Stochastic Optimization Chance Constrained LP - II Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Objectives To formulate chance constrained LP (CCLP) for reservoir operation To discuss linear decision rules To discuss deterministic equivalent of a chance constraint Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Introduction In reservoir planning and operation problems, future inflow, Qt, is a random variable and is not known with certainty Its probability distribution may be estimated from the historical sequence of inflows Storage St and release Rt, are functions of inflow, Qt St and Rt are also random variables In a constraint containing two random variables, if the probability distribution of one is known, the probabilistic behavior of the second can be expressed as a measure of chance in terms of the probability of the first variable If a constraint contains more than two random variables, we get into computational complications We need to understand the specific problem clearly to reformulate the problem, if necessary, and avoid those complications. Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Chance Constraint The constraint, relating the release, Rt, (random) and demand, Dt, (deterministic), is expressed as a chance constraint P [Rt ≥ Dt] ≥ 1 Probability of release equaling or exceeding the known demand is at least equal to 1; 1 is the reliability level Interpretation of this chance constraint is simply that the reliability of meeting the demand in period t is at least 1 Similarly, chance constraints for the maximum release and the maximum and minimum storage is  P [Rt ≤ Rtmax] ≥ 2 P [St ≤ K] ≥ 3 P [St ≥ Smin] ≥ 4   Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Chance Constraint… To use the above chance constraints in an optimization algorithm, first determine the probability distribution of Rt and St Probability distribution of Qt is known Probability distribution of Rt and St can be derived from that of Qt Since St , Qt and Rt are all interdependent through the continuity equation, it is not possible to derive the probability distributions of both St and Rt simultaneously. To overcome this difficulty and to enable the use of linear programming in the solution, a linear decision rule is appropriately defined Linear decision rule (LDR) relates the release, Rt, from the reservoir as a linear function of the water available in period t. Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Linear Decision Rule Simplest form of such an LDR is Rt = St + Qt - bt where bt is a deterministic parameter called the decision parameter. In this LDR, the entire amount, Qt, is taken into account while making the release decision. Depending on the proportion of inflow, Qt, used in the linear decision rule, a number of such LDRs may be formulated. General form of this LDR may be written as Rt = St + t Qt - bt 0  t  1 t = 0 yields a relatively conservative release policy with release decisions related only to the storage, St; t = 1 yields an optimistic policy where the entire amount of water available (St + Qt), is used in the LDR. Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Linear Decision Rule… Consider the LDR Rt = St + Qt - bt Storage continuity equation is St+1 = St + Qt - Rt Using above two equations St+1 = bt Random variable, St+1, is set equal to a deterministic parameter bt Thus, the role of the LDR in this case is to treat St, as deterministic in formulation Advantage: To do away with one of the random variables, St, and the distribution of the other random variable, Rt, may be expressed in terms of the known distribution of Qt. This implies that the variance of Qt is entirely transferred to the variance of Rt. Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Linear Decision Rule… For eg. assume that the evaporation loss Et is a linear function of the average storage volume, e0t + et (St + St+1)/2 where e0t is the fixed evaporation volume loss and et is the evaporation volume loss per unit average storage volume in period t Then the linear decision rule can be written as, Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Deterministic Equivalent of Chance Constraint Knowing the probability distribution of inflow, Qt , it is possible to obtain the deterministic equivalents of the chance constraints using the LDR, as follows: P [Rt ≥ Dt] ≥ 1 P [St + Qt - bt ≥ Dt] ≥ 1 P [bt-1 + Qt - bt ≥ Dt] ≥ 1 P [Qt ≥ Dt + bt - bt-1] ≥ 1 P [Qt ≤ Dt + bt - bt-1] ≤ 1- 1 The term Dt + bt – bt-1, is deterministic with bt and bt-1 being decision variables and Dt being a known quantity for the period t. FQt (Dt + bt – bt-1)  1 - 1 (Dt + bt – bt-1)  FQt-1 (1 - 1) Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Deterministic Equivalent of Chance Constraint… (Dt + bt – bt-1)  FQt-1 (1 - 1) FQt-1 (1 - 1) is the flow, qt, at which the CDF value is (1 - 1). Similarly, Deterministic equivalent of chance constraint P(Rt  Rtmax)  2 is Rtmax+ bt – bt-1  FQt-1 (2) Since the storage, St+1, is set equal to the deterministic parameter, bt, the chance constraints containing only the storage random variable are written as deterministic constraints (without using the probability distribution of inflows). Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Chance Constrained LP Complete deterministic equivalent of the CCLP can be written as Minimize K Subject to (Dt + bt – bt-1)  FQt-1 (1 - 1) for all t Rtmax+ bt – bt-1  FQt-1 (2) for all t bt-1  K for all t bt-1  Smin for all t bt  0 for all t K  0 for all t Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Chance Constrained LP… While solving this model, for a problem with 12 periods (months) in a year, we also set b0 = b12 for a steady state solution. Further, depending on the nature of LDR used, the decision parameters, bt, may be unrestricted in sign. For example, if we use the LDR, Rt = St - bt, the decision parameter, bt, may be allowed to take negative values. CCLP can also be applied for reliability based reservoir sizing Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Example: For the following chance constrained optimization problem, formulate the equivalent deterministic optimization problem using the LDR, Rt = St – bt. Storage continuity should be maintained. Neglect losses. Following table gives the F-1( ) values for the inflows and Rmax and Rmin values for different periods. Minimize K P [Smin ≤ St ≤ K] ≥ 0.8 t P [Rt ≤ Rtmax] ≥ 0.85 t P [Rt ≥ Dt ] ≥ 0.75 t t F-1(0.0) F-1(0.2) F-1(0.25) F-1(0.75) F-1(0.8) F-1(0.85) Rmax Rmin Smin 1 12 33 60 90 93 24 2 3 20 48 80 84 6 21 36 72 85 Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Example… Solution Deterministic equivalent of P [Smin ≤ St ≤ K] ≥ 0.8 t. This can be divided into two constraints as P [Smin ≤ St ] ≥ 0.8 and P [ St ≤ K] ≥ 0.8. Linear decision rule is given as Rt = St – bt. Then, continuity equation can be written as St+1 = St + Qt –R = St + Qt – St + bt = Qt + bt Hence, St = Qt-1 + bt-1 Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Example… Deterministic equivalent of P [Smin ≤ St ] ≥ 0.8 P [Smin ≤ Qt-1 + bt-1] ≥ 0.8 P [Qt-1 ≥ Smin - bt-1] ≥ 0.8 P [Qt-1 ≤ Smin - bt-1] ≤ (1- 0.8) P [Qt-1 ≤ 2 – bt-1] ≤ 0.2 FQt-1 (2 – bt-1)  0.2 (2– bt-1)  FQt-1-1 (0.2) Therefore, (2– b3)  6 t =1 (2– b1)  12 t = 2 (2– b2)  3 t = 3 Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Example… Deterministic equivalent of P [ St ≤ K] ≥ 0.8 P [St ≤ K] ≥ 0.8 P [Qt-1 + bt-1≤ K] ≥ 0.8 P [Qt-1 ≤ K - bt-1] ≥ 0.8 K - bt-1 ≥ FQt-1-1 (0.8) Therefore, K – b3 ≥ 72 t =1 K – b1 ≥ 90 t = 2 K – b2 ≥ 60 t = 3 Deterministic equivalent of P [Rt ≤ Rtmax] ≥ 0.85 P [St - bt ≤ Rtmax] ≥ 0.85 P [Qt-1 + bt-1 - bt ≤ Rtmax] ≥ 0.85 P [Qt-1 ≤ Rtmax - bt-1 +bt] ≥ 0.85 Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Example… Rtmax - bt-1 +bt ≥ FQt-1-1 (0.85) 90 – b3 +b1 ≥ 85 t =1 84 – b1 +b2 ≥ 93 t = 2 84 – b2 +b3 ≥ 80 t = 3 Deterministic equivalent of P [Rt ≥ Dt ] ≥ 0.75 P [St - bt ≤ Dt] ≥ 0.75 P [Qt-1 + bt-1 - bt ≤ Dt] ≥ 0.75 P [Qt-1 ≥ Dt - bt-1 +bt] ≥ 0.75 Dt - bt-1 +bt ≤ FQt-1-1 (0.25) 24 – b3 +b1 ≤ 21 t =1 20 – b1 +b2 ≤ 33 t = 2 20 – b2 +b3 ≤ 20 t = 3 Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc

Thank You Water Resources Planning and Management: M6L4 D Nagesh Kumar, IISc