explicit stochastic approach for planning the operation of...

Extreme Hydroloeical Events: Precipitation, Floods and Droughts (Proceedings of the Yokohama Symposium, July 1993). IAHS Publ. no. 213, 1993. 349

Explicit stochastic approach for planning the operation of reservoirs for hydropower production

S. P. SEMONOVIC & R. SRINIVASAN Department of Civil Engineering, The University of Manitoba, Winnipeg, MB Canada, R3T 2N2

Abstract A new reliability model for planning the operation of a multipurpose reservoir for hydropower generation and flood control, which considers the stochastic nature of inflows into account, has been developed and presented in this paper. The proposed solution algorithm maximizes the benefits accrued from hydropower generation and minimizes the economic losses incurred due to the reservoir not meeting the required reliabilities for hydro-electric energy supply and flood control. This algorithm uses a separable programming formulation to approximate the nonlinear energy function. An original method of incorporating the energy function in the reliability model is presented. This model determines the optimal reservoir release policy along with the optimal reliabilities of satisfying hydropower demand and flood control storage requirements. Therefore, this tool has some advantages in planning the operations of reservoirs in extreme hydrological events such as droughts. The model is applied and tested using Manitoba Hydro data.

INTRODUCTION

Most of the models developed for hydropower optimization are deterministic (Daellenbach & Read, 1976; Grygier & Stedinger, 1985; and Reznicek & Simonovic, 1990). Modelling with deterministic inputs is quite unrealistic as the inputs such as reservoir inflows, energy demands etc. are very difficult to find a priori for the whole planning period and hence, are uncertain.

Stochastic nature of inputs can be incorporated in the model either implicitly or explicitly. In an implicit model, stochastic nature of inputs is incorporated through sensitivity analysis. Optimization is performed with different scenarios of stochastic data to evaluate their impact on the operation policy. In an explicit model, stochastic nature of inputs is incorporated directly in the model formulation.

One of the explicit stochastic approaches used for modeling the reservoirs with hydropower generation is stochastic dynamic programming (referred to as SDP) (Little, 1955; and Stedinger et al., 1984). In spite of the computational burden, it has been emphasized that the stationary and non-stationary dynamic programming can find better operating policies if better hydrologie state variables are employed. SDP algorithms only identify the optimal policy within the class of policies examined.

Another approach is chance constrained programming (referred to as CCP) which addresses the problem of including risk directly in the optimization. In this approach, the stochastic nature of inputs is incorporated directly in the model through the use of cumulative probability functions of random variables. Furthermore, probabilistic constraints and preassigned tolerance levels are used to transform the

350 S. P. Simonovic & R. Srinivasan

stochastic optimization problem into its deterministic equivalent. CCP formulations neither penalize explicitly the constraint violations nor provide a recourse action to correct realized constraint violations as a penalty (Yeh, 1985). However, the capability of CCP to incorporate the stochastic nature of inflow in a linear program offers an advantage over the other stochastic planning models.

In CCP, the reliability of the optimal policy is settled a priori. In reality, the economic considerations make such a selection rather difficult. The view of linear chance-constraints allows the interpretation of a LP model as a system (Sengupta, 1972), where each probabilistic constraint can be viewed as a system component, each with its reliability i.e. its tolerance measure. This interpretation leads to the evolution of "systems reliability programming".

Application of a reliability programming (referred to as RP) approach for reservoir management is first attempted by Colorni & Fronza (1976). In this work, a single reliability constraint is used to represent the reliability of the whole system and the purpose is to determine the monthly contract volumes to be released by the reservoir. The approach has been extended to model a single multipurpose reservoir as well as reservoir systems without considering hydropower generation by Simonovic & Marino (1980, 1981, 1982).

The present paper introduces a reliability model for planning the operation of a reservoir with hydropower generation as the major purpose. Stochastic nature of reservoir inflows is incorporated in the model through the use of cumulative probability functions. A set of probabilistic constraints associated with the reliabilities of the reservoir performance has been defined as the major component of the model. The approach considers these reliability levels as decision variables.

The next section of the paper presents a model formulation. Description of the linearization of the energy function follows. Algorithm description and the results of its application to Manitoba Hydro case study conclude the paper.

MODEL FORMULATION

Reliability model for planning a multipurpose reservoir is developed using the flow continuity equation:

S, = St_j +• i, - R, - SP, - LO, (1)

where the index t pertains to the time period (t-l,t), St is the storage in the end of the time period t; Rt is the release through the turbines in the time period t; SPt is the spilled water from the reservoir in the time period t; LOt is the loss of water from the reservoir through evaporation, seepage etc. in the time period t; and it is the inflow into the reservoir in the time internal (t-l,t). In the present study, the loss component (LO,) is ignored.

Decision variables in the model are (1) release through the turbines in the load strip j in the time period t, Rt->; (2) head in the reservoir in the load strip j in the time period t, ht-i; (3) storage in the reservoir at the end of the time period t, St; (4) spill from the reservoir in the time period t, SPt; (5) exported power from the utility in the load strip j in the time period t, Et->; (6) imported power from other utilities in the load strip j in the time period t, R,-"; (7) reliability of hydropower generation in the planning

Stochastic approach for planning the operation of reservoirs for hydropower 351

period, (3; and (8) reliability of flood control in the planning period, a. Mathematical modelling of a multipurpose reservoir with stochastic inflows

requires a set of probabilistic constraints which are derived from the physical characteristics of the reservoir as shown in Fig. 1. The reservoir is basically used for flood control and hydropower generation. To provide for the flood control purpose the maximum storage constraint can be represented as:

P ( S, < S„ ) > a (2)

where Smax is the maximum storage in the reservoir and 0t is the minimum flood control volume to be maintained in the reservoir in the time period t, and a is the probability of exceedance of this constraint. This constraint ensures that the storage in the reservoir is less than (Smax - 0,) in a% of the time period t, and thus, a possibility of a flood is avoided in a% of the time. So the value of a is taken as the reliability of flood control.

Inflow

Maximum storage (S

Flood control storage ( 8 t)

Operating Range

for hydropower production

(v t )

Dead storage

Fig. 1 Conceptual diagram of the reservoir

Using (1) in (2):

R, - spt ^ ^ - e ) > a (3)

Since it is a random variable, (3) is rewritten as a chance constraint as below:

P (h ^ Smax - 6t - S,_7 + R, + SPt) > a (4)

The load-duration curve for each time period is approximated by a few linear strips (as fraction of the time period), and the load is assumed to be constant within each strip. Different energy prices are assigned to each of these strips. Let nsl be the total number of strips in a unit time period and it is assumed that all the time periods have equal number of strips.

For each load stripy, along with the decision variables Rtj and h£J, there is an

associated storage value St}. In each time period, the nsfh load strip is the last one and

the storage in this strip is taken as the storage at the end of the time period t, i.e.:


S, = S? V t (5)

and the deterministic equivalent of (4) is written as:

nsl

Sima - 0, - C ; + £ ^ + SP, > F-\a) V t (6) ;=/

where Ft (.) is the cumulative distribution function of inflows in the time period t. The second main use of the reservoir is the hydropower generation. Minimum

storage requirement (also called a rule curve) is established based on the demand for energy from the reservoir. Second probability constraint is derived from the fact that the storage in the reservoir in any time period should not go below the minimum storage requirement in that time period.

The second constraint can be written as:

P (S, > v,) > (3 (7)

where St is the storage at the end of the time period t, vt is the minimum volume required for hydropower production, and (3 is the probability of exceedance of this constraint. This constraint ensures that the storage in the reservoir is greater than vt in 13% of the time period t, and thus, the energy requirement can be met fully in /?% of time. So the value of (3 is taken as the reliability of hydropower production.

Analogous to (6), the deterministic equivalent of (7) is:

nsl

V, " £1 + £ # + SPr * *rV-/3) V t (8)

Besides these probability constraints, the reliability model includes a number of deterministic constraints too.

Constraint on the domestic energy demand: The utility has to satisfy domestic energy demand described by the load duration curve in each time step of the planning period. The utility has to import energy (It

j) during the periods of deficit i.e. when the energy produced (C^R^h^) is less than the domestic energy requirement (ENMINt->) or if there is a firm energy commitment, the utility can export energy (E ) in order to increase its revenue. The constraint which represents these two criteria is,

O • R1, • ft1, - E\ + it = ENMItft V j,t (9)

where O is the conversion factor in the energy equation. Even though the utility can increase its net benefit by exporting lots of energy

to other utilities, there is a maximum bound on the exportable energy in each load strip j (EMAX-i).

Ejt < EMAXJ V t,j (10)

where EtJ is the exported energy. Constraints on reliabilities: Constraints on the reliabilities of the system for

flood control and hydropower generation respectively, are:

0 < a < 1 (11)

0 < 13 < 1 (12)


Constraints on storage: The hydraulic head (the difference between the reservoir head and the tail water level referred to as TWL) on the turbines is a function of the storage in the reservoir and the release through the turbines. At any time period t, the relation between the reservoir head (h ) and the storage (S ); and the relation between the TWL (hd) and the release through the turbines (R,) are expressed respectively as:

f,(St)

-f2(Rt)

(13)

(14)

The nonlinear storage-stage curve as given by (13) is linearized piecewise in the operating range (the elevation range in which the utility would like to operate the reservoir for hydropower production) and is rewritten as:

h, ,nsl i • S" = h (15)

where hmin is the lower bound of the operating range of the reservoir and "Jr is the slope of the linearized storage-stage curve in the operating range.

Storage at the end of each time period is a decision variable and its benefits are included in the objective function. Even though the storage variables appear in (15), since there is some approximation involved in deriving (15), the flow continuity equation as given by (1) is introduced as an additional constraint:

i, = S, - 5,_7 + R, + SP, (16)

Since inflow (ij is a random component, right hand side expression of (16) may lie anywhere in the range of variation of inflow. Lower and upper bounds of inflow in any time period (shown as points A and B respectively in Fig. 2 are the deterministic equivalents of the lowest and highest probabilities specified in the cumulative

0.99

I

0.01

Ft (Pt)

Inflow (it)

Fig. 2 Cumulative distribution function of sum of random inflows.


distribution function of the sum of inflows in that time period. In the database of this case study, cumulative distribution functions have been

specified between 0.01 and 0.99 for each time period. Flow values at points A and B are FEt (0.01) and F a (0.99) respectively, are the upper and lower bounds on the right hand side expression of (16). Hence:

sn,l _ y;, +T£ % + Spt> F~j {0M) V , (17)

H

sf - sT-'j + £ % + spt ^ Fil W-") v f (18)

H For a planning period of 1 year with 12 monthly time steps (T=12) and

two load strips (the on-peak and off-peak) in each time step, the model contains 86 constraints and 122 decision variables.

Objective function: The objective of this model is to maximize the benefits accrued from hydropower generation and minimize the losses incurred due to not meeting the required reliability levels for each of the purposes served by the reservoir. The objective function has three components viz. the benefits accrued from hydropower production and the yearly risk loss functions for not meeting the reliability levels associated with flood control and hydropower production respectively.

The mathematical form of this objective function is written as:

max T

£1 nsl

Y, rj • (O • R1, • rt-^+rexpi • EJt-cimpi • it

- Z » - L203)

-LCt-SPt+Br-sf (19)

First component in (19) consists of the domestic energy supply (Gi*RtJ*htJ-EtJ) with a revenue coefficient of r> (in $ per unit domestic energy) associated with it; exported energy (Ej) from the utility with an associated revenue coefficient of rexp (in $ per unit energy exported); cost incurred when importing energy (ij) from other utilities with an associated cost coefficient of amp* (in $ per unit import energy); losses due to the spilled water (SPJ from the reservoir (since it is not used for hydropower generation) with an associated loss coefficient of LCt (in $ per unit storage spilled): and the benefit expected by keeping a storage (St

nsl) for future energy generation with an associated benefit coefficient of Bt (in $ per unit storage left in the reservoir). Other two components in (19) evaluate the probability of system failure in terms of monetary losses, a and /3 are the reliabilities of the reservoir system, and L,(.) and L2(.) are the risk-loss functions for flood control and hydropower generation respectively.

Information about the types of data necessary to develop a risk-loss function for flood control can be seen in Simonovic & Marino (1981). But there is no information available in the literature to develop a risk-loss function from hydropower generation. Until realistic risk-loss functions are developed, logarithmic risk functions, which are convex functions of risks are assumed in this study.

A separable programming (referred to as SP) approach has been introduced in this model to overcome the nonlinearity in the objective function and the constraints.


The objective function given in (19) in its present form is quadratic and indefinite. By introducing two new variables Xt-i and Y(J:

Xjt = 0.5 • (flf + li) (2°)

yf = 0.5 • (rt, - fy (2i) the product term in (19) is replaced by:

Ri • H-~ wr-- inr- (22) After substituting (22) in (19), the expression (19) is still quadratic and

indefinite, but separable. Hence a SP algorithm can be used to obtain approximate optimal policy. Concept behind the SP is to approximate each of the nonlinear single valued functions by piecewise linear functions. The two functions {Xt-i}

2 and {YtJ}2 in

(22) are concave functions and hence the separable program is a convex program. So any local maximum of the objective function within the feasible region is an optimal solution to the problem and the approximating problem can have no local maxima other than its global maximum.

ALGORITHM

For solving the optimization problem represented by the objective function (19) and the set of constraints (2 through 18), a three level algorithm has been developed. The program basically follows this algorithm by combining: (a) nonlinear search; (b) separable programming; (c) linear programming which are shown as Level 1, Level 2 and Level 3

respectively in Fig. 3. The main task performed at the first level is the evaluation of reliabilities. Initial

reliability pair a0 and j30.is given as an input data. Using this pair, the algorithm computes two additional pairs of reliabilities. For these three pairs of reliabilities, three objective function values are computed (performed at Level 3). The three objective function values are compared at the first level and if all the three values are within the specified accuracy range (0.1% here), then the program is terminated and the optimal values are printed. If the three values are not within the desired range, the worst pair of reliabilities is dropped and a new pair is computed using Complex Box multivariable search. The objective function values for this new set of reliabilities are computed and compared. This iterating procedure is continued until either the objective function values converge within the desired accuracy range or the number of iterations exceeds the maximum.

At the second level, for a given pair of reliabilities, the probabilistic constraints are converted to their deterministic equivalents. Then the coefficients of the decision variables in the constraints and the objective function and the right hand side coefficients of all the constraints are evaluated. Separable programming algorithm is incorporated in this level which computes the coefficients corresponding to the decision variables R(J and ht-i, in the objective function given by (19) and the energy constraint given by (9).


Level 1

r ~i

INPUT Nonlinear Search

Evaluation of reliabilities

Nonlinear Search

Comparison of objective

function values

OUTPUT

Separable Programming

Computation of the coefficients of decision

variables

Level 2

Linear Programming Routine

Optimal policy evaluation

for chosen reliabilities

Level 3

1

Linear Programming Routine

Evaluation of optimal

value of objective function

Fig . 3 Program architecture.

At the third level, a powerful linear programming solving routine is incorporated. This routine takes the input from the second level and evaluates the optimal values of the objective function and the decision variables for a given pair of reliabilities. For the three pairs of reliabilities, three objective function values are evaluated and sent to the first level.

MANITOBA HYDRO CASE STUDY

The Manitoba Hydro system consists of 13 hydropower generating stations and 3 thermal power generating stations. It is an interconnected utility and has an agreement with its neighbouring utilities. According to that agreement, the utility will export energy to other utilities when the reservoir storage and power plant conditions permit excess energy production and will import energy from other utilities during deficit periods, while fully satisfying the domestic energy demand all the time.

Manitoba Hydro and Winnipeg Hydro operate an integrated system to supply Manitoba. The hydroplants are across the rivers Winnipeg, Laurie and Nelson. The input data such as catchment characteristics, inflow sequence, load pattern etc. for all the reservoirs in the Manitoba Hydro system are grouped as if those inputs are given for a single reservoir. For the first input, the catchment characteristics of Lake Winnipeg, the largest lake in Manitoba, is assumed to represent the characteristics of all the reservoirs in the Manitoba Hydro system.

The effect of tail water level (referred to as TWL) on the hydraulic head on the turbines is neglected. However, even for a zero discharge the TWL is 168.64 feet


(referred to as ftj which is assumed as the datum TWL in the analysis. The operating range is revised taking this datum TWL into account and the revised range is (711-168.654) ft to (715-168.64) ft i.e. 542.36 ft to 546.36 ft.

Inflow component in (1) is the sum of inflows into all the catchments in the Manitoba Hydro system. Monthly flow record from 1931 and 1989 is used in this study. The cumulative distribution functions (CDFs) of monthly inflows need to be given as input to the program.

Energy demand in all the power stations is summed to get the energy demand for the Manitoba Hydro system, which is taken as the minimum domestic energy demand. Each time period consists of two load strips viz. the on-peak and off-peak strips. Amount of energy to be exported is decided by the utility based on the availability of water, firm energy commitments etc., which has a maximum bound of 1500 MW in a single time period in the present study. The proportion of time for the on-peak and off-peak periods are 0.55 and 0.45 respectively. So the maximum exportable energy in a month (24*30 hours) in these strips is 594 000 MW h (0.55* 24*30*1500) and 486 000 MW h (0.45*24*30*1500) respectively.

Energy equation in the load stripy in the time period t, is written as:

Energy = — : [MWh] ^li>

where Rtj is the release through the turbines in kefs in the load strip j in the time period

t; htJ is the head over the turbines in ft in the load strip j in the time period t; t> is the proportion of the time period t for the load strip j in hours; e is the efficiency of the turbines (assumed to be 0.9 here); and 11.8 is the conversion factor. So the monthly energy conversion factors for the on-peak and off-peak strips, from (36), are 30.203 and 24.712 respectively.

Storage-benefit coefficients (BJ and spill-loss coefficients (SP,) are assumed as 100 000 $/kcfs and 150 000 $/kcfs respectively. So the algorithm tends either to maximize the energy production or to store the water for future use while trying to minimize the spill.

The reliabilities of reservoir system performance for flood control and hydro-power production are 0.7789 and 0.8307 respectively. Optimal objective function value is 890.2 million $. Corresponding optimal release policy is given in Table 1.

Table 1 Optimal release policy for Manitoba Hydro case study.

Month

October

November

December

January

February

March

Releases

On-peak

38.2

45.7

58.6

61.0

49.0

45.9

in kefs

Off-peak

25.6

32.7

45.7

47.5

37.4

34.5

Month

April

May

June

July

August

September

Releases

On-peak

36.0

35.0

32.5

30.6

33.8

32.1

in kefs

Off-peak

23.6

19.4

17.7

15.5

17.6

17.4


Sensitivity analysis: Use of the reliability model under extreme hydrological conditions such as droughts is illustrated through an example here.

The minimum storage (vj requirement also represents the storage level above which the operation of the reservoir is carried out for hydropower generation. When there is a higher minimum volume requirement, available storage for hydropower generation decreases. For the same energy demand, because of the reduced available storage the reliability of the reservoir for meeting this demand decreases. Results of this sensitivity analysis is shown in Fig. 4.

Similar studies could be carried out with low flows as the inflow pattern in the data for the reliability model, reliability of the reservoir for meeting a given energy demand could be evaluated. This reliability of the decision maker in taking suitable preventive measures to reduce the negative consequences of failure, should a failure occur.

0.85-

0.75-

0.B5-

~~-*^

• * ^

> ; - •

~ > . -

0.05 0.1 Minimum required storage

0.15 0.2

ALPHA - + - BETA

Change in reliabilities with minimum storage bound

Fig. 4 Sensitivity of the reliability to the minimum storage requirement.

CONCLUSIONS

A new reliability model for planning the operation of a single multipurpose reservoir for hydropower production and flood control, has been developed. This model has been applied to Manitoba Hydro system. Logarithmic risk-loss functions and linearized storage-stage relations are used in the model. A separable programming algorithm is introduced to approximately linearize the hydropower function and a new method of incorporating this algorithm in the reliability model is presented.

The model finds a policy which maximizes the benefits accrued from hydro-


power generation and which minimizes the economic losses incurred due to not meeting the required reliability levels from the various purposes served by the reservoir. This model finds the optimal values of the reliabilities in addition to the optimal release policy.

Acknowledgements Support for the presented research has been provided by Manitoba Hydro Research and Development Committee.

REFERENCES

Colorni, A . , & Fronza, G. (1976) Reservoir management via reliability programming, Wat. Resour. Res., 12(1), 85-88.

Daellenbach, H. G., & Read, E. G. (1976) Survey on optimization for the long-term scheduling of hydro-thermal power systems, presented to the ORSNZ conference, New Zealand.

Grygier, J. C , & Stedinger, J. R. (1985) Algorithms for optimizing hydropower system operation, Wat. Resour. Res., 21(1), 1-10.

Little, J. D. C. (1955) The use of storage water in a hydroelectric system, J. Oper. Res. Soc. Am., 3, 187-197. Reznicek, K . K., & Simonovic, S. P. (1990) An improved algorithm for hydropower optimization, Wat.

Resour. Res., 26(2), 189-198. Sengupta, J . U. (1972) Stochastic Programming Methods and Applications, North-Holland, Amsterdam. Simonovic, S. P., & Marino, M. A. (1980) Reliability programming in reservoir management, 1. Single

multipurpose reservoir, Wat. Resour. Res., 16(5), 844-848. Simonovic, S. P., & Marino, M. A. (1981) Reliability programming in reservoir management, 2. Risk-loss

functions, Wat. Resour. Res., 17(4), 822-826. Simonovic,. S. P., & Marino, M. A. (1982) Reliability programming in reservoir management, 3. System of

multipurpose reservoirs, Wat. Resour. Res., 18(4), 735-743. Stedinger, J . R, Sule, B. F. & Loucks, D. P. (1984) Stochastic dynamic programming models for reservoir

operation optimization, Wat. Resour. Res., 20(11), 1499-1505. Yeh, W. W-G, (1985) Reservoir management and operation models: A state of the art review, Wat. Resour.

Res., 21(12), 1797-1818.

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