Since demand changes due to various factors such as daily weather conditions, it is necessary to plan the plan promptly every time. However, as demand increases and it is difficult to secure a water source, the operating conditions become strict, and it is necessary to make effective use of limited resources. Therefore, the operation center 14 is provided, data is collected by the information networks 18-1 to 18-8 to make a plan, and the planned value is transmitted to each facility, whereby the entire water system is comprehensively managed.

FIG. 2 is a diagram showing an outline of the operation plan. First, the demand performance data 22 is used to analyze the demand forecast parameters (21). From the weather and other factors, the predicted value of demand at each reservoir is calculated based on this parameter (23). Next, the planned water withdrawal amount is determined in consideration of the forecast demand amount and the sales contract with other groups such as adjacent cities (24). The pipe network data 26 is a relatively small change in the setting conditions necessary for distribution calculation, and is pipe network connection information, facility upper and lower limits such as flow rate and water storage amount, and cost coefficient for water supply and purification. Setting conditions that change relatively frequently are
It is set by the operator via the input terminal 27. For example, the initial storage amount of each distribution reservoir, the priority order of smoothing, the designated flow rate value, and the like. Using these series of data and conditions as input, a distribution plan is made (25), and planned values such as the amount of water stored in each reservoir and the flow rate of each pipeline are determined. The present invention provides an algorithm for highly and rapidly planning a distribution plan that occupies a central position in an operation plan. The contents will be described below.

A processing flow of the distribution plan 25 according to the present invention is shown in FIG. First, various conditions necessary for calculation are set, and the problem is formulated (31). Here, the input items are organized.

1) Predicted demand amount ... The demand amount of each distribution reservoir on an hourly basis, which is the output of the demand forecast 23.

2) Planned water intake: Total water intake to be planned from the total demand and the water distribution amount and water intake amount under the sales contract with other groups,
It is an output of the water intake plan 24.

3) Pipe network connection information: information representing how each facility, branch point, and pipeline are connected.

4) Facility conditions: upper and lower limit values of water storage amount in distribution reservoir, upper and lower limit values of pipe flow rate, number of water supply pumps and discharge amount thereof.

5) Cost coefficient: Purification cost and water supply cost per unit flow rate.

6) Flow rate switching time: The switching time when the flow rate is controlled by a pump or a valve.

7) Designated flow rate: This is a flow rate at a designated place which is not allowed by other flow rates for the convenience of operation.

8) Smoothing priority order: Settings for smoothing processing described later.

9) Initial water storage amount ... The water storage amount at the initial time of the plan, which is collected from each facility through the information networks 18-1 to 18-8.

After setting the conditions, the distribution plan is formulated as a minimum cost flow problem and calculated (32). Here, the minimum cost flow that satisfies only the conditions for material storage and the upper and lower limits of water storage and flow rate is obtained. Although such a minimum cost flow can be calculated at high speed by mathematical programming, it is often not applicable to actual operation because the flow rate fluctuates over time. Therefore, the solution is improved so as to satisfy other various conditions required for operation, centering on smoothing of the flow rate (33). The operation planner judges the planned value obtained by performing the above processing (34), and if it is judged to be inappropriate, the setting condition is changed and the processing is repeated again.

Here, the concept of handling the allocation plan as a minimum cost flow problem will be described. FIG. 4 is a diagram showing a model used in that case. Consider the distribution plan as a problem that determines the flow from source 36 to sink 37 in a multi-layered network as shown. 38-1 to 38-3 are networks corresponding to the water system at each time. 39-1 to 3 are water intake sources, 40-1 to 3 are branch points, 41-1 to 3 and 42-1 to
3 represents a reservoir. Arcs 51-1 to 52 in the layer
-1 to 53-1 to 3-4 and 54-1 to 5-3 represent the flow rates of the pipelines with no downflow delay. 44-0 of the arc between layers
An arc connecting nodes corresponding to the same reservoir, such as 3, 45-0 to 3, represents the amount of water stored in each reservoir. Therefore, the arcs 44-0 and 45-0 from the source represent the initial water storage amount, and the arcs 44-3 and 45-3 to the sink represent the final water storage amount. The water source will also be treated as a large reservoir and treated in the same way. At this time, the upper limit amount of water intake can be set by the arc 43-0 from the sink.
The arc between the layers, which connects the nodes corresponding to the different facilities, represents the flow rate of the pipeline that causes the delay. If the network expanded in this way is used as a model, the water storage volume can be treated exactly like the discharge volume. By adding the cost coefficient and upper and lower limits to each arc and calculating the minimum cost flow on the expansion network, it becomes possible to calculate a dynamic plan value that effectively uses the reservoir water storage volume.

The minimum cost flow problem modeled on the extended network described above can be formulated as a multistage linear programming problem as described below. First, the flow rate of each arc is x
_i (t) (i = 1 ... n, t = 1 ... T). The subscript i corresponds to the actual facility, and n is the total number of intake sources, distribution reservoirs, and pipelines. The subscript t corresponds to the time, and T is the planned unit time. The condition for material storage at each node on the extended network is (total inflow) = (total outflow) +
Since it is demand,

[0031]

[Equation 1]

It is represented by Where x (t) is x _i
An n-dimensional vector having (t) as elements, A ₁ and A ₂ are m × n
The matrix, b _t, is an m-dimensional vector representing the demand amount. However, the number of nodes in the layer of the extended network, that is, the total number of water intake sources, distribution reservoirs, and branch points is m. Also, x
(0) is an initial water storage amount, b _t (t = 1 ... T) is a forecast demand amount, and is a constant input in advance. Furthermore, x
If the vector formed by arranging all the elements of (t) and b _t (t = 1 ... T) is X and b, then Equation 1 is

[0033]

[Equation 2]

It becomes Here, A is a matrix having a staircase structure as shown in FIG. The small matrices A ₁ and A ₂ arranged in the diagonal direction are −
It is a sparse matrix having 1, 0, 1 as elements, and the others are all zero matrices. Similarly, if the upper and lower limits of the flow rate of each arc and the cost coefficient are vectors l, h, and c, it goes without saying that they are inequality conditions.

[0035]

[Equation 3]

Must be satisfied. If the total cost is z,

[0037]

[Equation 4]

[0038] As above, the minimum cost flow problem is
It was formulated as a multi-stage linear programming problem of finding X that minimizes the cost expressed by the equation 4 under the constraint shown by the equation 3.

Next, a method of solving the minimum cost flow problem by the multistage linear programming will be described. The following is a part corresponding to the process 32 of FIG. For the minimum cost flow problem that has only the material balance condition as the constraint equation, the non-zero elements of the coefficient matrix and the basis inverse matrix are all 1 or -1. This has the following advantages on a computer.

(1) Most of the calculations can be expressed as addition and subtraction, and division can be eliminated. Therefore, if all coefficients and constants are integers, all variables used on the computer can be defined as integers. This means that the product-sum operation of real numbers can be converted into the summation of integers, which greatly reduces the calculation time.

(2) When performing iterative calculation on a computer, the problem of error management must always be considered.
However, since they are all integers and there is no division, the error is always zero, which eliminates the need for checking for error management. If programming is performed in consideration of the above two points, the speed can be significantly increased.

Further, focusing on the staircase structure of the coefficient matrix of the multi-stage linear programming problem, a fast solution method based on the simplex method and applying the basis decomposition method is applied. In the case of the ordinary simplex method, the initial feasible solution is obtained by setting a problem with an artificial variable added, but in the case of the multistage problem as described above, the problem becomes even larger. Therefore, the procedure shown in FIG. 7 is used to prevent the problem scale from increasing. Now, a problem with the final time being k will be represented as P (k). First, with k = 1, solve P (1) (6
1). Next, the number of time steps is increased by 1, and the initial value of the added variable x (k) is set to the final time x of the solution of P (k-1).
It is set to (k-1) and the sensitivity analysis is applied to add an artificial variable only to those that do not satisfy the condition to obtain an initial feasible solution of P (k) (62). Then, P (k) is optimized. This operation is repeated to increase the number of time steps until k = T (64). In this way, the problem can be solved with the addition of artificial variables minimized.
The above is the general flow of the multi-stage linear programming.
Does not necessarily have to be performed for all the times k, and the process 62 can proceed to the process 63 by obtaining an initial solution of the problem to which several times have been added.

A method of solving P (T) when k = T in the process 63, that is, P (T) will be described below. In the simplex method, a large amount of calculation is required for the processing related to the base inverse matrix. Therefore, by expressing these processing in a form of sequentially performing the inverse matrix calculation of a small matrix, the storage capacity and the amount of calculation are reduced. . Therefore, first, the calculation of the simplex multiplier and the relative cost coefficient will be described.

Now, in the problems of the expressions 2 to 4, let us consider a basis matrix in a special state where the number of bases shown in the expression 5 is exactly m from each time.

[0045]

[Equation 5]

At this time, the calculation of the simplex multiplier can be expressed in a form in which the inverse matrix calculation of the diagonal small matrix is sequentially performed. However, in general, there is also the exchange of bases over time, so that the structure as shown in Formula 5 is not obtained. Therefore, the basis matrix B is expressed as follows.

[0047]

[Equation 6]

Here, among the column vectors forming the original basis matrix, B _{1 is} a matrix that collects the column vectors that are not included in the special basis matrix, and B _{1 is} the matrix that is also included in the original basis matrix. If the above matrix is B ₂ , then F will have the following structure by appropriate substitution.

[0049]

[Equation 7]

Using the above expression, the inverse matrix of F can be calculated as follows.

[0051]

[Equation 8]

Based on the above preparations, the procedure for calculating the simplex multiplier will be described. If the simplex multiplier is π and the cost coefficient vector corresponding to the basis variable is c _B , then using Equation 6,

[0053]

[Equation 9]

The relationship can be derived. Here, corresponding to the decomposition of the basis matrix in Equation 7,

[0055]

[Equation 10]

[Mathematical formula-see original document] From Equation 8 and Equation 9,

[0057]

[Equation 11]

It becomes From number 9,

[0059]

[Equation 12]

Therefore, when the calculation is performed for each submatrix in consideration of the structure shown in Expression 5, the following equation is obtained.

[0061]

[Equation 13]

However, the simplex multiplier π and the cost coefficient vector c _B ′ are expressed as a combination of m-dimensional vectors as follows.

[0063]

[Equation 14]

[0064]

[Equation 15]

From Equation 13, the following recurrence formula is obtained for the simplex multiplier.

[0066]

[Equation 16]

Utilizing the above relationship, the simplex multiplier of a large-scale matrix can be decomposed into repeated small matrix inverse matrix calculations, and both the storage capacity and the calculation amount can be greatly improved.

Next, the processing procedure of the P (T) solution will be described. In this solution, the variable to be optimized is x _B
It is characterized in that the calculation amount is reduced by repeatedly solving the problem limited to x _N (t) for t = 1 ... T. FIG. 7 shows a processing flow chart. First, the stage t to be optimized is set to the final time T (71). Next, the simplex multiplier and the relative cost coefficient are calculated (72). The simplex multiplier is obtained by the calculation procedure described above. On the other hand, the relative cost coefficient vector λ, where N is a non-basic matrix and c _N is a cost coefficient vector corresponding to a non-basic variable,

[0069]

[Equation 17]

However, like B, N is also a large-scale matrix, and therefore πN is not easy to calculate. However, in a limited problem, the calculation of the relative cost factor is performed in the t-th stage, namely

[0071]

[Equation 18]

Only need to be performed. However, N _{t, t−1} and N _{t, t} are small matrices obtained by dividing N in the same manner as in the _equation 5.

Thus, if the following relation is satisfied for the calculated relative cost coefficient of the t-th stage and the non-basic variable of the t-th stage and all the upper and lower limit elements j, the optimization is completed at time t ( 73).

[0074]

[Formula 19]

If it is not optimal, for j that does not satisfy the above condition,

[0076]

[Equation 20]

Select the number s which becomes (74),

[0078]

[Equation 21]

Thus, the solution is updated (75). Here, θ is the minimum number at which any of the updated variables reaches the upper and lower limits. If the variable that reaches the upper and lower limits first is non-basic, no basis transformation occurs, so there is no need to recalculate the relative cost coefficient, and the optimality is checked again (76). The variable that first reached the upper and lower limits is
In the case of the basis, the basis variables are exchanged, and the basis inverse matrix and the correction matrix are updated (77). The number 2
The calculation of the base inverse matrix in 1 can be speeded up by the same calculation as the simplex multiplier. The above process is repeated until the condition of the optimality, Formula 19, is satisfied, and when optimized, the time is decremented by one (78) and the same is repeated until t = 1 (79). If there is a base conversion even once in all stages (80), the process from the beginning is repeated again.
As described above, the algorithm that attempts to thoroughly reduce the calculation amount enables the minimum cost stream to be processed at high speed.

In the above linear programming method, when a fixed cost coefficient is used for each variable, no feasible solution exists when it cannot be allocated, so that calculation becomes impossible and the planned value cannot be obtained. Therefore, by adding a large cost to the area outside the upper and lower limits, in practice, there is always a feasible solution, and it is possible to calculate even when it becomes impossible to allocate due to severe conditions such as water shortage or pipeline damage. To do so. FIG. 8 shows the cost function used in the minimum cost flow calculation. (A) shows the cost function of the pipe flow rate (b) reservoir water storage amount. All are stair costs consisting of multiple sections. If the absolute values of these cost coefficients satisfy the following conditions, it is possible to obtain a planned value that serves as a guideline for operation.

(1) | Maximum value of cost coefficient within upper and lower limit of pipeline | << | Cost coefficient outside upper and lower limit of reservoir operation | The transportation cost to send to a certain reservoir rises when the lower limit of the reservoir is divided If the cost is lower than that, the lower limit of the reservoir will be broken even though there is a margin for water intake.

(2) | Cost coefficient outside upper and lower limit of reservoir operation |
<| Distribution reservoir upper / lower cost coefficient outside the limit | In normal operation of a reservoir, the upper and lower limits are often set within a range with a margin left over from the capacity of facility design for safety.
However, if it is unavoidable, the operation should be performed while allowing the state within the upper and lower limits of the equipment beyond the upper and lower limits of the operation. In order to reflect such operational convenience, the absolute value of the cost coefficient outside the operational lower limit is made slightly smaller than the absolute value of the cost outside the facility upper and lower limit.

(3) | Cost coefficient outside upper and lower limits of distribution reservoir equipment |
<< ｜ Cost coefficient outside upper and lower limit of pipeline | When allocation is not possible, we would like to limit only the reservoir to the variable that divides the upper and lower limits. This is because a solution that does not satisfy the upper and lower limits of the reservoir is more useful for obtaining information about when and where water is insufficient than a solution that flows backward through the pipe or exceeds the flow rate limit. If, for example, if backflow for 1 hour is allowed and upper and lower limit cracks in the reservoir are eliminated for 10 hours, the desired solution cannot be obtained simply by using | the cost coefficient outside the upper and lower limits of the reservoir | .

(4) | Pipeline lower / lower bound cost coefficient | << |
Artificial Variable Cost Factors | An artificial variable must be larger than any other cost.

In addition, for operational convenience, it may be necessary to collect as much water as possible until the limit value is reached, and to secure as much water storage amount as possible in the distribution reservoir from the viewpoint of safety. Under these conditions, by adding a positive cost to the storage volume of the intake source and a negative cost to the storage volume of the reservoir, the storage volume of the intake source is as small as possible and the storage volume of the reservoir is possible. It will be as much as possible and will be reflected in the planned value. Also,
When it is desired to specify the flow rate, the specified value becomes the minimum as shown in (c), and the specified value can be obtained by adding a sufficiently large absolute cost.

By using the above-described algorithm and cost function, the minimum cost flow problem on the extended network can be solved very quickly. However, the solution obtained here is the minimum cost flow that satisfies only the minimum conditions for the consistency of the material balance and the upper and lower limits of the flow rate.
In a pipeline in which the flow rate is controlled by a valve, a flow rate with drastic time fluctuation is not desirable. In addition, for a pipeline that is pumped by a pump, a flow rate corresponding to the pump discharge amount may be required. At this stage, these conditions are not taken into consideration. Incorporating the mathematical conditions into the above minimum cost flow problem not only increases the number of conditions all at once, but also makes the inverse matrix calculation complicated and requires error management, resulting in loss of speed. Therefore, the desired planned value is obtained by improving the above minimum cost flow problem.

The method of improving the solution, which is the portion corresponding to the process 33 in FIG. 3, will be described below. First, a target flow rate is set for each pipeline that requires smoothing. For example, for the pipeline that adjusts the flow rate by operating the valve,
A planned value with little fluctuation in flow rate is desirable, and it is the average flow rate during a certain time period. In addition, in the pipeline where water is sent by the pump, it becomes a discrete value according to the pump discharge amount. These are calculated based on the minimum cost stream.

Next, the process of heuristically searching for a correctable portion is repeated. Even if a flow is added on a path where the sum of cost coefficients is 0, as a result of the addition, the cost does not change as long as the flow rate of each arc forming the path is within the upper and lower limits. Utilizing this, a path with cost 0 is searched for on the extended network, and an appropriate flow is added to bring the path closer to the target amount. At this time, if the path to be modified is a loop in which the start point and the end point match, the condition of the material balance at each node is satisfied even if the path is modified.
However, searching all suitable loops on a large scale network still requires a huge amount of calculation time. Therefore, by limiting the shape of the loop to two types of loops, the search time is shortened and smoothing processing is performed at high speed. Examples of these two types of loops are shown in FIG. 9 and FIG. 1, respectively.
It shows in 0.

Now, it is assumed that the flow rate of the conduit u is corrected so as to approach the target value. A method of configuring the loop 101 of FIG. 9 will be described. Select the appropriate two reservoirs s and t,
A route including the pipeline u is searched for in the route from the distribution reservoirs s to t. Next, the time i at which the flow rate of the pipeline u is the lowest and the time j at which the flow rate is the highest are selected. 91-i and 91-j are times i of the extended network.
And the spatial network corresponding to j. Also, 92-
95-i, 96-98-i, 92-95-j, 96-9
8-j are nodes and arcs corresponding to the reservoirs, branch points, and pipelines of the searched routes on these two different layers. Furthermore, 99-i and 99-i + 1 are the water storage amount of the reservoir s,
100-i and 100-i + 1 are arcs corresponding to the amount of water stored in the reservoir t. The loop 101 is a connection of these arcs on the extended network. That is,
In the two different layers corresponding to the time to be corrected, the paths corresponding to the searched routes are connected to the two layers by connecting the paths passing through the arcs corresponding to the water storage amounts of the two reservoirs at both ends. Add a correction amount along this loop. If the loop direction and arc direction are opposite, the correction amount is reduced. Therefore, the transportation direction of the route to be searched does not necessarily have to be a fixed direction. The loops as described above pass through corresponding arcs in the bed in opposite directions, thus offsetting the cost of flow. If the two storage reservoirs have the same storage cost coefficient, the loop cost will always be zero. The modification on this loop is to increase / decrease the flow rate of the pipeline from the reservoirs s to t at time i and j, and to change the reservoir water volume at both ends accordingly from time i to j-1. Equivalent to.

A method of constructing the loop 102 of FIG. 10 will be described. Similar to the case of the loop 101, appropriate two distribution reservoirs s and t are selected, and a route including the pipeline u is searched for in the route from the distribution reservoirs s to t. Further, among the searched routes, a route with a cost of zero is selected. On the extended network, the time to correct the flow rate is i,
The path corresponding to the searched route in the layer corresponding to the time i and the arcs 99-i, i + corresponding to the amount of stored water from the nodes corresponding to the two reservoirs at both ends to the sink node.
, E and 100-i, i + 1, ..., E are connected to form a loop 106. If the storage cost coefficients of the two reservoirs are the same, the cost of this loop will also be zero. The modification on this loop is equivalent to modifying the flow rate of the pipeline from the reservoirs s to t at no cost to the time i, and changing the reservoir water volume at both ends from the time i to the final time accordingly. To do.

Next, FIG. 11 shows a processing flow showing the order of performing these corrections. The correction is performed based on the priority order of the smoothing target pipelines input in advance. Here, the pipeline to be smoothed is always limited to a pipe that directly flows into or out of the reservoir. In general, pumps and valves in water supply are mostly responsible for controlling the outflow from distribution reservoirs, so this limitation has little effect on operational planning.

First, the input smoothing priority and target flow rate are set (111). The priority is set to the highest priority for the pipeline for which the flow rate is specified. The target flow rate will be investigated starting from a distribution reservoir directly connected to the smoothing target pipeline (hereinafter referred to as a direct connection distribution reservoir) and changing the end distribution reservoir in order. For each pair of the starting point reservoir and the ending point reservoir, first, all the paths that connect them are listed (112), and the paths that do not include the smoothing target pipeline and the pipelines that have a higher priority than the smoothing target pipeline are included. The path is deleted (113). For each remaining path, of the flow rates of the smoothing target pipeline at each time point, the time point where the deviation from the target flow rate is the largest in the positive direction and the largest time point in the negative direction is selected, and as shown in FIG. Configure and modify the loop. Repeat on the same path while changing the time until it cannot be corrected. (114) If the cost of the path is 0, the time most distant from the target flow rate is selected, and the path as shown in FIG. 10 is configured and corrected. When all the paths have been corrected, the listed paths are deleted (115), and another pair of reservoirs is processed. In the series of processes described above, the process ends if there are no correction points, and otherwise the same process is repeated from the beginning.

FIG. 12 shows an example of the planned value obtained by applying the present invention. (A) is the amount of water stored in a reservoir in the water system, (b), (c) is the flow rate of the pipeline directly connected to the reservoir,
Inflow side and outflow side, respectively. In both cases, the flow rate is adjusted by pumping water. The data used for the calculation is the case where the inflow rate is significantly limited by the equipment inspection of the inflow side pump from 23:00 to 2:00. Even in such a special case, it can be allocated by effectively utilizing the stored water volume. Further, during the standard switching time set at 17:00, 23:00, and 6:00, there is little fluctuation in the flow rate, and the planned value is in accordance with actual operation.

[0094]

EFFECTS OF THE INVENTION By applying the present invention to a water supply operation plan, it is possible to make a dynamic plan effectively utilizing the reservoir water storage amount. The planned values are not only the most economical operation, but also those that meet the requirements of each facility in the water system, such as the water purification process at the water purification plant, valve operation and pump operation at distribution reservoirs and pipelines. Can be Furthermore, even a large-scale water system operation plan for a large city can be calculated in a planning time that can be practically used by a control computer.

[Brief description of drawings]

FIG. 1 shows a water system diagram of a general water supply system and its operation mode.

FIG. 2 is a schematic diagram of a water supply operation plan.

FIG. 3 is a flow chart of a distribution plan according to the present invention.

FIG. 4 is a diagram showing a multi-layered extended network used as a model when a distribution plan is treated as a minimum cost flow problem.

FIG. 5 is a diagram showing a structure of a coefficient matrix representing an equality condition when the minimum cost flow problem is formulated as a multistage linear programming problem.

FIG. 6 is a schematic flowchart showing a solution when the minimum cost flow problem is formulated as a multistage linear programming problem.

FIG. 7 is a detailed flowchart showing a method for solving a multistage linear programming problem.

FIG. 8 is a diagram showing an example of a cost function used in an operation plan.

FIG. 9 is a diagram showing an example of a correction path on an extended network when using a heuristic smoothing processing method.

FIG. 10 is a diagram showing an example of a correction path on an extended network when a heuristic smoothing processing method is used.

FIG. 11 is a flowchart showing a heuristic smoothing processing method.

FIG. 12 is an example of a planned value of a water supply operation plan that is made by applying the present invention.

─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Takushi Nishitani 1099 Ozenji, Aso-ku, Kawasaki-shi, Kanagawa Hitachi, Ltd. System Development Laboratory (72) Inventor Takeo Kasai 5-2-1 Omika-cho, Hitachi City, Ibaraki Prefecture Issue Hitachi, Ltd. Omika Factory (72) Inventor Nihei Tate 5-21 Omikacho, Hitachi City, Ibaraki Hitachi Ltd. Omika Factory (72) Inventor Shigeyuki Shimauchi 4-6 Kanda Surugadai, Chiyoda-ku, Tokyo Address Hitachi, Ltd. (72) Inventor Akira Hayashi 4, 6 Kanda Surugadai, Chiyoda-ku, Tokyo Address: Hitachi Ltd. (72) Inventor ▲ Takahashi Teruaki 1-1, Minato-cho, Naka-ku, Yokohama, Kanagawa (72) Inventor Susumu Sano 1-1, Minato-cho, Naka-ku, Yokohama-shi, Kanagawa Yokohama-shi water (72) Inventor Masahito Watanabe 1-1 Minato-cho, Naka-ku, Yokohama-shi, Kanagawa Yokohama-shi Waterworks Bureau (72) Inventor Taku Hayasaka 1-1-1, Minato-cho, Naka-ku, Yokohama-shi, Kanagawa Yokohama Waterworks Bureau (56) References JP-A-1-305407 (JP, A) JP-A-3-3015 (JP, A) (58) Fields investigated (Int.Cl. ⁷ , DB name) G05D ^7/ 00-7/06

Claims (3)

(57) [Claims] 1. A water supply pipe network, a water intake source, a water treatment plant, a distribution reservoir, a spatial network with branch points as nodes, water pipes, and water pipes as arcs. The nodes are connected by an arc representing the amount of water storage, and the virtual water intake source is a source node and the virtual consumption points are sink nodes. , The source node and each intake source and reservoir of the spatial network at the initial time are connected by an arc that represents the initial storage amount, and the sink node and each intake source and distribution reservoir of the spatial network at the final time are an arc that represents the final storage amount. Using a multi-layered extended network connected by, the constraint condition is the material balance at each node and the amount of water transfer at each arc, the amount of water transfer, and the upper and lower limits of the amount of water storage.
The minimum cost flow on the extended network that minimizes the total cost obtained from the cost coefficient that represents the cost per unit amount of water transfer amount, water transfer amount, and water storage amount at each arc is obtained by mathematical programming, and the operational requirement In the water supply operation planning method, the minimum cost flow is calculated by improving the minimum cost flow so as to satisfy the other constraint conditions. Is the flow rate of each arc that constitutes the expanded network, the amount of water transfer, the amount of water transfer, and the amount of water storage. Formulated as a multi-stage linear programming problem, with the inequality for the upper and lower limits of quantity as the constraint condition, and the total cost determined by the flow rate and cost coefficient at each arc as the evaluation function, the small scale for each stage corresponding to each planned time After dividing the problem into variables,
It is divided into non-basic variables that take upper or lower limits, and non-basic variables that take values determined by the equality condition and the equality condition. Using the inverse matrix, the relative cost coefficient that represents the amount of change in total cost when each non-basic variable is changed by a unit amount is obtained, and the non-basic variable with the largest proportion that reduces the total cost is It is characterized by finding the minimum cost flow by the simplex method with division calculation, which repeats the procedure of changing the combination of the basic variable and the non-basic variable to reduce the total cost by changing it to the upper and lower limit values. Water supply operation planning method. 2. A water supply pipe network for water supply, a water intake source, a water treatment plant, a reservoir, a spatial network using branch points as nodes, water pipes, and water pipes as arcs. The nodes are connected by an arc representing the amount of water storage, and the virtual water intake source is a source node and the virtual consumption points are sink nodes. , The source node and each intake source and reservoir of the spatial network at the initial time are connected by an arc that represents the initial storage amount, and the sink node and each intake source and distribution reservoir of the spatial network at the final time are an arc that represents the final storage amount. Using a multi-layered extended network connected by, the constraint condition is the material balance at each node and the amount of water transfer at each arc, the amount of water transfer, and the upper and lower limits of the amount of water storage.
The minimum cost flow on the extended network that minimizes the total cost obtained from the cost coefficient that represents the cost per unit amount of water transfer amount, water transfer amount, and water storage amount at each arc is obtained by mathematical programming, and the operational requirement In order to satisfy the other constraint conditions, the minimum cost flow is improved so that a dynamic planned value of the amount of water transfer, the amount of water transfer, and the amount of stored water is drafted. The improvement involves searching for a flow path that connects two reservoirs, including the pipeline for which the amount of water to be introduced and the amount of water to be delivered are to be corrected, and the time when the amount of water to be delivered to the pipeline to be corrected is the highest and the lowest when compared to the target amount. On two different layers of the expansion network corresponding to different times, a path consisting of one or a plurality of arcs corresponding to the same flow path is provided and an arc corresponding to the water storage amount of two reservoirs is provided between the two layers. A path consisting of A method of planning a water supply operation, characterized in that a loop, which is a closed path, selected and connected to each other, is configured, and that the flow rate of each arc is corrected by the same amount along the loop repeatedly. 3. A water supply pipe network, a water intake source, a water treatment plant, a reservoir, a spatial network using branch points as nodes, water pipes and water pipes as arcs, and a unit time for each time of operation planning. The nodes are connected by an arc representing the amount of water storage, and the virtual water intake source is a source node and the virtual consumption points are sink nodes. , The source node and each intake source and reservoir of the spatial network at the initial time are connected by an arc that represents the initial storage amount, and the sink node and each intake source and distribution reservoir of the spatial network at the final time are an arc that represents the final storage amount. Using a multi-layered extended network connected by, the constraint condition is the material balance at each node and the amount of water transfer at each arc, the amount of water transfer, and the upper and lower limits of the amount of water storage.
The minimum cost flow on the extended network that minimizes the total cost obtained from the cost coefficient that represents the cost per unit amount of water transfer amount, water transfer amount, and water storage amount at each arc is obtained by mathematical programming, and the operational requirement In order to satisfy the other constraint conditions, the minimum cost flow is improved so that a dynamic planned value of the amount of water transfer, the amount of water transfer, and the amount of stored water is drafted. The improvement involves searching for a flow path that has zero cost among the flow paths that connect the two reservoirs, including the pipeline where you want to correct the amount of water transfer or water transmission, and on one layer of the expansion network that corresponds to the time you want to correct. Then, the path consisting of one or more arcs corresponding to the flow path, and the
From the node corresponding to one reservoir to the sink node,
It improves by iteratively constructing a loop that is a closed path by selecting and connecting paths each consisting of an arc corresponding to the water storage amount and correcting the flow rate of each arc by the same amount along the loop. A water supply operation planning method characterized by the above.