Control system for flow of turbined water from a plurality of hydroelectric plants

Abstract

A system for controlling the flow of turbined water from a plurality of hydroelectric plants arranged in series along a watercourse with an open channel flow, defining upstream of each plant, a plurality of head races subject respectively to hydraulic flow and level constraints. The flow of water turbined by each of the plants is controlled by a flow setpoint. The system includes regulation of a global electrical production power set-point for the plurality of hydroelectric plants by a flow regulation setpoint taking into account the flow setpoint of each of the plants. The flow regulation setpoint determined by the regulation is weighted for each of the plants by weighting coefficients as a function of the respective hydraulic characteristics of the head plurality of races.

Claims

1. A system for driving a turbined water flow rate of a plurality of hydroelectric plants arranged in series along a watercourse with an open channel flow, defining upstream of each of the hydroelectric plants a plurality of respective reaches subjected to hydraulic flow rate and level requirements, said plurality of hydroelectric plants comprising at least three hydroelectric plants, wherein the turbined water flow rate for each of said plants is controlled by means of a flow rate setpoint on said plant determined from a run-of-the-river flow rate setpoint taking into account a level regulation of the reaches and the inflow rates in said reaches, wherein the system comprises a regulation of an overall power output setpoint for said plurality of hydroelectric plants by means of a regulation flow rate setpoint taken into account by the flow rate setpoint of each of said plants and in that said regulation flow rate setpoint determined by said regulation is weighted for each of said plants by means of weighting coefficients as a function of the respective flow rate and level hydraulic requirements of the reaches defined upstream of said plants in order to meet said flow rate and level hydraulic requirements.

2. The system according to claim 1, wherein the overall power output setpoint corresponds to the sum of a power setpoint of a power program and a balancing power for the power grid to which the plurality of plants is connected.

3. The system according to claim 1, wherein the regulation of the overall power produced by the plurality of hydroelectric plants to meet an overall power output setpoint controls an overall effective power setpoint corresponding to the sum of the effective power setpoints of each plant, the effective power setpoints of each plant being determined by means of the flow rate setpoint of each of said plants.

4. The system according to claim 1, wherein the weighting coefficients .sub.i are dynamic and vary over time.

5. The system according to claim 4, wherein a sum of the weighting coefficients .sub.i applied to the power regulation flow rate setpoint of the plants except for the last plant is equal to the number n of the plants: $\underset{i=1}{\overset{n-1}{.Math.}}{}_i=n.$

6. The system according to claim 4, wherein each of the weighting coefficients .sub.i is determined by minimizing a criterion corresponding to an equation involving the weighting coefficient .sub.i to be determined and the weighting coefficient .sub.i1 applied to the immediately upstream plant, the weighting coefficient .sub.1 of the first plant upstream of the queue being fixed.

7. The system according to claim 4, wherein the weighting coefficients .sub.i are determined by taking into account a level deviation between a nominal level setpoint of the reach i and a prediction of the level of said reach i at an optimisation horizon.

8. The system according to claim 7, wherein taking into account the level deviation takes into account the weighting coefficient .sub.i associated with a plant and the weighting .sub.i1 associated with the plant immediately upstream of said plant in a linear relationship corresponding to said level deviation.

9. The system according to claim 1, wherein the weighting coefficients .sub.i are constant over time.

10. The system according to claim 9, wherein for m most upstream plants of the queue, with m1, the weighting coefficient .sub.i of a plant is higher than a weighting coefficient .sub.i1 of the plant immediately upstream of said plant:
.sub.i1<.sub.i, and for n-r most downstream plants of the queue, with r1 and n the number of plants, the weighting coefficient .sub.i of a plant is lower than the weighting coefficient .sub.i1 of the plant immediately upstream of said plant:
.sub.i1>.sub.i.

11. The system according to claim 9, wherein determining a coefficient .sub.i takes into account for m most upstream plants of the queue with m1 (respectively for n-r most downstream plants of the queue with r1 and n the number of plants), a ratio of a permitted volume variation for the reach i and a permitted volume variation for the first upstream reach (respectively for the last downstream reach).

12. The system according to claim 9, wherein for m most upstream plants of the queue, with m1 (respectively for n-r most downstream plants of the queue, with r1 and n the number of plants), the weighting coefficient .sub.i of a plant is determined as a function of the weighting coefficient .sub.i1 of the plant immediately upstream of said plant and the ratio of a permitted volume variation for the reach i to a permitted volume variation for the first upstream reach (respectively for the last downstream reach), said ratio being weighted by the weighting coefficient associated with the first upstream reach (respectively by the weighting coefficient associated with the next-to-last downstream reach), whereas for the n-r-m plants between said most upstream plants of the queue and said n-r most downstream plants of the queue, the weighting coefficient corresponds to a same maximum value.

13. A plurality of hydroelectric plants arranged in series along a watercourse with an open channel flow, defining upstream of each of hydroelectric plants a plurality of respective reaches, said plurality of hydroelectric plants comprising at least three hydroelectric plants, and comprising a system for driving a turbined water flow rate according to claim 1, wherein the turbined water flow rate for each of said plants is controlled by means of a flow rate setpoint on said plant determined from a run-of-the-river flow rate setpoint taking into account a level regulation of the reaches and the inflow rates in said reaches, wherein the system comprises a regulation of an overall power output setpoint for said plurality of hydroelectric plants by means of a regulation flow rate setpoint taken into account by the flow rate setpoint of each of said plants and in that said regulation flow rate setpoint determined by said regulation is weighted for each of said plants by means of weighting coefficients as a function of the respective flow rate and level hydraulic requirements of the reaches defined upstream of said plants in order to meet said flow rate and level hydraulic requirements.

Description

PRESENTATION OF THE FIGURES

(1) Further characteristics, purposes and advantages of the invention will become clearer from the description that follows, which is purely illustrative and non limiting, and which should be read with regard to the appended drawings in which:

(2) FIG. 1, already commented, is a scheme illustrating the arrangement of a plurality of hydroelectric plants arranged in series along a watercourse defining upstream of them a plurality of respective reaches;

(3) FIG. 2, already commented, is a scheme illustrating a driving system of the state of the art for a plurality of hydroelectric plants arranged in series along a watercourse according to the state of the art;

(4) FIG. 3 is a scheme illustrating a system for driving a plurality of hydroelectric plants arranged in series along a watercourse according to possible embodiments of the invention.

DETAILED DESCRIPTION

(5) The first aspect of the invention is concerned with a system for driving a turbined water flow rate of a plurality of hydroelectric plants arranged in series along a watercourse with an open channel flow, defining upstream of each of them a plurality of respective reaches. These are hydroelectric plants arranged run-of-the-river, along a river or a tributary, in a configuration referred to as an open channel flow by opposition to penstock, the water circulation of which is not with an open channel.

(6) FIG. 1 illustrates a possible configuration of hydroelectric plants in which the invention can be implemented. Driving is implemented for at least three of said reaches, preferentially for each of said reaches i. As previously, the description will be made in an illustrating and a non-limiting way in particular with regard to this configuration, by repeating the notation set out above. The invention relates on the other hand also to a plurality of hydroelectric plants arranged in series along a watercourse with an open channel flow, defining upstream of each of them a plurality of respective reaches 1, 2, 3, 4, said plurality of hydroelectric plants comprising at least three hydroelectric plants, and comprising a turbined water flow rate management system according to the invention.

(7) FIG. 3 schematically illustrates a system for driving hydroelectric plants in series run-of-the-river according to a possible embodiment. The invention is typically implemented in the case of an existing driving system for hydroelectric plants in series run-of-the-river as set forth in FIG. 2 detailed above. The common elements will therefore not be all necessarily detailed again.

(8) As previously, the turbined water flow rate for each of said plants is controlled by means of a flow rate setpoint QCU.sub.i for said plant, determined from a run-of-the-river flow rate setpoint QF.sub.i taking into account a regulation of the level QCHi of the reaches and the inflow rates in said reaches, that is the inflow rate Qe and the sum of the flow rates of the supplies Qa.sub.i of the upstream reaches.

(9) The system comprises a regulation of the electrical power for said plurality of hydroelectric plants to comply with an overall power output setpoint Pc by means of a power regulation flow rate setpoint QRGP taken into account by the flow rate setpoint QCu.sub.i for each of said plants.

(10) The overall power output setpoint Pc can correspond to the sum of a power setpoint of a power program Pc0 and a balancing power, varying at each instant, of the power grid to which the plurality of plants is connected, in particular in the case where the electric power capability of the plurality of plants is sufficiently significant. Therefore, the aim is to provide services to the power grid by adjusting its production to the variations of remote control level N and to the variations in frequency so as to meet at any instant and automatically the production-consumption balance. For the plurality of plants, this service consists in producingin addition to the power program setpoint Pc0powers corresponding to what is commonly called the primary regulation and the secondary regulation while meeting the dynamic criteria defined by the power grid manager. Besides the power program setpoint Pc0, the overall power output setpoint Pc can also comprise a possible secondary regulation, whereas the primary regulation is managed by means of a local regulation peculiar to each production group equipping the plant Ui.

(11) The primary regulation aims at reaching the production-consumption balance via controlling to the electrical frequency of the power grid. It is thus proportional to the frequency difference f between the electrical frequency on the grid and a fixed frequency.

(12) The purpose of the secondary regulation is double: resorbing the residual frequency deviation induced by the primary regulation and correcting budget deviations of the regulation zones. For this reason, this regulation is implemented at the overall level of each zone by resorting to a secondary regulating power available from the production groups taking part in the regulation. Thus, Pr is the electric power corresponding to the participation stated by the operator of the queue of plants to this secondary regulation. A remote control signal N between 1 and 1 is applied to this power Pr by the power grid manager for balancing supply and demand within the power grid it takes in charge. Thus, the overall power output setpoint Pc can be written as:
Pc=Pc0+N.Pr

(13) The overall power regulation by the plurality of hydroelectric plants to meet the overall power output setpoint Pc controls an effective overall power setpoint Pce corresponding to the sum of the effective power setpoints of each plant Pcei, the effective power setpoints of each plant being determined by means of the flow rate setpoint QC.sub.ui of each of said plants. More precisely, the overall effective power setpoint Pce is subtracted from the overall power setpoint Pc before regulation.

(14) Indeed, a turbined water flow rate by a plant is controlled by means of a flow rate setpoint QC.sub.ui, which is converted into an effective power setpoint Pce.sub.i for effectively controlling the production groups of the plant. Yet, this conversion involves flow rate/power charts, which are necessarily marred by errors. This closed loop of the overall power regulation enables errors introduced by the charts to be rejected.

(15) The difference between the overall power setpoint Pc and the effective overall power setpoint Pce passes through a power regulation corrector which determines, from this difference, an overall power regulation flow rate setpoint QRGP which is taken into account by the flow rate setpoint QCu.sub.i of the plants. The power regulation corrector is for example a proportional-integral corrector. The sampling time is for example in the order of 5 seconds.

(16) The overall regulation flow rate setpoint QRGP determined by the power regulation corrector is weighted for each plant by means of weighting coefficients .sub.i peculiar to each of said plants and as a function of the respective hydraulic characteristics of the reaches defined by said plants, to give a regulation flow rate setpoint QRGP.sub.i peculiar to each plant Ui:
QRGP.sub.i=.sub.iQRGP
The weighting coefficients .sub.i assume at least two different values, preferably at least three different values. They can be dynamic, that is variable over time or otherwise be static, that is constant.

(17) This regulation flow rate setpoint QRGP.sub.i is used with other flow rate setpoints to give a plant flow rate setpoint Qcui for the plant Ui. In addition to the regulation flow rate setpoint QRGP.sub.i, the plant flow rate setpoint QC.sub.ui typically takes into account the parallel anticipation flow rate QAPi, the series anticipation flow rate QASi and the level regulation flow rate QCHi. Thus, there can be
QCu.sub.i=QAP.sub.i+QCH.sub.i+QAS.sub.i+QRGP.sub.i

(18) A demodulation flow rate QDDi can be added thereto aiming at meeting the demodulation criterion according to which the flow rate turbined by the last plant Un has to correspond to the flow rate Qe completed by the possible tributaries Qa.sub.i of the reaches i:

(19) ${\mathrm{Qt}}_n={\mathrm{QAP}}_n+\mathrm{.Math.}=\mathrm{Qe}+\underset{k=1}{\overset{n}{.Math.}}{\mathrm{Qa}}_i+\mathrm{.Math.}$

(20) where designates a permissible tolerance, in the order of 2% relative to the inflow rate in the queue Qe. The demodulation function is made by resorting to a downstream-queue flow rate regulation between the parallel anticipation flow rate QAP.sub.n to be followed by the last plant Un and the turbined flow rate Qtn to control of said last plant Un. The demodulation flow rate term QDD calculated is added to the flow rate setpoints of the last plant Un, or even the last plants Un to Unk. This parallel action amounts to transferring the downstream demodulation requirement to upstream and to using the intermediate reaches to absorb the demodulation requirement.

(21) Several methods can be used to define the weighting coefficients .sub.i, including two methods that are set forth hereinafter. The first one involves dynamic weighting coefficients .sub.i, whereas the other one involves constant weighting coefficients .sub.i.

(22) Dynamic Weighting Coefficients .sub.i

(23) In this method, an optimisation module has the object to distribute the queue power to be produced on each of the plants by calculating the coefficients .sub.i while meeting the hydraulic requirements, essentially the level and flow rate requirements.

(24) The essential level requirement relates to meeting the permitted tidal ranges Mi, that is the difference between the maximum permitted level in the reach I and the level setpoint Zc0.sub.i, on the equivalent levels zeq.sub.i of the reaches. The fixed level setpoint Zc0.sub.i is a constant level setpoint of the reach i, corresponding to a water level imposed for the reach i, on a long period of time, generally several years. Bounded variations, the permitted tidal ranges, are thus possible, but the fixed level setpoint Zc0.sub.i makes up an overall target level for reach i. The optimisation module has however the purpose to exploit at best the available tidal ranges to maximise the frequency-power secondary reserve capacity Pr applied to the queue of plants Ui and to minimise the number of possible restatements. The requirement on the flow rates essentially consists of meeting the downstream demodulation of the queue.

(25) Each of the weighting coefficients .sub.i is determined by minimising a criterion corresponding to an equation involving the weighting coefficient .sub.i to be determined and the weighting coefficient .sub.i1 applied to the immediately upstream plant, the weighting coefficient .sub.1 of the first-upstream plant U1 of the queue being fixed, and in particular being possibly zero in the case where no tidal range is permitted in reach 1.

(26) Criterion to be Minimised

(27) The optimisation is made according to a prediction horizon T.sub.opt typically between 3 minutes and 1 hour, and preferably between 15 minutes and 30 minutes. The criterion to be minimised at each calculation step, typically every five minutes, is the following one:

(28) $J=\underset{i=1}{\overset{n}{.Math.}}{}_{\mathrm{zi}}^2.Math.{\mathrm{.Math.}}_i^2+{}_D.Math.{}^2+{}_{\mathrm{RR}}.Math.\underset{i=1}{\overset{n-1}{.Math.}}{\left({}_i-{}_{i\mathrm{init}}\right)}^2$

(29) (in the following, there is 1in unless otherwise specified).

(30) On the other hand, if Mmax.sub.i and Mmin.sub.i have to appear to take the reference levels Href.sub.i, reference measurement of the level in reach i generally measured downstream of reach i into account, the requirements to be met are the following ones:

(31) $-{\mathrm{.Math.}}_i+\frac{M{\max }_i-M{\min }_i}{2}-\frac{M_i}{2}{A}_i+{}_i.Math.B_i+{}_{i-1}.Math.C_i\frac{M{\max }_i-M{\min }_i}{2}+\frac{M_i}{2}+{\mathrm{.Math.}}_i$$\mathrm{where}{A}_i+{}_i.Math.B_i+{}_{i-1}.Math.C_i=\mathrm{zc}{0}_i-z\stackrel{}{e}{q}_i\left(t_0+T_{\mathrm{opt}}\right),$
which corresponds to the water level deviation in reach i;
Q.sub.Demodulation+Q++oQ.sub.Ddemodulation+Q=[QCu.sub.n(t.sub.0+T.sub.opt)QAP.sub.n(t.sub.0+T.sub.opt)],

(32) $\underset{i=1}{\overset{n-1}{.Math.}}{}_i=n,$ .sub.1=0, a bounding of the weighting coefficients .sub.i .sub.min.sub.i.sub.max and a bounding of the variation in the weighting coefficients .sub.i between each time step |.sub.i|.sub.max for 1in1, 300Q.sub.Ddemodulation+300,

(33) with: the convention .sub.n.QRGP=Q.sub.Demodulation=QRGP.sub.n the convention .sub.0=0, representing the absence of power regulation on the plants external to the queue being driven; Mmin.sub.i=M.sub.i+Href.sub.izeq.sub.i; Mmax.sub.i=2.M.sub.iMmin.sub.i; tolerance on the demodulation quality; .sub.D, constant priority factor (set to prioritise demodulation); .sub.RR, constant return spring coefficient (a priori low not to compete with J.sub.1 and/or J.sub.2 when and/or .sub.i are non zero); .sub.Zi, A.sub.i, B.sub.i, C.sub.i and Q are real numbers updated at each optimisation step; .sub.i.sub._.sub.init corresponds to a known initial value of .sub.i.

(34) On the other hand, in order that the overall regulation flow rate setpoint QRGP is wholly reflected on the queue by means of the regulation flow rate setpoints QRGP.sub.i, the sum of the weighting coefficients applied to the power regulation flow rate setpoint of the plants except for the last plant (the coefficient of which is dictated by demodulation) is equal to the number n of the plants:

(35) $\underset{i=1}{\overset{n-1}{.Math.}}{}_i=n$
Initial Values of the Weighting Coefficients .sub.i

(36) The initial values of the weighting coefficients .sub.i are determined beforehand and correspond to the set of coefficients provided to the optimisation module when initialised, or when re-initialised. It is reminded however that the first, fixed, coefficient .sub.1, is preferably chosen zero and that the last coefficient .sub.n depends on the demodulation. The other initial weighting coefficients .sub.i can be determined in different ways.

(37) One of them consists in determining the initial set from the hydropeaking overflow rates QCM.sub.i for each of the reaches i, that is the variations in the turbined flow rates due to the adaptation of electrical production to fluctuations in electricity demand. More precisely, they can be determined as a function of the proportion of the hydropeaking overflow rates QCM.sub.i that the plant Ui represents. For example, there can be thereby:

(38) 0${}_{i\mathrm{init}}=n.Math.\frac{{\mathrm{QCM}}_i}{\underset{i=1}{\overset{n}{.Math.}}{\mathrm{QCM}}_i}$

(39) This determination allows a continuity relative to the existing configurations by virtue of the hydropeaking overflow rates QCM.sub.i taken into account. With this first method, an example of coefficient obtained for a queue of ten plants is the following set:

(40) [0; 0.3710; 0.8207; 1.1804; 1.4727; 1.6863; 1.6863; 1.6863; 1.0961; 0].

(41) Another method consists in deducing the initial weighting coefficients .sub.i from a static optimisation aiming at minimising the criterion

(42) $\underset{i}{\max }{\left(\frac{{}_{i-1}-{}_i}{S_i}\right)}^2$

(43) With this second method, an example of coefficient obtained for a queue of ten plants is the following set:

(44) [0; 0.2941; 0.6143; 0.9022; 1.1664; 1.7047; 2.2928; 2.0237; 1.0019; 0].

(45) Expression of [zc0.sub.izeq.sub.i(t.sub.0+T.sub.opt)]

(46) The weighting coefficients .sub.i are thus determined by taking into account a level deviation between a fixed nominal setpoint with the level zc0.sub.i of reach i and a prediction of the equivalent level zeq.sub.i of said reach i at an optimisation horizon T.sub.opt:
[zc0.sub.izeq.sub.i(t.sub.0+T.sub.opt)].

(47) By noting t.sub.0 the present instant, for a considered reach i, the predicted weighted equivalent level zeq.sub.i(t.sub.0+T.sub.opt) is calculated from the flow rate setpoint QCu.sub.i of plant Ui and the flow rate setpoint QCu.sub.i1 of plant Ui1 upstream of plant Ui:

(48) ${\mathrm{zeq}}_i\left(t_0+T_{\mathrm{opt}}\right)={\mathrm{zeq}}_i\left(t_0\right)+{}_{t_0}^{t_0+T_{\mathrm{opt}}}\frac{{\mathrm{QCu}}_{i-1}\left(t\right)-{\mathrm{QCu}}_i\left(t\right)}{S_i}\mathrm{dt}$

(49) with S.sub.i the apparent area of reach i, that is the free area of reach i, considered as constant. It is to be noted that it is an approximation since the variation in the water level in reach i is considered in the present instant to and the optimisation horizon T.sub.opt. But since the estimation of the variation in the equivalent level zeq.sub.i*S.sub.i is a very good image of a volume variation in reach i, this approach is quite valid.

(50) The trajectories of the flow rate setpoints QCu.sub.i of each plant i are estimated in the future from t.sub.0 to t.sub.0+T.sub.opt with T.sub.opt=5 minutes:
QCu.sub.i(t)=QAP.sub.i(t)+QAS.sub.i(t)+QCH.sub.i(t)+QRGP.sub.i(t)
QCu.sub.i1(t)=QAP.sub.i1(t)+QAS.sub.i1(t)+QCH.sub.i1(t)+QRGP.sub.i1(t)

(51) If the reasoning is made on a time basis in the interval [t.sub.0; t.sub.0+T.sub.opt]: the parallel anticipation flow rate QAP.sub.i(t) is considered as constant in a first approach (it is also possible to estimate a variable parallel anticipation flow rate QAP.sub.i(t) by considering a simple linear interpolation); the level regulation flow rate QCH.sub.i(t) is considered as constant in a first approach. This approximation is relatively good when |zc0.sub.izeq.sub.i|<M.sub.i, that is most of the time. Indeed, in this case, the level regulation of reach i works very little because the weighted level is accompanied by the setpoint, and QCH.sub.i(t) therefore evolves very little on the interval taken into account, in the order of five minutes; the series anticipation flow rate QAS.sub.i(t) is considered as a constant in a first approach because it is

(52) $\underset{k=1}{\overset{i-1}{.Math.}}{\mathrm{QCH}}_k\left(t\right)$

(53) As regards taking the power regulation into account, there is as a power regulation flow rate QRGP.sub.i:
QRGP.sub.1(t)=.sub.i.TI.sub.RGP(t.sub.0)+.sub.i.K.sub.i RGP.sub.t.sub.0.sup.t.sup.0.sup.+t(Pc0(u)+N(u).PrPce(u)).du+.sub.i.K.sub.P RGP.(Pc0(t)+N(t).PrPce(t)

(54) with TI.sub.RGP(t.sub.0) an integral term of the power regulation at the instant t.sub.0, and K.sub.i RGP and K.sub.P RGP gains of the power regulation corrector. The term Pc0(t)+N(t).PrPce(t) can be considered the following way: Pc0(t)+N(t).Math.Pr constant in a first approach between to and t.sub.0+T.sub.opt. However, a prediction of the inflow rate Q.sub.e enables Pc0(t) to be estimated on the interval considered. It is also possible to contemplate to apply statistical methods to estimate N(t) in the interval [t.sub.0; t.sub.0+T.sub.opt]; Pce(t) is considered as converging to Pc0(t)+N(t).Pr as a ramp of 5 MW/min.

(55) Then, the same reason is applied to QAP.sub.i1(t), QAS.sub.i1(t), QCH.sub.i1(t) and QRGP.sub.i1(t). Finally, it is thus obtained that
[zc0.sub.izeq.sub.i(t.sub.0+T.sub.opt)]=[A.sub.i+.sub.i.B.sub.i+.sub.i1.C.sub.i].

(56) Thus, taking the level deviation into account takes the weighting coefficient .sub.i associated with plant U.sub.i and the weighting .sub.i1 associated with plant U.sub.i1 immediately upstream of said plant U.sub.i into account in a linear relationship corresponding to said level deviation [zc0.sub.izeq.sub.i(t.sub.0+T.sub.opt)].

(57) Expression of [QCu.sub.n(t.sub.0+T.sub.opt)QAP.sub.n(t.sub.0+T.sub.opt)

(58) By making the same reasoning as above, [QCu.sub.n(t0+Topt)QAP.sub.n(t0+Topt)] is written as Q.sub.Demodulation+Q and corresponds to meeting the demodulation criterion at the prediction horizon.

(59) Designing the Level Setpoint

(60) The level setpoint Zc.sub.i applied to the level regulator of plant Ui is calculated so as to accompany the natural evolution of the weighted level zeq.sub.i in reach i and thus to minimise the action of the level regulation. For this, an additional term Zci can be added to the level setpoint Zc0i of the level regulation of plant Ui:
Zc.sub.i=Zc0.sub.i+Zc.sub.i

(61) with:

(62) ${\mathrm{Zc}}_i=\frac{1}{S_i}\left({\mathrm{QRGP}}_{i-1}-{\mathrm{QRGP}}_i\right)\mathrm{dt}$

(63) However, the implementation of the determination of dynamic weighting coefficients can turn out to be complex, and thereby it can be preferable to use weighting coefficients i which are constant over time, and which are however optimised in order to meet the same requirements.

(64) Constant Weighting Coefficients .sub.i

(65) The aim is to calculate coefficients .sub.i which are constant and optimal in terms of the time of emptying and filling the reaches i (i=1, 2, . . . , n) by operating the queue according to the law of communicating vessels, that is by transferring the volume of water contained in the upstream reaches to the downstream reaches when the regulation flow rate setpoint QRGP is positive and vice-versa, by transferring the volume of water contained in the downstream reaches to the upstream reaches when the regulation flow rate setpoint QRGP is negative. Thus, in the queue, there can be distinguished: emitting reaches: the upstream reaches operating in emptying/filling; transmitting reaches: the intermediate reaches, playing a role either as direct transmission (without water storage) as long as the demodulation does not act on themselves, or as absorption when this acts on them; receiver reaches: the downstream reaches operating in filling/emptying, that is conversely to the upstream reaches.

(66) The evolution of the water volume V.sub.i in a reach i is governed by the equation:

(67) $\frac{d\left({\mathrm{Vo}}_i+V_i\right)}{\mathrm{dt}}={\mathrm{QF}}_{i-1}+u_{i-1}-\left({\mathrm{QF}}_i+u_i\right)$

(68) where the volume V0.sub.i corresponds to the static level Zc0.sub.i in reach i held by virtue of the run-of-the-river flow rates QFj(j=i and i1) including during hydraulic disturbances, typically unforeseen supplies not measured in a reach. The control variables u.sub.j=QRGP.sub.j+QDD.sub.j enable the objectives and requirements of queue driving and demodulation to be met.

(69) In the absence of demodulation, one can write:

(70) $\frac{d{V}_i}{\mathrm{dt}}=\left({}_{i-1}-{}_i\right)\mathrm{QRGP}$

(71) with V.sub.i=S.sub.i(Z.sub.iZc0.sub.i) and S.sub.i the area of reach i.

(72) By considering the initial state V.sub.i(0)=0 and the final state V.sub.i(T), there is:
V.sub.i(T)=(.sub.i1.sub.i).sub.0.sup.TQRGPdt)

(73) Thus, at the instant t=T, the ratio R of the volume variations between any two reaches with indices i and x can be written as:

(74) $R=\frac{V_i\left(T\right)}{V_x\left(T\right)}=\frac{{}_{i-1}-{}_i}{{}_{x-1}-{}_x}$

(75) By designating T the emptying (respectively filling) time of reach i for a modification of its half-band equivalent level Mvi (respectively Mvi) of permitted tidal range volume, there is V.sub.i(T)=M.sub.vi.

(76) The emitting reaches i are characterised by .sub.i1<.sub.i and the receiving reaches i are characterised by .sub.i1>.sub.i. In other words, the emitting reaches are being emptied if the regulation flow rate setpoint QRGP is positive whereas the receiving reaches are being filled, whereas the emitting reaches are being filled if the regulation flow rate setpoint QRGP is negative whereas the receiving reaches are being emptied.

(77) Thus, for the m most upstream plants of the queue associated with the emitting reaches, with m1, the weighting coefficient .sub.i of a plant U.sub.i is higher than the weighting coefficient .sub.i1 of the plant immediately upstream of said plant U.sub.i:
.sub.i1<.sub.i,

(78) and for the n-r most downstream plants of the queue associated with the receiving reaches, with r1 and n the number of plants, the weighting coefficient .sub.i of a plant U.sub.i is lower than the weighting coefficient .sub.i1 of plant U.sub.i1 immediately upstream of said plant U.sub.i:
.sub.i1>.sub.i.

(79) As regards the intermediate transmitting reaches, they are characterised by .sub.i1=.sub.i=.sub.max. The maximum value of the coefficients .sub.max is determined irrespective of the emptying time if the volume budget is perfectly met, and can be for example arbitrary set to 1 to control flow rate saturation of the hydroelectric plants U.sub.i.

(80) The aim of .sub.i calculation is to minimise the number of restatements, that is to minimise the times the power program setpoint Pc0 is modified. It amounts to minimising the volume variations or to maximising the emptying/filling time T.

(81) To maximise the emptying/filling time T and ensure optimality of the coefficients .sub.i: the number of transmitting reaches is chosen so as to equilibrate the volume balance of the water transfer to the queue, that is such that the sum of the volumes corresponding to the permitted tidal ranges Mvi of the emitting reaches is as close as possible to that of the receiving reaches; the emitting and receiving reaches are synchronous by simultaneous saturation which is ensured by
V.sub.i(T)=(.sub.i1.sub.i).sub.0.sup.TQRGPdt

(82) by setting: R=Mv.sub.i/Mv.sub.x for the emitting reaches and R=(Mv.sub.i)/(Mv.sub.x)=Mv.sub.i/Mv.sub.x for the receiving reaches.

(83) By applying these two principles on a group of emitting or receiving reaches (R=Mv.sub.i/Mv.sub.x), from

(84) $R=\frac{V_i\left(T\right)}{V_x\left(T\right)}=\frac{{}_{i-1}-{}_i}{{}_{x-1}-{}_x}$

(85) one obtains

(86) ${}_i={}_{i-1}-\frac{{\mathrm{Mv}}_i}{{\mathrm{Mv}}_x}\left({}_{x-1}-{}_x\right)$

(87) The determination of a coefficient .sub.i thus takes into account a ratio of a volume variation permitted for the reach to a volume variation permitted for a reach upstream of said reach. The boundary conditions are defined on the one hand by reach 1 which does not receive a power regulation flow rate setpoint QRGP upstream, that is .sub.0=0 where .sub.0 represents the absence of power regulation on the plants external to the driven queue, and by the last reach n which is subjected to the demodulation criterion, that is .sub.n=0.

(88) Consequently, x=1 is taken for the emitting reaches and x=n is taken for the receiving reaches. The transmitting reaches are located between the indices m and r. Thus, for i=1, 2, . . . , n: there is:
for the emitting reaches: .sub.i=.sub.i1+.sub.i.Mv.sub.i/Mv.sub.1;
for the receiving reaches: .sub.i=.sub.i1.sub.n1.Mv.sub.i/Mv.sub.n; and
for the transmitting reaches: .sub.i=.sub.i1=.sub.max.

(89) Consequently, for the m most upstream plants of the queue, with m1 (respectively for the n-r most downstream plants of the queue, with r1 and n the number of plants), the weighting coefficient .sub.i of a plant U.sub.i is determined as a function of the weighting coefficient .sub.i1 of the plant U.sub.i1 immediately upstream of said plant U.sub.i and the ratio of a volume variation permitted for reach i to a volume variation permitted for the first upstream reach with index 1 (respectively for the last downstream reach with index n), said ratio being weighted by the weighting coefficient .sub.1 associated with the first upstream reach 1 (respectively by the weighting coefficient .sub.n1 associated with the next-to-last downstream reach with index n1), whereas for the plants n-r-m between said m most upstream plants of the queue and said n-r most downstream plants of the queue, the weighting coefficient corresponds to a same maximum value .sub.max.

(90) To determine .sub.1 and .sub.n1, it is sufficient to use the recurrence of the equations below: for the emitting reaches

(91) 0${}_i={}_1\left(1+\frac{{\mathrm{Mv}}_2}{{\mathrm{Mv}}_1}+\frac{{\mathrm{Mv}}_3}{{\mathrm{Mv}}_1}+\mathrm{.Math.}+\frac{{\mathrm{Mv}}_i}{{\mathrm{Mv}}_1}\right)$ for the receiving reaches:

(92) ${}_{n-i}={}_{n-1}\left(1+\frac{{\mathrm{Mv}}_{n-1}}{{\mathrm{Mv}}_n}+\frac{{\mathrm{Mv}}_{n-2}}{{\mathrm{Mv}}_n}+\mathrm{.Math.}+\frac{{\mathrm{Mv}}_{n-i+1}}{{\mathrm{Mv}}_n}\right)$ for the transmitting reaches:
.sub.m=.sub.r=.sub.max

(93) Hence, by setting i=m in the equation of the emitting reaches and ni=r in the equation of the receiving reaches, as a result:

(94) ${}_1=\frac{{}_{\max }}{1+\underset{k=3}{\overset{m}{.Math.}}{\mathrm{Mv}}_k/{\mathrm{Mv}}_2}$$\mathrm{and}$${}_{n-1}=\frac{{}_{\max }}{1+\underset{k=r+1}{\overset{n-1}{.Math.}}{\mathrm{Mv}}_k/{\mathrm{Mv}}_n}$
are obtained.

(95) This static optimisation of the weighting coefficients .sub.i has the advantage of offering control limiting variations in the turbined flow rate, and which: limits level fluctuations in reaches thus avoiding a degradation in holding the levels ensured by level regulation, limits operations of actuators and valving thus reducing their mechanical fatigue.
Designing the Level Setpoint

(96) The level setpoint Zc.sub.i applied to the level regulator of plant Ui is calculated so as to accompany the natural evolution of the weighted level zeq.sub.i in reach i and thus to minimise the action of the level regulation. For this, an additional term Zc.sub.i can be added to the level setpoint Zc0.sub.i of the level regulation of plant Ui:
Zc.sub.i=Zc0.sub.i+Zc.sub.i
For the emitting reaches, the additional term is:

(97) ${\mathrm{Zc}}_i=\frac{1}{S_i}\left({\mathrm{QRGP}}_{i-1}-{\mathrm{QRGP}}_i\right)\mathrm{dt}$

(98) with S.sub.i designating the area of the reach i.

(99) For the transmitting reaches and the receiving reaches, the level setpoint deviation becomes:

(100) ${\mathrm{Zc}}_i=\frac{1}{S_i}\left({\mathrm{QRGP}}_{i-1}+{\mathrm{QDD}}_{i-1}-{\mathrm{QRGP}}_i-{\mathrm{QDD}}_i\right)\mathrm{dt}$

(101) with Zc.sub.i is bounded between a minimum level and a permissible maximum level. It is to be noted that the demodulation flow rate setpoint QDD.sub.i corresponds, for the plants of the receiving reaches, to a same overall setpoint noted QDD whereas for the plants of the transmitting reaches, it is a local demodulation setpoint.

(102) Demodulation

(103) As explained above, demodulation consists in meeting the demodulation criterion on the last plant

(104) ${\mathrm{Qt}}_n={\mathrm{QAP}}_n+=\mathrm{Qe}+\underset{k=1}{\overset{n}{.Math.}}{\mathrm{Qa}}_k+$

(105) where designates a permissible tolerance, in the order of 2% relative to the inflow rate in the queue Qe. The demodulation function is made by resorting to a queue downstream flow rate regulation between the parallel anticipation flow rate setpoint QAP.sub.n to be followed and the turbined flow rate Qtn to control. The demodulation flow rate term QDD calculated is added to the flow rate setpoints of all the power plants of receiving reaches. This parallel action amounts to transferring upstream the downstream demodulation requirement and to using transmitting reaches to absorb the demodulation requirement.

(106) The overall demodulation flow rate term QDD is determined by a proportional-integral type regulation on the difference between the parallel anticipation flow rate QAP.sub.n and the flow rate Qt.sub.n turbined by the last plant Un.

(107) To control evolution of the levels in transmitting reaches, a regulation of the central downstream level setpoint is implemented so as to re-centre the levels of the reaches. It calculates a local demodulation flow rate term QDDi which is added to the flow rates of the plants of the transmitting reaches. Thus, the local demodulation flow rate term QDDi of the plant of reach i is determined by a proportional-integral type regulation on the difference between the fixed level setpoint Zc0.sub.i+1 of the plant of the downstream reach i+1 and the level setpoint Zc.sub.i+1 applied to the level regulation of the plant of the downstream reach i+1.