WO2006021125A1 - System for controlling hydroelectric power plants - Google Patents

Abstract

A method and device of controlling at least one water level at a predetermined point influenced by at least one hydroelectric power plant (P) is disclosed, wherein the discharge through the power plant is regulated. The method employs a model predictive control (MPC) algorithm. In a preferred embodiment, a cascade of power plants is controlled simultaneously. The method is specifically adapted for handling large amounts of discharge and the specific constraints in connection with the hydraulics of hydroelectric power plants and avoids large discharge variations.

Description

System for controlling hydroelectric power plants

Field of the invention

The present invention relates to the control of water levels and flow in distrib¬ uted water flow structures, in particular in rivers with single or cascaded hydroe¬ lectric power plants. In particular, the invention relates to a method and a device for controlling at least one water level at a predetermined point influenced by at least one hydroelectric power plant. The invention further relates to a computer program product implementing such a method.

Background of the invention

River power plants are man-made constructions, which are built into the course of a river to generate electrical energy. Such plants have major impacts on the water level and flow as they retain the water ahead of them and set the dis¬ charge through their facilities. If these impacts are not limited as much as pos¬ sible, the riparian habitat is exposed to the power plant activities, the ecological equilibrium is affected and river navigation is hindered.

For these reasons, the authorities usually impose conditions on the operation of power plants. For the time being, these conditions mostly imply keeping the wa¬ ter level at a pre-specified point upstream of the river power plant within certain bounds around a reference value. This is achieved by adjusting the discharge through the power plant facilities. Prior-art control concepts employ a Pl control¬ ler with an additional feed-forward term for each power plant. As these control¬ lers are local and no or little coordination and exchange of information exists between the power plants in the cascade, they do not consider the conse-

BESTATIGUNGSKOPIE quences of their control actions to downstream power plants. Thus they often impose significant variations on water discharge in order to keep the water level at the prescribed point constant. For cascades of power plants, these fluctua¬ tions in discharge are unpredictably amplified above the natural discharge varia¬ tions in the river.

Summary of the invention

It is an object of the present invention to provide a method for controlling the headwater level of a hydroelectric power plant which is robust and which avoids strong fluctuations in discharge or water level. This object is achieved by a method according to claim 1.

The present invention is further directed at a computer program product accord¬ ing to claim 15, which implements the method of the present invention, and to a device according to claim 16, which is specifically adapted for executing a method according to the present invention.

Advantageous embodiments of the invention are laid down in the dependent claims.

Thus, according to the present invention a method of controlling at least one water level at a predetermined point influenced by at least one hydroelectric power plant (the so-called concession level) is provided. The method controls said at least one water level by applying control moves which comprise adjust¬ ing at least one discharge through said power plant. A model predictive control algorithm is employed for deriving said control moves. The application of model predictive control leads to a robust control whose parameters are easily ad¬ justed and which avoids large fluctuation of discharge.

The method according to the present invention is a supervisory method which is, at least in principle, applicable to any distributed flow structure. The term "distributed water flow structures" includes, but is not limited to, river hydroelec¬ tric power plants (lying either on the course of the river or using a by-pass channel), hydroelectric power plants using a reservoir lake, cascades of combi¬ nations of the above, irrigation channels, potable water supply networks etc. The supervisory control system would be used to appropriately manage the wa¬ ter resources, with different objectives in each of these cases. These include, but are not limited to, the minimization of water level and/or discharge fluctua¬ tions, the maximization of the economic profit produced by the use of the instal¬ lation (electric energy production, delivery of potable water), the minimization of water losses, the respect of environmental constraints imposed by the authori¬ ties etc.

Among the various distributed flow structures, hydroelectric power plants are unique in several respects. For example, a hydroelectric power plant generally comprises two different means for outflow from the headwater: a first means employed for the generation of electric energy, in particular, one or several tur¬ bines; and a second means bypassing the first means, in particular, one or sev¬ eral weirs. Generally, these means will be controlled independently, and they need not be in the same position along the flow structure. In contrast to an irri¬ gation system or a potable water supply, where water is removed "laterally", the outflows from these means are combined into the same water flow structure (usually a river reach) downstream from the power plant. In addition, the weirs can be substantially upstream from the turbines, in particular in a so-called channel power plant. Power plants are also different from structures such as irrigation canals or potable water supplies in that they need to handle large amounts of discharge (typically, several hundred cubic meters per second, as compared to typically only a fraction of a cubic meter per second up to a few cubic meters per second for an irrigation system or a water supply network) and that they must be able to cope with large unexpected variations of inflow, e.g., in the case of heavy rainfall. For a successful control algorithm, suitable control objectives must be found, which will generally be specific for a power plant. Just by the way of example, upper and lower limits of the concession level may be imposed, a minimum discharge through the power plant may be required, a minimal and maximum discharge through the turbines may exist, and the rate of change of the discharge may not be arbitrarily large. Therefore, control method¬ ologies developed for irrigation or for potable water supply networks are not readily applicable to power plants.

In particular, the method according to the present invention may comprise the following steps:

(a) determining a system state from at least one measurement;

(b) determining a sequence of future control moves for a predetermined time horizon in a manner to minimize a predetermined cost function and to satisfy predetermined constraints, wherein said sequence of future control moves is based on predictions of a system behavior based on an internal plant model;

(c) applying the first control move of said sequence of future control moves; and

(d) repeating steps (a) to (c).

In other words, in this embodiment the method employs a receding horizon strategy with a finite time horizon. Effects beyond the finite time horizon may be considered by applying an optional terminal weight, e.g., a Riccati weight (see below). The system state is preferably determined from the measurement by state estimation, e.g., by employing a Kalman filter. Optionally, balanced model reduction may be employed for reducing the number of system states.

Usually, the hydroelectric power plant comprises one or more turbines and one or more weirs. Then preferably the model predictive control algorithm is adapted to keep a discharge through either the turbines or the weirs fixed. To this end, the algorithm takes such a fixed discharge explicitly into account.

Preferably, the model predictive control algorithm takes one or more of the fol¬ lowing constraints into account: a maximum and/or a minimum concession level; a minimum and/or a maximum discharge through the turbines; a maximum rate of change of the discharge through the turbines; a minimum discharge through the weirs; and a maximum rate of change of the discharge through the weirs. Each of these constraints may be a hard or soft constraint.

The model predictive control algorithm may additionally be adapted to take a maximization of economic profit explicitly into account. This may be achieved by making the cost function depend on time in a manner to reflect actual or ex¬ pected demand and/or a time variation of the unit price of electricity generated by the power plant over a predetermined time horizon. By the way of example, the value of the cost function may be dependent on the unit price of electricity in a manner that an increase in discharge through the turbines leads to a stronger reduction of the cost function at times where the unit price of electricity is higher than at times where this unit price is low, e.g., by making the weight of the cor¬ responding term of the cost function explicitly time-dependent.

In order to avoid large variations in discharge, which are generally undesired, the model predictive control algorithm is preferably adapted to minimize the magnitude of the control moves.

The model predictive control algorithm may be readily adapted to take expected future disturbances upstream of said hydroelectric power plant into account.

The method of the present invention may advantageously be applied to a plural¬ ity of hydroelectric power plants forming a cascade. These plants are then con¬ trolled simultaneously by the model predictive control algorithm.

Advantageously, the model predictive control algorithm is adapted to take lat¬ eral inflow (e.g., at river junctions) and/or lateral outflow (e.g., at river branchings) between the hydroelectric power plants into account. This may be important, e.g., if a cascade of river power plants is controlled, and if the river receives additional inflow between power plants from a feeder river, or if the river splits up at some point between power plants. In particular, a branching may be modeled as described below in connection with Fig. 5, i.e., at the point of the branching, flow is discretized in a manner such that a discretization point is chosen at the branching, while the water levels are discretized in a manner such that a discretization point is chosen just upstream of the branching.

The model predictive control algorithm is preferably based on a discrete-time, discrete-space state space model of a river reach which is linearized around an operating point, rather than a transfer function formulation. In particular, the state-space model may be derived by linearizing and discretizing in time and space the well-known Saint Venant equations.

The state-space model generally comprises a state vector describing the river reach, an input vector of input variables which influence the state vector, and an output vector which may be used for defining the control objective. For a hy¬ droelectric power plant, the input vector will generally comprise an inflow dis¬ charge and the outflow discharges through the turbines and weirs. The output vector will generally comprise the concession level. The state vector will gener¬ ally comprise water levels and discharges at the spatial discretization points. Advantageously, the spatial discretization points for water levels and discharges are placed alternately along the river reach. For state estimation, a Kalman filter may be employed.

The Saint Venant equations are advantageously discretized in predetermined discretization time steps, wherein said discretization time steps satisfy the one- dimensional Courant criterion (see Eqs. (41) and (42) below). Since these time steps will be generally too short to allow for either for a realistic simulation or for a computation of a control move in sufficient time, the model predictive control algorithm advantageously computes the control moves only at discrete simula¬ tion time steps which are a multiple of said discretization time steps. The input vector is then kept constant during each simulation time step, which considera¬ bly simplifies simulation and accelerates computation. The method may comprise a step where the terminal state of the finite time se¬ ries of computed states is weighted by a Riccati weight, to obtain an infinite- horizon solution to the control problem that guarantees stability and improves performance.

In order to minimize discharge variations, the input vector in a state space model may comprise a change of said discharges through the turbines and/or weirs instead of or in addition to the actual values of the discharges.

To improve performance and robustness, the method may comprise a step of changing the operating point whenever a discharge at a predetermined position upstream of said power plant changes by more than a predetermined threshold.

The model predictive control algorithm of the method according to the present invention can be readily implemented in hardware (e.g., in a custom-built digital signal processor) or in software. In particular, the invention also encompasses a computer program product comprising computer program code means for con¬ trolling one or more processors of a computer such that the computer performs the following steps:

(a) receiving an input value representing a water level at a predetermined point influenced by at least one hydroelectric power plant;

(b) from said input value, determining a system state;

(c) for said system state, determining a sequence of future control moves for a predetermined time horizon in a manner to minimize a predetermined cost function and to satisfy predetermined constraints, wherein said sequence of future control moves is based on predictions of a system behavior based on an internal plant model of said hydroelectric power plant;

(d) providing as an output the first control move of said sequence of future control moves;

(e) repeating steps (a) to (c). The above-described specific embodiments of the method according to the pre¬ sent invention may be implemented in such a computer program product, muta¬ tis mutandis.

A device for controlling at least one hydroelectric power plant according to the present invention comprises: measuring means for determining a water level at a predetermined point influenced by at least one hydroelectric power plant; a controller for deriving control moves which comprise adjusting at least one discharge through said power plant, said controller receiving an output of said measuring means; regulating means for regulating a discharge through said hydroelectric power plant according to said control moves,

The controller is specifically adapted to employ model predictive control for de¬ riving said control moves. To this end, the controller may comprise one or more processors and one or more memories, where the memory comprises computer code means for controlling the processors of a computer such that the proces¬ sors execute a model predictive control algorithm.

Such a device lends itself to being adapted to execute a control method accord¬ ing to one of the above-described specific embodiments.

Brief description of the drawings

The invention will be described in more detail in connection with a exemplary embodiments illustrated in the drawings, in which

Fig. 1 shows a diagram of an MPC controller for a single power plant;

Fig. 2 shows a diagram illustrating the general concept of Model Predic¬ tive Control;

Fig. 3 shows a diagram illustrating the receding horizon concept;

Fig. 4 shows a diagram illustrating the division of a river into compart¬ ments;

Fig. 5 shows a diagram illustrating the choice of discretization points close to a branching;

Fig. 6 shows a diagram illustrating system states and inputs for a chan¬ nel power plant model;

Fig. 7 shows a diagram illustrating generic river reaches of which a cas¬ cade is composed;

Fig. 8 shows a diagram illustrating system states and inputs for a cas¬ cade model of the Untere Aare;

Fig. 9 shows a diagram illustrating a river reach between two channel power plants;

Fig. 10 shows a diagram illustrating a sinusoidal input disturbance;

Fig. 11 shows a diagram illustrating variables following a sinusoidal input disturbance;

Fig. 12 shows a diagram illustrating a ramp disturbance;

Fig. 13 shows a diagram illustrating variables following a ramp distur¬ bance;

Fig. 14 shows another diagram illustrating variables following a ramp dis¬ turbance; and

Fig. 15 shows another diagram illustrating variables following a sinusoidal input disturbance.

Detailed description of preferred embodiments

According to the present invention, Model Predictive Control (MPC) [D.Q. Mayne et al., Constrained model predictive control: stability and optimality", Automatica, vol. 36, pp. 789-814, 2000] is applied for a supervisory control sys¬ tem for water level and flow control of distributed water flow structures. In par¬ ticular, in the following an embodiment is described in which MPC is applied to the control of river power plants.

The specific case to which this embodiment is applied concerns a cascade of five hydroelectric power plants between Aarau and Beznau in the course of the river Untere Aare, in Aargau, Switzerland. The operation of the plants has to be regulated in such a way that certain constraints on the river's water level are met, while incoming water flow disturbances are damped. The prior-art local control scheme which is currently employed has proven inadequate to deal with this problem, and water flow disturbances are actually amplified during the propagation through the cascade. As environmental constraints are becoming tighter, the currently employed prior-art controllers cannot meet the specifica¬ tions, and a new advanced control strategy is needed.

As opposed to local Pl controllers, the supervisory control scheme according to the present invention takes into account information about all power plants in the cascade and coordinates the control actions at the different plants. The su¬ pervisory controller has to be able to determine the discharges through all the power plants of the cascade such that the discharge variations are damped and the water level constraints imposed by the authorities are met. A possible ap¬ proach to achieve this objective is to use an internal model to predict the future behavior of the system and to derive from these predictions the control moves which best fulfill the control objectives.

Figure 1 schematically shows how an MPC controller can be applied to a ge¬ neric river power plant. Suitable sensors, as they are well known in the art, de¬ termine an incoming discharge q_in of the river 1 upstream of the power plant P and the so-called concession level h_c (generally, the headwater level in a loca¬ tion close to the power plant P). These variables and a reference water level h_rβf are fed to the MPC controller 2, which employs a river model designated as

RM to compute the discharges q_out based on an MPC algorithm.

The application of MPC to the control problem of any physical process is di¬ vided into the following stages:

(i) the derivation of a process internal model that, starting from a measured ini¬ tial condition at a certain time instant t will be used to predict the future behavior of the process, and (ii) the formulation of a constrained optimal control problem.

Based on the predictions and the problem formulation, a sequence of control moves is calculated by solving the underlying constrained optimization problem. From the sequence of control moves, only the first one is applied to the proc¬ ess. At the next time step, a new measurement is acquired and the above pro¬ cedure is repeated shifted in time. This strategy is referred to as the receding horizon strategy.

A particular contribution of the technology according to the present invention is the application of a supervisory MPC controller for the water level and discharge control problem of a cascade of river power plants. For this, a number of novel approaches to the water level control system have been introduced. These in¬ clude:

A. The derivation of a linear discrete-time and discrete-space state-space model of the cascade that is based on the Saint-Venant equations. Al¬ though the Saint-Venant equations are the state of the art for modelling one-dimensional river hydraulics, they usually serve as a basis for deriving transfer function models in the frequency domain. The main advantage of the state-space model is the straightforward identification of the required parameters, which are obtained from river geometry and steady state measurements. The model has been developed in a modular form and can be extended to describe any arbitrary river topology and power plant configuration. It also incorporates the future influence of incoming distur¬ bances before propagating through the cascade. Although it is linearized around a specific operating point, its parameters are adapted to operating point changes. An additional technique that has been applied for the first time to the water level control problem is the balanced model reduction, which significantly reduces the state-space model dimensions and the computational complexity of the constrained optimization problem.

B. The formulation of a constrained optimal control problem over a receding horizon, for the supervisory water level control problem of the cascade. The control problem formulation includes various specific strategies that are applied for the first time to the control problem of river hydraulics. These include:

(a)The Δu formulation of the control problem that allows the minimiza¬ tion of the water discharge variations, (b) The introduction of both soft and hard constraints on the controlled and the manipulated variables. (c)The tuning strategy for the cost function.

(d)The use of a Riccati weight on the terminal state of the finite horizon, to obtain the infinite horizon solution to the control problem that guarantees stability and improves performance.

C. The use of the truncated (reduced) model for the estimation of the states of the river from the existing measurements and the implementation of a tailored adaptation algorithm for the Kalman filter.

The general concept of MPC is schematically depicted in Fig. 2. The figure shows a control sequence for control of a power plant 4 (designated as P) by a controller (optimizer) 3, employing a plant model PM and receiving a reference value r(k). Before any calculation can be done, the internal model PM of the plant on which the predictions will be based is derived and the prediction hori¬ zon N is chosen. Additionally, the control objectives are expressed in a cost function and the constraints on plant states and inputs are defined. The step¬ wise procedure then is given as follows:

• At the time step k, the sequence of future control moves for the horizon N is determined such that the cost function is minimized and the con¬ straints are satisfied using the predictions of the system behavior based on the internal plant model.

• The first control move u(k) of the determined control sequence is ap¬ plied.

^• From measurements the system states at time step k + l are identified (by state estimation using a Kalman filter). • From the updated plant states, the optimal control sequence over the shifted horizon is determined anew.

Moving the optimization window after each control step is referred to as the re¬ ceding horizon concept. This is graphically shown in Fig. 3. The upper part of this figure shows a diagram with predicted outputs 5 as a function of time. A desired reference (control objective) is denoted by "ref". In order for the outputs 5 to approach the reference "ref, a control sequence 6 of future control moves is computed for times k+1 , ..., k+N, based on the situation at time k. Only con¬ trol move 7 at time k, however, is actually applied. Then at time k+1 the situa¬ tion is reassessed based on a measurement M of the actual output at time k+1 , new control moves are computed for times k+2, ..., k+N+1 , and the sequence is repeated.

Components of MPC

For the problem formulation of MPC, three components are necessary: an in¬ ternal model of the system, an objective function and constraints. In this section, these components are shortly described.

In order to predict the future behavior of a process, a linear and discrete-time internal model of the process is needed. The state space representation of such a system is given as x(k + 1) = Ax(k) + Bu(k), (-_| ) y(k) = Cx(k), (2) where A is the system matrix, B the input matrix and C the output matrix. Here, ^χ(k) is the state vector, u(k) the input vector, y(k) the output vector and k the discrete time variable.

In quadratic optimal control the objective is to find an input sequence which minimizes the quadratic cost function Λ^r-i

Mx(OI u(O), ..._: U(N - I)) 4 ∑[χ^τ(k)Qx(k) + u^τ (k)Ku(k)] + x^r(N)Q_tx(N) k=0 (3) for a given state χ(0) yielding the optimizer

U* (X(O)) = [«* (0) , ..., u* (N - I)}^1' = arg min J_jV(a:(0), u(ϋ), ..., w(JV - I)).

«⁽0)...._:«(Λ^r-i) (4)

The weight matrix Q has to be positive semi-definite and n positive definite. With a finite N, the problem is referred to as the finite horizon problem. The weight matrix Q_t is called the terminal weight. By applying this weight the ac¬ cumulated costs for the time steps N..... oo can be taken into account if the sys¬ tem is in a control-invariant set at time step N₁ and Qt is the solution of the algebraic Riccati equation. This term is not mandatory but may improve control performance.

Constraints on states and inputs can be given by the physics of the system or as operational constraints set by the operator. The constraints on the states ^χ(^k) and on the inputs u(k) are defined as x(k) < x(k) < x(k), k = 0, ..., N, (5) u(k) < u(k) < u(k), k = 0, ..., N - l, (6) with lower limits z(^k), ^(*0 and upper limits *(^k), ^(^fc).

Problem Formulation

In order to formulate the control problem as a standard Quadratic Program (QP), the future state sequence is formulated explicitly

and denoted as

X = S^xx(0) + S¹¹U. (8) With this notation the cost function is given as

Mx(O), U) = X^TQX + U^TUU, (₉) where Q = blockdiag{Q, ..., Q, QJ and £ = blockdiagjft, ...,%}_ using Eq. (8) this results in

J_N(x(0), U) = (S^xx(0) + S^UU)^T Q(S^xx(0) + S^UU) + U^TUU = U^T(S^uTQS^u + it)U + 2x^T(0)S^xTQS^uU+

+ x^τ(0)(S*^τQS*)x(0). ₍₁₀)

To find the optimal control sequence u^* _t the complete problem formulation is

U^* = arg inin [J_N(x(0), U)] ^ ^ ,

subject to

K ≤ S^xx(0) + S^uU < X, (12) U < U < TJ. (13) where K- and X contain the limits on the states at each time step within the horizon

x(0) x(0) x(l)

X = a(i) X = x(N) x(N) (14) and LL and U the limits on the inputs

2έ(0) ύ(0) M(I) ύ(l)

U = U = u(N - 1) H(N - 1) (15)

Soft Constraints

In order to avoid infeasibility problems that might arise from the violation of the hard constraints, a common strategy is to relax or 'soften' the constraints, i.e. allow the constraints to be slightly violated if necessary. This is only possible for operational constraints whereas physical constraints are always hard con¬ straints. A straightforward way to soften constraints is to introduce slack vari¬ ables. The inequalities (5) and (6) are reformulated as

x{k) - ε_x(k) < x(k) < x(k) + ε_x(k), k = 0, ..., N, (₁₆)

0 ≤ ε_x(k), k = 0, ..., N (_{1 7}) and u(k) - ε_u(k) < u(k) < u(k) + ε_u(k), k = 0, ..., N - I, (<| 8)

0 < e_tt(fc), k = 0, ..., N - l. (₁₉)

The slack variables £χ, ε_u are only non-zero if the original constraints (5), (6) are violated and their values are heavily penalized in the cost function. The pe¬ nalization of the slack variables leads to an extension of the cost function (3) resulting in

JV-I

J_N(x{0), U) = ∑[x^τ{k)Qx(k) + u^τ(k)Ku(k)+ fe=0

+εl(k)Q_εε_x(k) + εl(k)Tl_εε_u{k)} +

+x^T(N)Q_tx(N), (20) with Qε and ftε as diagonal weight matrices for the slack variables.

Terminal State Weight

For MPC, an infinite horizon is required to prove stability. In the control problem formulation, quadratic costs were assigned to the states ^χ(^k) and to the inputs ^u(^k) up to a horizon of N time steps.

It is assumed that after N steps, the system state ^χ(^N) is close enough to the origin such that the constraints on the system states and on the inputs are al¬ ways inactive in the future and can be neglected. Then, the terminal weight Qt is used to include the accumulated costs for k = N^, ... , ∞ and thus all future penalties on states and control moves. Assuming inactive constraints beyond the horizon N, such a terminal weight can be calculated by Dynamic Program¬ ming as shown in [Mayne et al., I.e.]. The terminal weight Qt is found as the so¬ lution of the discrete algebraic Riccati equation

Q_t = A^TQ_tA + Q - A^TQ_tB(B^TQ_tB + n)^~lB^TQ_tA, (₂₁ ) with the system matrix A and the input matrix ^B as defined in Eqs. (1 ) and (2). The optimal feedback control law which minimizes the cost-to-go from N to in¬ finity in the unconstrained case is _u(k) = -(B^τ Q_tB + K)-¹JB⁷Q_tA) XJk).

^{^} ^ ^' (22)

This linear feedback controller is the Linear Quadratic Regulator (LQR) yielding the state update equation

x(k + 1) = {A - BKu)Mk), VA; > N. (₂3)

In the constrained case, the feedback law Eq. (22) would lead to violations of the constraints on inputs or states, because these constraints were ignored for the calculation in Eq. (21 ). The conditions on states and inputs for which the linear feedback law Eq. (22) holds can be formulated as ^χ(k) e Xfeb, Vfc ≥ -V, (24) u(k) e U_[sb, Vfc ≥ N. (25) with ^fsb and <5l/_fsb being the feasible sets of ^χ(^k) and M^) for which all con¬ straints are met. The state ^χ(^N) has to lie within a control invariant set Xd, such that x(N) e X_ci → x{k) G X_kb, u{k) G f/feb, Vfc > N (26) for the system of Eqs. (22), (23).

One way to ensure that this condition is met is to choose ^N large enough such that ^χ(^N) is guaranteed to lie in a control invariant set ^d. Another way to en¬ force ^χ(^N) ^e ^ci is to impose a terminal set constraint, i.e. explicitly include the constraint ^X(^N) ^{€ χ}α in the problem formulation. The terminal set constraint is chosen as the largest possible control invariant set ^, which is called the maximum output admissible set [D.Q. Mayne et al., I.e.]. This set can be calcu- lated off-line as shown in [I. Kolmanovsky, E. G. Gilbert, "Theory and Computa¬ tion of Disturbance Invariant Sets for Discrete-Time Linear Systems," Mathe¬ matical Problems in Engineering, Vol. 4, pp. 317-367, 1998]. Recalling

«(0)

U = u(N - 1) (27) for the control sequence, the resulting control problem is to find the optimal se¬ quence of control moves U*

U^* = αrg min [J₀₀(X(O), U))

° (to) subject to the constraints on the states and on the inputs Eqs. (5)-(6) for k = 0..... N - 1 and the linear feedback law u(k) = -K LQX(Ie), Vfc > N. (29)

If the minimization of J∞ under these conditions is feasible, the resulting control law which beyond the horizon N corresponds to an LQR control law is stable as shown in [D.Q. Mayne et al., I.e.].

Modular Modelling of the Physical System

Single River Reach

A model of a single river reach between two power plants is derived from the Saint Venant equations o = ^ dz ₊ ^ dt (30)

^H(^ZΛ) is the water height measured from the river bed, S(M) the wetted cross- sectional area and Q(^z> 0 the discharge at the position z at the time instant t . The parameter If(z, t) is the friction slope and ¹^) is the river slope.

The partial differential equations (30) and (31 ) are discretized in time and space and linearized around an operating point characterized by Q₀(z) and H₀(^) Q{z, t) = Q_a(z) + AQ(z, t) (32)

H(z, t) = H₀(Z) + AH(z, t) (33)

The river top width ^w(^z- 0 is assumed to be constant over time for a certain op¬ erating point W₀ (z). we also assume that the cross section areas are rectangu¬ lar

S(z, t) = W₀(z)H(z, t) (34)

For the space discretization the river reach of length L is equally divided into ⁿ compartments of length ^dL = ^Llⁿ. This is illustrated in Fig. 4. The calculation points for water levels and discharges are shifted by half a compartment length and are placed alternately along the river. The inflow Qm and the outflow <w are located at the same position as h and ^«+1 respectively.

Applying a zero order hold (ZOH) time discretization, the linear, discrete-time state-space model ξ(k + l) = Θξ(k) + Φti(fc) (35) y(k) = Υξ(k) (36) is obtained with

as the discrete state-space vector and the discrete input vector, respectively.

To model a branching, the derived equations are adjusted for some calculation points close to the branching. In Fig. 5 the discretization in the vicinity of the branching is given. The variables for which the calculation is adjusted are K,, <A and Ql. For the rest of the variables, the previously derived equations are ap¬ plied without changes.

For the calculation of ^hn the total discharge Qi at the downstream side is needed, which is the sum of the discharges ^Qi and ^Ql. The adjustments for the calculations of ^Qi are more involved. The share of ^Qn-ι which will pass the calcu¬ lation point of ^Ql is needed and is approximated by 9n-i(*0 - ^Qli(^k). Additionally, the water level upstream of the calculation point, resulting from the water flow without a branching, is needed. This water level is approximated by Ki.

Single Power Plant

The spatial discretization of the Saint Venant model for a channel power plant is given in Fig. 6. The application of the Saint Venant model to this structure is described in the following.

The state space representation Eqs. (35) and (36) of the obtained model is re¬ stated as ξ(k + l) = Θξ(k) + Vu(k), ₍₃₈) y(k) = Tξ(k), (39) where Φ) contains the water levels and the discharges at the sampling points. The inputs ^u(^k) are the inflow into the river reach Qm and the discharges at the weirs Ct and the turbines ^QLt. The output v(^k) of the system is the concession level. Thus, the state, the input and the output vectors are

Note that all water levels and discharges are normalized and correspond to de¬ viations from the operating point.

Blocking Control Moves

To assure open-loop stability of the Saint Venant model, the one-dimensional

Courant criterion must be fulfilled: dz

Δt < — . c (41 ) For the Saint Venant model, At is the length of one time step, dz is the com¬ partment length and c is the wave propagation celerity

with the average water height above ground H_aVm To fulfill the Courant criterion for a Saint Venant model of a realistic river reach, time steps of at most one second are needed. This is too short for the MPC scheme as a long prediction horizon would be necessary to cover a time interval of physical relevance (tens of minutes). To overcome this problem, the inputs are assumed constant for m time steps. One control step thus comprises m simulation steps of the Saint

Venant model. For that purpose £(^m) is written as a function of f (°) ξ(l) - Θ£(0) + Φu(0), (43) ξ{2) = θ²£(0) + θΨ«(0) + Ψ«(l), (44)

ξ(m) = θ^mξ(0) + θ^m-¹Ψ«(0) + . . . + ΘΦi/(m - 2) + Φϊi(m - l). (45)

All inputs are assumed constant during these m time steps

«(0) = ti(l) = . . . = u(m - l). (46) yielding

ζ(m) = θ^mξ(O) + (θ"^1"1* + . . . + ΘΦ + Φ)u(O). (47)

Hence, the system and the input matrices for a blocking control move of ^m time steps can be defined as e_m = e^m, (48) τn-1 Φ_m = ∑ Θ'Φ,

^ (49) and the state space representation results in ξ{k + l) = θ_mξ(k) + *m<k), (50)

V(k) = Tξ(k), (₅₁ ) where one time step now corresponds to one control step of length m ^■ At as opposed to one simulation step of length At in the original Saint Venant model. With this new time discretization, k denotes the discrete control steps. In the further considerations, this definition of k will be used. Fixed Internal Model Inputs

MPC assumes that all elements in the input vector are manipulated variables. This does not hold for ^u(^k) in Eq. (40). The input Qm is regarded as a distur¬ bance, which is given as the measured inflow to the river reach, and therefore cannot be manipulated. Additionally, either the discharge at the weirs C* or at the turbines iLt is fixed, whereas the other discharge is the manipulated vari¬ able which is used for control.

In order to adhere to these settings, we separate the fixed inputs from the ma¬ nipulated variable, which is the discharge at the weirs or the turbines depending on the operating regime. Using

the system Eqs. (50), (51) results in ξ{k + i) = θ_me(fc) + Φrⁿ«^mon(*) + /. (53) y(k) = Υξ(k), ₍₅₄₎

where / is obtained from the fixed inputs u^ftx(k)_t which are assumed constant for the entire horizon, yielding

/ = ψ£V^« ₍55)

Consequently, the system is transformed into an equivalent linear system. To dispose of the affine part, the steady state solution &>, u_s of the system Eqs. (53), (54) is calculated first. For the steady state solution, the system equations ξ_s = θ_m& + Φ£"X + /, (56)

Vs = T& (57) hold. To obtain the steady state solution, Eqs. (56) and (57) are written as

/-θ_m -*rⁿ l [ & 1 ₌ [ / 1

T 0 J L u_s J [ y. J ^• ₍53₎ This system is under-determined because there are more variables, namely £«, u_s and Vs, than equations. An approach which yields a solvable algebraic sys¬ tem is to set

Vs = Υξ_s = 0 (59) as an additional condition. The steady state values ^ and u_s are then obtained from Eq. (58).

Subtracting Eqs. (56), (57) from (53), (54), respectively, the affine part can be disposed and a linear system results ξ(k + i) - ξ_s = e_m(ξ(k) - ξ_s) + *rⁿ(.y<^man(k) - u_s), (60) y(k) - y_a = T(ξ(k) - ξ_s). ₍₆₁)

From this it can be seen that imposing 2/« = ° leads to the desirable outcome that the output of the linear system is equal to the output of the affine system.

Δu Formulation of the Internal Model

An important control objective is to dampen variations in the discharge. This is achieved by minimizing the changes of the manipulated variable u^mαn(k). The change of the manipulated variable ^δu(^k) is used as system input rather than the absolute value. This can be done by including the discharge u^mαn{k) - u_s j_n the state vector as an additional state, which is the sum of the previous dis¬ charge and the current change in the discharge. This yields the augmented state space formulation

& + 1) - ξ_s i _ r ^β _m ΦΓ 1 r m - ξ. l , r * t; m»an , _δu(k) x(fc+l) x(k)

(62)

This model in state space representation x(k + 1) = Ax{k) + Bδu(k), (54) y{k) = Cx{k) (65) is used for the MPC problem formulation.

Cascade of River Reaches

The river reaches between the power plants are self-contained systems, and models for these river reaches are derived as previously shown, and combined to model an entire cascade. The river elements that are required for this modu¬ lar approach are given in Fig. 7 and allow to compose any combination of power plants with or without man-made channels.

For each river reach 3 the system ξ_j(k + l) = e&(k) + V_jUj(k) (₆₆) _yj(k) = Υ_jξ_j(k) (67)

results, where &(*) and ^uΛ^k) correspond to the vectors given in Eq. (37). The output %^■(*) of the system is the concession level ^hcA^k).

Cascade of Power Plants

The considered cascade example lies in the course of the Untere Aare. The structure of this cascade is given in Fig. 8. The plants Aarau, Rϋchlig, Wildegg- Brugg and Beznau are channel power plants, whereas the plant Rupperswil- Auenstein is a basic river power plant. Between Wildegg-Brugg and Beznau, an additional inflow from the rivers Reuss and Limmat has to be taken into ac¬ count.

Depending on the operating regimes, the manipulated discharge at the power plant 3 denoted as ^qT^an is either the discharge through the weirs ^q7 or the dis¬ charge through the turbines ^qJ. The other discharge remains constant and is denoted as ^qi . For a basic river power plant, the weirs and the turbines are located at the same place. Thus, it is possible to model this as a single overall discharge which always corresponds to the manipulated discharge ^qT^an. For _nfιx these plants, no fixed discharges h exist.

The model of the cascade can be built from the models of the different river reaches. For each river reach * a system

φ 77? on IX

+ ^™ itn δu_t(k) + w. fix (68)

0

results, which is derived from Eqs. (62), (63) by leaving out the steady state elaborations and keeping the affine part. These models are connected such that the outflows out of a river reach correspond to the inflows into the successive river reach. For the cascade in Fig. 8, the state vector, the input vector and the vector of fixed inputs are

The outputs v(k) of the system are the concession levels of all power plants

The affine state space representation of the cascade model is χ(k + l) = Aχ(k) + Bδu(k) + f, (72) y(k) = Cχ(k), (73) where the system matrix A, the input matrix B, the output matrix C and the affine part / have been built from the system matrices of the individual river reach models Eqs. (68), (69) according to the vectors given in Eqs. (70), (71 ).

To dispose of the affine part /, the equations to calculate the steady state vec¬ tors x<>, δu_s and Vs are

X_s = Aχ_s + Bδu_s + f, (74)

Vs = Cχ_s. (75)

As the discharge changes in steady state δu_s are zero, these equations can be written as

I ^c J^Xs= LJ- (76)

The steady state vector Xs also contains the steady state values of the total manipulated discharges ^qT^an. The rows of (^{J ~ A}) corresponding to these total manipulated discharges are zero. Therefore the system Eq. (76) is under- determined. The condition

Vs = Cχ_s = 0 (77) turns Eq. (76) into a solvable algebraic system. With the steady state solution x_* a linear system for the cascade is derived

resulting in x{k + 1) = Ax(k) + Bδu(k). (80) y(k) = Cx(k) (81 ) as internal model for MPC. Additional Considerations

Change of Operating Point

The Saint Venant model is derived from the Saint Venant equations by lineari¬ zation around an operating point and discretization in time and space. As most cascades of river power plants have a length of several kilometers the propaga¬ tion of a disturbance through the cascade takes several hours. Thus, different operating points in different parts of the river are applied. This is possible be¬ cause the river reaches between power plants are self-contained systems and the operating point for one river reach can be chosen independently of the op¬ erating points of the others.

The desired steady state value of the concession level is the same for all oper¬ ating points, namely the reference value prescribed by the authorities. Addition¬ ally, the steady state discharges along a river reach without branchings and junctions are the same in the whole reach. From the overall discharge and the concession level, a steady state water level line in such a river reach can be determined. The discharge and the corresponding water levels are used as op¬ erating point values. Thus the number of operating points is determined by the chosen resolution of the overall discharge.

For a river reach bounded by channel power plants, the situation is more com¬ plex because branchings and junctions have to be considered. In Fig. 9 such a river reach is shown. In steady state, the total inflow Qm = C + QL and the total outflow lout = Ct + ^QΪut are equal to the discharge Qc, measured at the point of the concession level.

The distributions of the inflow Qm and outflow Qout among the weirs and turbines are not distinct. For a specific overall discharge Qc, there are different possible combinations of the discharges C, <4, Ct and oLt. For a given discharge resolution, these combinations are bounded. The effort to determine these op- erating points can be remarkably reduced by taking into account that in steady state the river reach can be cut into two independent parts at the point of the concession level. Depending on the discharge at the concession level Qc and the distribution of this discharge among weirs ^Qf_n and turbines ^QL upstream, the parameters of the upstream part can be determined without any consideration of the distribution at the downstream power plant. The reverse holds for the downstream part of the reach. The motivation for this is that the concession level h_c and the discharge Qc at the same point are independent of the dis¬ charge distribution among weirs and turbines and can be used as boundary conditions for the two parts.

If there are lateral in- or outflows in a river part, the overall discharge Qc is dis¬ tributed among the discharges C, <L, and ^Qin and in steady state

Hence, additional in- and outflows increase the number of possible discharge distributions for a given overall discharge. Apart from the increase in the num¬ ber of possible combinations, the concept of determination of the steady state parameters for the operating points remains the same as above.

Because the number of different operating points is bounded for a specific dis¬ charge resolution, the parameters for the operating points can be determined a priori from steady state measurements at the natural river and stored in an ap¬ propriate data structure. If a change in the operating point is necessary, the pa¬ rameters of the new operating point are retrieved and the matrices of the inter¬ nal model of MPC are constructed.

As the operating point depends on the discharge, the criterion for a change in operating point is the discharge at the point of the concession level. If the dis¬ charge at this point deviates by more than a certain threshold from the current operating point discharge, then an adaptation of the internal model matrices to the new operating point is initiated. Balanced Model Reduction

A disadvantage of the derived state space internal model is the large number of states. A linear system with a reduced number of states is computationally pref¬ erable, as long as it accurately models the input-output behavior of the original system.

For balanced model reduction, standard algorithms are available [M. G. Safonov, R. Y. Chiang, "A Schur Method for Balanced-Truncation Model Reduction," IEEE Transactions on Automatic Control, Vol. 34, No. 7, pp. 729- 733, 1989]. The corresponding linear transformations between the full state vec¬ tor ^χ(^k) and the reduced state vector ^r(^k) are written as

x(A;) = T_rxr{k). (₈₄)

This balanced model reduction is applied to the original system transforming it into r{k + l) = A_rr{k) + B_τδu{k), (₈₅) y(k) = C_rr(k). (86)

The system matrix A-, the input matrix B_r and the output matrix CV of the re¬ duced internal model are obtained as

A = T_xrAT_rx, (98)

Predicted Disturbances

The internal model presented so far assumes that the fixed inputs stay constant over the entire horizon, which results in a constant affine part /. If additional information about future disturbances is available, this can be incorporated in the problem formulation leading to improved control performance. Even if no exact predictions of future disturbances are possible, a rough estimate of the future discharge evolution can be extracted from available discharge measure¬ ments further upstream of the river.

To obtain a linear system, an auxiliary state variable which is always equal to one is introduced, and the affine part is included in the system matrix. The affine system

[ Si⁺^ ] - [ ^eo ^■ T } [ „-#'- ., H T ] *<*> ÷ /⁽*⁾. «^») vW = t ϊ o ] [ _o,J⁽*>_ _1} (91)

is reformulated as the linear system

H(k + D θ_m Φ;;Γ f(k) ξ(k) u^man(k) o 1 0 u^man(k- 1) + 1 δυ(k), (92)

1 o 0 1 1 0

Tf(k+l) A,(k) x_f(k) ξ(k) y(k) T O O ] u"'^an(k- 1) (93)

Cr 1

X_f(k)

In this case, the system matrix Λf(*O is time-variant. In the general problem formulation for MPC above it was assumed that the system matrix was constant over the entire horizon. For a time-varying system matrix Af, Eq. (7) has to be reformulated. The calculation of ^xf(^Q)>-i^χΛ^N) as functions of ^x/(°) and δu(Q),...,δu(N- l) becomes

X₁(O) /

*/(i) Af(I)

X₁(O) +

L ^χf(N) J A_f(N) -...-AfQ)Af(I)

which is denoted as

X_/ = S_/* x_f {0) + S? δU ₍₉₅₎ and replaces Eq. (8). The rest of the problem formulation can be adopted from the section entitled "Problem Formulation" above.

Proposed Control System

Control Objectives

The primary objective of the supervisory control system is to dampen the dis¬ charge variations. This corresponds to keeping the changes in the discharges at the weirs and turbines as small as possible. A secondary objective is to avoid large deviations of the concession level from the reference. As these are con¬ tradictory demands, a trade-off between the two criteria results.

Moreover, the authorities impose limits within which the concession level may vary. These limits may be violated only for a short time or under extraordinary circumstances like floods, heavy rainfalls or emergency cases at the power plant. Hence, the limits on the concession level are accounted for by using soft constraints.

Concerning the turbines, physical limits on the minimal and maximal discharge and on the maximal rate of change of the discharge exist. The turbines can pro¬ vide a discharge that is bounded between zero and a certain maximum value. The maximal rate of change of the discharge is given by the maximal accelera¬ tion or deceleration of the turbines, which is roughly ± 200m³/s/min. These are physical constraints and therefore hard constraints which cannot be violated.

Similarly for the weirs, the discharge and the change in discharge are limited. For channel power plants, the authorities set a lower limit on the discharge through the weirs to reduce the impact of the plant on nature. It is assumed that an upper limit for the weir discharge does not exist. The weirs handle all the water which does not pass the turbines. To change the discharge, the weirs have to be opened or closed. These movements limit the maximal change in discharge. Typically, this limit is roughly ± 50m³/s/min.

Control Problem Formulation

Single Power Plant

Given the control objectives and the constraints that were described above, we formulate the optimal control problem for a single power plant. The concession level h_c(k) and the total manipulated discharge, denoted here as ^£«(&), are con¬ tained in the state vector ^χ(^k) and therefore their limitations can be handled as state constraints. Since the imposition of hard constraints on either the water level or the discharge might lead to infeasibility, the operational constraints on ^hΛ^k) are softened with a slack variable. The constraints then are

h_c(k) - ε_h(k) ≤ h_c(k) < h_c(k) + ε_h(k), ₍₉₆)

0 < ε_k{k), (97) x_u{k) < x_u{k) < Z_n(A:), (98)

The slack variable ^εh(k) is determined by the MPC algorithm. In the ideal case, ^εh(k) is _Zero and the original limits !h(k) and M^fc) apply. To keep ^ε/»(^fc) small, a high penalty Qε is assigned to it in the cost function. There are no slack vari¬ ables for the constraints on ^χΛ^k) and Su{k)_t because these are hard con¬ straints. With SU being the control sequence, i.e., the inputs at each time step within the horizon

and δU* the optimal control sequence, the problem formulation is δU*

subject to the model Eqs. (64), (65) and the constraints (96)-(99).

For one power plant, Q has just one non-zero element on the diagonal in the position of the concession level and Tl is a scalar. Because there is only one slack variable, Qe is also a scalar.

Cascade of Power Plants

For the control of a cascade of power plants constraints are present on each concession level, on the total manipulated discharges and on the changes in the manipulated discharges. The concession levels and the total manipulated discharges are contained in the state vector ^χ(^k) and therefore are state con¬ straints. The constraints on ^hc(k) are operational constraints and are formulated as soft constraints h_c(k) - ε_h(k) < h_c(k) < h_c(k) + ε_h(k)_: (₁₀2)

0 < e_h(k). (₁₀3) where ^W is a vector with the slack variables as elements. The constraints on the total manipulated discharges, aggregated in the vector ^χu(k)_ι and on the in¬ puts ^δu(^k) are hard constraints defined as

^χ _u{k) ≤ ^χ _u(k) < ^χ _u{k), (104) δu(k) < δu(k) < δu(k). (105)

The problem formulation is

N-I δU^* = argmm VV(Jt)Qz(Jt) + δu^τ (k)Tlδu(k) + εl(k)Q_εε_h(k)}

S ^mU tO (106) subject to the model Eq. (80), (81 ) and the constraints (102)-(105). The weight matrix Q has non-zero elements only on the diagonal in the positions of the concession levels whereas Tl and Qε are both diagonal matrices of dimension 5, which is the number of the cascaded plants considered. Estimation of Non-Measurable Quantities

Single Power Plant

In the concept of MPC the values of the state vector describing the state of the plant are measured before each control step. In the case of a cascade of river power plants these values correspond to the water levels and discharges along the river. Since not all these values are measurable, they are estimated from the available measurements of the concession level and the headwater level of the power plants.

To explicitly incorporate model and measurement errors, a Kalman filter [R.E. Kalman, "A new approach to linear filtering and prediction problems", Transac¬ tions of the ASMA - Journal of Basic Engineering 82 (Series D), pp. 35-45, 1960] is used for estimation. Assuming Gaussian white noise on model states and measurements, the Kalman filter minimizes the steady state error covari- ance

P(oo) = Um £{(*(*) - x(k))(Φ) ^~ mf}, ₍₁₀₇₎ where ^W is the estimation of ^χ(^k). The system is extended to x(k + l) = Ax{k) + Bδu(k) + v{k) (108) y{k) = Cx(k) + φ(k) (109) where model errors and measurement errors are brought in as noise ^v(^k) and ψ(^k), respectively. Under the assumption that the noise is zero mean and noise values at different times steps are uncorrelated, the symmetric, positive semi- definite covariance matrices of ^v(^k) and ^<f(^k) are

£?M*)i,^r(Q> = I _{0 [{ k ≠ ι} , E{_Ψ(k)ψ^T(l)} = { _{0 ή} . _{k ≠ L (1 10)}

The Kalman filter then is given as x^*(k + l) = Ax(k) + Bδu(k) (Ϊ Ϊ I ) x(k) = x^*(k) + K(k) [y(k) - Cx*(k)} _{(1 12}) with ^χ*(k) as extrapolated state vector. The covariance matrix of the errors in ^x*(*0 indicated by P^*i^k), the gain matrix ^κ(^k) and the noise covariance matrix

H^k) of the errors in H^k) are

P*(Λ + l) = AP(k)A^T + A _{(1 1}3)

K{k) = P^*(k)C^T [CP^*{k)C^T + Φ]^'1 (-I ₁₄)

P(A;) = P*(k) - K(k)CP^*(k). _{(1 1 5)}

For given initial conditions ^x*(®) and ^p*Φ) this algorithm can be carried out it- eratively at each control step.

The state estimator adapts the covariance and the gain matrices at each control step such that the model errors are filtered. The longer the algorithm is running the better are these adaptations because more measurements could be taken into account. At each control step the formulae Eqs. (113)-(115) are applied to update the matrices. In these formulae the state space matrices A, B and C are used. These state space matrices have been derived by linearizing the Saint-Venant equations around an operating point. When the operating point changes, these matrices are altered and the covariance and the gain matrices do not correctly filter the model errors any more because the algorithm assumes that the values in the state vector are always referenced to the same operating point. Therefore the covariance and the gain matrices have to be adapted to the new operating point. This is done by going a certain number of time steps back into the past and redoing the calculation of the covariance and gain matrices using the state space matrices of and the measured output values referenced to the new operating point.

Cascade of Power Plants

As the river reaches between two power plants are self-contained systems, a separate state estimator for each river reach can be used. One single state es¬ timator for the overall system would lead to higher computation effort because of the larger dimensions of the involved matrices. For each river reach i, a system is extracted from Eqs. (80), (81 ), and Gaussian white noise is assumed on the model states and the measurements resulting in

X₁[Ic + 1) = A_ιx_t(k) + B_tδu_ι(k) + i/_τ(k), (<| «| 6)

The covariance matrices ^Λ» and Φ_* of the noise ^u>(^k) and Ψτ(^k), respectively, are defined the same way as in the previous section. For each of these systems the Kalman filter algorithm described for a single Plant can be applied.

As will be seen in the simulation results, the measurements of the concession level and of the headwater level of each power plant are sufficient to obtain ac¬ curate estimations of all system states. Since the measurement equipment for these two water levels is already installed, no additional costs arise to obtain measurements.

Results and Comparison

This section describes closed-loop simulations of the power plant cascade in the course of the Untere Aare (Fig. 8) to evaluate the performance of the devel¬ oped model-based supervisory controller. To simulate the considered river part, the state-of-the-art river simulation software FLORIS [available from Scietec Flussmanagement GmbH, Linz, Austria, www. scietec . at] is applied. The topographic data of the channel power plant Beznau was taken to build a cas¬ cade of five river power plants. The structures of these plants and the distances between them have been adapted to the real cascade in the Untere Aare, but the cross section geometry data were replicated from Beznau for all plants.

The MPC tuning parameters employed in all simulations in this section are shown in Table 1. The simulation time step is 72s, meaning that roughly every minute a new control step is calculated and applied. The horizon is set to 50 time steps (=1 hour). This approximately corresponds to the propagation delay from the upstream to the downstream end of the cascade and thus allows to cover the propagation of an incoming disturbance through the whole cascade. Table 1 : MPC tuning parameters for the cascade

The control parameters are tuned with focus on discharge damping of the cas¬ cade as a whole which corresponds to the damping of the discharge at the last power plant. Therefore, the largest weight (1.0) is assigned to changes in dis¬ charge of the last power plant (Beznau). As the discharge changes at the other power plants are of minor interest, the respective penalty for the fourth power plant is ten times lower (0.1 ) and the changes in the discharge of the first three power plants are penalized with even smaller weights (0.01 ).

To fully utilize the available storage volume for disturbance damping, the weight on the concession level deviations is very small (0.002), which merely ensures that deviation of the concession level is eventually driven back to zero and does not remain at the limits in steady state. The slack variables are heavily penal¬ ized (10.0) in order to not violate the imposed concession level bounds during regular operation.

The closed-loop performance is not sensitive to the exact choice of these pa¬ rameters. As the tuning parameters do not depend on the operating point, they are the same for any given overall discharge, resulting in a straightforward tun¬ ing. For comparison with the control concept currently applied, local Pl controllers with additional feed-forward terms for all river reaches were provided by a local company and connected to FLORIS. All parameters of the Pl controllers were tailored to the modelled power plant cascade by the local company as well.

Sinus Disturbance at Low Discharge

The first simulations are run at a nominal steady state discharge of 200m³/s. This low discharge is chosen as a case study because this hydraulic situation is very sensitive to incoming disturbances and thus difficult to control. At this low discharge, the turbines are in charge of concession level control. The weir dis¬ charges are constant during the entire simulation. The initial discharge distribu¬ tion among weirs and turbines is shown in Table 2.

Table 2: Discharge distribution at 200m³/s

The discharge variations with respect to the initial value of 200m³/s that are im¬ posed as disturbance are shown in Fig. 10. Such discharge variations, which are approximately sinusoidally shaped with an amplitude of ± 30m³/s appear frequently in the considered power plant cascade and are caused by the too large thresholds in the controllers of the upstream power plants. Figure 11 shows the resulting concession levels and the turbine discharges at the five power plants (P1-P5). No disturbance predictions were incorporated. Results for the local Pl controllers are shown as dash-dotted lines, results for the supervisory MPC controller as solid lines. All Pl controllers apply feed¬ forward terms, such that the weir discharge at one power plant follows approxi¬ mately the outflow of the upstream plant as described. Because the main con¬ trol objective for the Pl control concept is to keep the concession levels con¬ stant, it adjusts the power plant discharges accordingly. As a consequence, the discharge variations that are imposed as disturbance are even amplified. To achieve a damping of the discharge variations, the Pl controllers would have to be tuned less aggressively (e.g. for plant P3). But a specific tuning that would yield concession levels varying exactly within the prescribed limits for this dis¬ turbance would likely cause too large level deviations for more intense distur¬ bances.

The MPC controller explicitly takes the level constraints into account and utilizes the allowed deviations of the concession levels well. The disturbance damping is improved from one power plant to the next one, such that at the fifth power plant P5 the initial discharge variations of ± 30m³/s are reduced to about ± 5m³/s Because of the comparably small storage volume in the second river reach (between P1 and P2), the second power plant P2 achieves only a mar¬ ginal damping. Nevertheless, it keeps its concession level within the specified constraints, whereas they are severely violated with the Pl controller. Already at the beginning of the disturbance, MPC lowers the levels for later compensation of the propagating disturbance, while the Pl controller can only react on distur¬ bances that have already arrived at the respective river reach.

Ramp Disturbance at Low Discharge

Another typical disturbance is a ramp shaped increase in discharge which oc¬ curs in case of rainfalls. Also for this test case, the most difficult operating range is at very low steady state discharge of 200m³/s. A ramp of + iθθm³/s within one hour is imposed to simulate heavy precipitations (Fig. 12). Figure 13 shows the resulting concession levels and discharges. Again, no dis¬ turbance predictions were incorporated. Results for the local Pl controllers are shown as dash-dotted lines, results for the supervisory MPC controller as solid lines. With Pl controllers, the concession level at plant P2 again deviates heav¬ ily while the level constraints are well met by the MPC controller. The dis¬ charges resulting from Pl control are steepened from one power plant to the next one which results in a large overshoot at the fifth power plant. The MPC controller in contrast achieves a considerable damping of the incoming ramp disturbance.

The evolution of the concession levels in Fig. 13 illustrates the difference be¬ tween the two applied control strategies. While the main objective of the Pl con¬ trollers is to keep the concession level constant, the MPC controller explicitly aims at utilizing the allowed level deviations to dampen the discharge variations by not immediately bringing the concession levels back to their references. The MPC tuning in this section focuses on discharge damping, but also with MPC, a more aggressive control which brings the concession levels faster back to zero is possible. This can be achieved by increasing the penalty on the concession level deviations. The consequence of more aggressive concession level control would be a slightly larger overshoot of the discharge at the fifth power plant which is necessary to drain the retained water more quickly and bring the con¬ cession levels down to zero.

The small peaks that can be observed with the MPC scheme in the discharge of P1 and P2 result from the assumption of constant incoming disturbances during the whole MPC horizon. Figure 14 in contrast shows simulation results for an MPC controller which incorporates predictions of incoming disturbances. It is assumed that the future disturbances can be perfectly predicted from upstream measurements 1h in advance. Due to these disturbance predictions, the dis¬ charge of the turbines at the fifth power plant P5 already starts to rise at t = 0 h. The incoming increase in discharge between t = Ih and t = 2h is smoothed out and slowed down by the cascade and takes approximately four hours at P5. The evolution of the resulting concession levels also shows the very early con¬ trol actions which lower the levels from the very beginning anticipating the in¬ coming disturbance.

Sinus Disturbance at High Discharge

In this simulation, the same sinusoidally shaped disturbance as shown in Fig. 10 is imposed on the cascade at a higher discharge of iθθθm³/s. Table 3 shows the distribution of the initial steady state discharge among weirs and tur¬ bines. For a given absolute amplitude of a disturbance, the damping is generally easier at higher discharges, since the relative amplitude of the disturbance is smaller. Additionally, at high discharges the turbines are fully utilized and the weirs are in charge of the concession level control. As the weirs are located close to the concession levels, they have a direct impact compared to the tur¬ bines which simplifies the control.

Table 3: Discharge distribution at i000m³/s

The simulation results for the Pl and MPC controllers are shown in Fig. 15. Re¬ sults for the local Pl controllers are shown as dash-dotted lines, results for the supervisory MPC controller as solid lines. For MPC, the control performance is very similar to the situation at 200m³/s. j_ne intermediate power plants P2 to P4 achieve a marginally better damping at this discharge, while the discharge variations at the fifth power plant are slightly larger compared to the same dis¬ turbance at 200m³/s Except for plant P2, the Pl controllers are obviously well tuned to keep the concession levels constant. Even though they achieve a cer¬ tain discharge damping, the aggressive concession level control causes dis¬ charge variations that are still significantly larger than with MPC.

Conclusions and Benefits

The present invention proposes the application of Model Predictive Control (MPC) for a supervisory control system for water level and flow control of dis¬ tributed water flow structures, such as river and reservoir-lake hydroelectric power plants, structured either as stand-alone systems or as cascades of com¬ binations of the two, irrigation channels, potable water supply networks etc. The supervisory control system is used to appropriately manage the water re¬ sources, with objectives that vary from case to case. These include, but are not limited to, the minimization of water level and/or discharge fluctuations at prede¬ termined points, the maximization of the economic profit produced by the use of the installation (electric energy production, delivery of potable water), the mini¬ mization of water losses, the respect of environmental constraints imposed by the authorities etc.

The specific application used for the demonstration of the supervisory control system was the water level control of cascaded river power plants. The key in¬ novations are (i) the implementation of a modular modelling strategy for the power plants cascade that can be easily used to model arbitrary river topologies and (ii) the application of model-based optimal control for the water level control of single and cascaded river power plants.

More precisely, an internal model which captures the significant river hydraulics and the impact of the power plants was developed. For this, the Saint Venant equations linearized around an operating point and discretized in time and space were used as an internal model for MPC. The main advantage of this internal model is the straightforward identification of the required parameters, which are obtained from river geometry and steady state measurements. Addi¬ tionally, all important hydraulic features are captured while the resulting internal model complexity is acceptable. The internal model was developed in a modular form and is therefore straightforwardly applicable to arbitrary river structures. As the modules of the internal model are self-contained systems, the parameter adaptation for different operating points is carried out for each module sepa¬ rately.

For the control system, MPC is applied to the control problem for a single chan¬ nel power plant. The control objectives were defined and the constraints present in the system were identified and mathematically described. A standard optimi¬ zation problem was formulated, consisting of a quadratic objective function sub¬ ject to linear constraints, where the operational constraints were softened using slack variables. As the system states are not directly measurable, a Kalman filter was applied to estimate these states from the available measurements, and an appropriately tailored adaptation algorithm was implemented to account for the changes of the operating point.

To enlarge the considered horizon and to elaborate on the stability of the pro¬ posed closed-loop system, a terminal weight on the system state was intro¬ duced under the assumption that the controller acts in a control invariant set after a certain horizon. This can be enforced by imposing a terminal set con¬ straint. Balanced model reduction was applied to lower the control problem di¬ mension, which reduced the computation effort for solving the underlying Quad¬ ratic Program. Incorporating predicted future disturbances, e.g. disturbance measurements far upstream of the considered cascade, allowed for more pre¬ ventive actions and further improved control performance.

For a cascade of power plants, an internal model was composed of sub-models for the different river reaches between successive power plants. To reduce the computation effort, a separate Kalman filter was applied to each river reach. The control objectives and the constraints that had been stated for a single power plant were adapted to formulate the MPC problem for the entire cascade.

The developed MPC control system was compared with the currently imple¬ mented Pl-type system demonstrating the achieved enhancements. In particu¬ lar, the damping of disturbances was significantly improved by coordinating the control moves in the entire cascade and considering the interactions of the power plants. In the MPC control concept, anticipated disturbances may be taken into account and compliance with the constraints is guaranteed, which is not possible with the currently employed Pl controllers that were designed to control water levels without considering discharge damping. The MPC tuning is straightforward and can easily be adapted to special hydraulic situations or emergency cases.

List of reference symbols

1 river

2 controller

Q 'in ' Q out discharge

K concession level ^href reference water level

MPC controller implementing model predictive control

RM river model

3 optimizer

4 power plant

P power plant

PM plant model r(k) reference u(k) control move

M measurement

5 predicted outputs 6 determined control sequence

7 applied control move k time step

N horizon ref reference

Λ, , Λ₃ , /J₅ , ... , ft_2#M , /I_2n+1 water level

Q₂-Q₄ Q2n-2><l2n discharge

/ length of a compartment h₂',h₂ ^r,h_n ^b water level q[,q[,qU discharge discharge (outflow)

discharge ^h _ci'^h _c2'^hc₃'^h _c4>^h _cs concession level ql,q]_n discharge (inflow)

W weir

T turbine q discharge t time h water level

P1 , P2, P3, P4, P5 power plant

Claims

Patent claims

1. A method of controlling at least one water level ( h_c ) at a predetermined point influenced by at least one hydroelectric power plant (P) by apply¬ ing control moves which comprise adjusting at least one discharge (q_out;qζ_ut,q_o'_ut ) through said power plant, characterized in that said method employs a model predictive control algorithm for deriving said control moves.

2. The method according to claim 1 , said method comprising the following steps:

(a) determining a system state from at least one measurement;

(b) determining a sequence of future control moves for a predeter¬ mined time horizon ( N ) in a manner to minimize a predetermined cost function and to satisfy predetermined constraints, wherein said se¬ quence of future control moves is based on predictions of a system be¬ havior based on an internal plant model (PM);

(c) applying the first control move of said sequence of future control moves;

(d) repeating steps (a) to (c).

3. The method according to claim 1 or 2, wherein said hydroelectric power plant (P) comprises one or more turbines (T) and one or more weirs (W).

4. The method of claim 3, wherein said model predictive control algorithm is adapted to keep a discharge (<C;qC ) through either said turbines (T) or said weirs (W) fixed.

5. The method according to claim 3 or 4, wherein said model predictive control algorithm takes one or more of the following constraints into ac- count, where each constraint may be a hard or soft constraint: a maximum and/or a minimum water level at said predetermined point upstream of said hydroelectric power plant; a minimum and/or a maximum discharge through said turbines; a maximum rate of change of the discharge through said turbines; a minimum discharge through said weirs; and a maximum rate of change of the discharge through said weirs.

6. The method according to one of the preceding claims, wherein said model predictive control algorithm is adapted to minimize the magnitude of said control moves.

7. The method according to one of the preceding claims, wherein said model predictive control algorithm is adapted to take expected future disturbances upstream of said hydroelectric power plant into account.

8. The method according to one of the preceding claims, wherein said model predictive control algorithm is adapted to take a time depend¬ ence of a unit price of electricity and/or a time dependence of demand for electricity into account.

9. The method according to one of the preceding claims, wherein a plural¬ ity of hydroelectric power plants (P1 , P2, P3, P4, P5) forming a cascade are controlled simultaneously by said model predictive control algo¬ rithm.

10. The method according to claim 9, wherein said method comprises de¬ termining, for each of a plurality of reaches between said power plants, a system state of said reach from at least one measurement, wherein each system state is based on a model selected from a plurality of pre¬ determined generic river reach models.

11. The method according to claim 9 or 10, wherein said model predictive control algorithm is adapted to take lateral inflow (q¹* ) and/or lateral outflow between said hydroelectric power plants (P1 , P2, P3, P4, P5) into account.

12. The method according to one of the preceding claims, wherein said model predictive control algorithm is based on a discrete-time discrete- space state space model which is linearized around an operating point.

13. The method of claim 12, wherein said method comprises a step of changing said operating point whenever a discharge at a predetermined position upstream of said power plant (P) changes by more than a pre¬ determined threshold.

14. The method of claim 12 or 13, wherein said state space model is a model derived from the Saint Venant equations discretized in predeter¬ mined discretization time steps and predetermined discretization space steps, wherein said discretization time steps satisfy the one- dimensional Courant criterion, and wherein said model predictive con¬ trol algorithm computes said control moves at discrete simulation time steps which are a multiple of said discretization time steps by keeping said input vector constant during each simulation time step.

15. A computer program product comprising computer program code means for controlling one or more processors of a computer such that the computer performs the following steps:

(a) receiving at least one input value representing a water level (h_c ) at a predetermined point influenced by at least one hydroelectric power plant (P);

(b) from said input value, determining a system state;

(c) based on said system state, determining a sequence of future con¬ trol moves for a predetermined time horizon (N ) in a manner to mini- mize a predetermined cost function and to satisfy predetermined con¬ straints, wherein said sequence of future control moves is based on predictions of a system behavior based on an internal plant model (PM) of said hydroelectric power plant (P) and is derived from a model predictive control algorithm ;

(d) providing as an output the first control move of said sequence of future control moves;

(e) repeating steps (a) to (d).

16. A device for controlling at least one hydroelectric power plant (P), said device comprising: measuring means for determining a water level (h_c ) at a prede¬ termined point influenced by at least one hydroelectric power plant (P); a controller (2) for deriving control moves which comprise adjust¬ ing at least one discharge through said power plant, said controller (2) receiving an output of said measuring means; regulating means for regulating a discharge (q_out ) through said hydroelectric power plant (P) according to said control moves, characterized in that said controller (2) is specifically adapted to employ model predictive control for deriving said control moves.