Mathematical formulation of the single reservoir SDP Model

8. APPLICATION OF A STOCHASTIC DYNAMIC PROGRAMMING

8.1 Mathematical formulation of the single reservoir SDP Model

Typically the system of power plants along the Victoria Nile (Figure 7.1), is modelled within the framework of a periodic Markovian decision process. The SDP model employs the method of successive approximations originally developed by White (1963) and extended and applied by others to derive operation rules for a single multipurpose lake-reservoir systems (Su &

Deininger, 1974; Oven-Thompson et al., 1982; Bras et al., 1983; Nandalal and Bogardi, 2007).

This approach is particularly suited to the serially correlated nature of inflows or net basin supplies which have been analyzed in detail for Lake Victoria in Chapter 6. A key element of the solution procedure is to select the two state variables for which release decisions are sought.

These are the reservoir storage at the beginning of a particular period and the total inflow realized in the preceding period. This strategy recognizes the reality of not being able to predict, at present, the exact inflow into Lake Victoria at the beginning of each period due to the large basin size involved and underlying complexity of its hydrological regime.

Similar versions of the procedure that consider storage at the beginning of the month and an inflow or forecast for the same month have been applied elsewhere (e.g. Alarcon & Marks, 1979; Tejada-Guibert et al. 1993; Kim and Palmer, 1997; Castelletti et al. 2005, 2007; Soncini- Sessa et al., 2007b) but were not pursued due to the limitations cited under this case study.

The mathematical formulation of the single reservoir/lake model begins with the variable definitions described in Table 8.1.

Table 8.1 Variable definitions of the SDP mathematical formulation

Variable Definition

T number of time periods per cycle.

t index showing the time intervals in a cycle in the real time direction (t = 1 ) i.e. one cycle per year

N is a stage counter in the successive approximations procedure. The total number of stages in the algorithm is denoted by N. Generally, the value of N is a multiple of cycles i.e. one cycle per year in this case. The cycles are repeated several times in the sequential decision process hence N = t*L where L = the number of cycles to be evaluated in the stochastic dynamic programming calculations. In a backward moving SDP algorithm, N takes on the values N, N-1,…,1.

S

i storage state variable representing the lake-reservoir volume at the beginning of period t. This variable can assume the value of M discrete values i = 1, …, M.

Qj^ inflow state variable representing the net inflow or net basin supply to the lake-reservoir during the period t -1. The net inflow takes on N discrete values j = 1, …, N.



^t^, ^t ¹^, ^t



t i i

f S S^ R is the function of the SDP model for any period t which represents the energy generated downstream of a lake-reservoir during stages n = 1,….N. It is a function of the beginning and end of period discrete reservoir storages

S S

_i^t

,

_i^t^¹

,

respectively and R^t,which is the total volume of release from the reservoir during period t.



^, ¹



n t t

i j

F S Q^ is the objective function of the SDP model which takes into account the accumulated expected energy generation by optimal operation of the system over stages n, n + 1,…,N, assuming reservoir storage is at current discrete level

S

_i^t and previous period net inflows were Q^t_j^¹. / 1

t t

k j

P Q Q ^  is a transition probability which represents the probability of the k^th discrete value of inflow event

Q

_k^t occurring during period t,

conditioned on the previous period inflows Q^t_j^¹occurring within discrete class j.

To obtain a current release policy for a single lake-reservoir, the optimization procedure begins at some known point in future and works backwards in time, over stages N, N-1,….., 1, searching for an optimal release decision that maximizes the total annual hydropower generated from that point to the end of the horizon. Although the operational horizon is theoretically infinite, the algorithm is initiated with an arbitrarily large value of annual cycles hence increasing finite values of N are tried until the algorithm converges to a stationary operating policy.

For example, at the end of the horizon, it is assumed that the expected power generated from that point on i.e. F^N^¹ (..) = 0. This means that at this point in time the power plants downstream of the lake reservoir are not usable any more. Based on a number of trials with varying number of years, it was established that the SDP algorithm requires a maximum of 5 years to assure convergence. Adopting a notation similar to Labadie (1994), the generalized DP formulation is:



^, ¹



^maxt ^/ ¹



^, ¹^,





¹^,



N t t t t t t t N t t

i j k j t i i i k

R Q

F S Q^ 



P Q Q ^  f S S^ R F ^ S^ Q  (8.1) where:

R S S

 

_i^t _i^t^¹

 Q

_k^t (8.2)

Equation 8.2 is the inverted form of the reservoir/lake mass balance relationship and with it, optimization is performed directly over the end-of-period volume with releases being treated as the random variables.

Strict bounds are maintained on reservoir storage levels in the optimization process but there is a potential risk of failure to satisfy constraints on release. Should releases exceed the allowable limits, a penalty term is subtracted from the objective function to discourage violation of release constraints (Labadie, 1993, 2003). Alternatively, program CSUDP can allow violations up to a pre-specified maximum risk of failure to exceed downstream release constraints.

The SDP optimization algorithm proceeds by aggregating the total expected value of the objective function at each stage until it reaches the accumulated value at the end of the first

year (e.g. when t = 1). This marks the end of the first iteration for the hypothetical example. It then sets the value of the objective function for the beginning of the 2^nd cycle to the accumulated value computed during the previous iteration. An outline of the procedure is illustrated in Figure 8.1. Convergence of the algorithm can be determined once one or both of the following two criteria are satisfied:

a) When the best final storage volume, given an initial storage volume does not differ for each season of the year in each of the successive iterations. In addition stabilization of the expected annual increment of total power generated from operation of the system must be demonstrated as the backward moving iterations proceeds (Loucks, et al., 1981, 2005).

This implies that ^{F S Q}ⁿ



ⁱ^t^, ^t^j^¹



^Fⁿ^¹



^{S Q}ⁱ^t^, ^t^j^¹



a constant value^, regardless of the storage volume and time period.

b) Once the derived operation policy does not change from year to year (Chow et al., 1975).

It has been demonstrated by Ross (1983) that if discounting is not considered in the application and the algorithm converges then the solution must be optimal.

Figure 8.1 Flow diagram for the SDP model (Nadalal and Borgadi, 2007).

Start

Discretize inflow to reservoir and compute inflow transition probability matrices for each stage

Discretize storage space of reservoir

Compute power generated at downstream power plants for all feasible initial and final storage levels and inflow combinations

Perform recursion 𝐹^𝑁(𝑆_𝑖^𝑡, 𝑄_𝑗^𝑡−1) = max

𝑅^𝑡 ∑ 𝑃[𝑄𝑘𝑡/𝑄_𝑗^𝑡−1]

𝑄_𝑘^𝑡

[𝑓𝑡(𝑆_𝑖^𝑡, 𝑆_𝑖^𝑡+1, 𝑅^𝑡) + 𝐹^𝑁+1(𝑆_𝑖^𝑡+1, 𝑄𝑘𝑡)]

Optimise benefit for remaining stages

Convergence criteria satisfied?

Formulate table of final storage levels for all sets of conditions (initial storage and inflow classes in each stage)

Stop Next cycle

Yes

Repeat, starting at last stage and proceeding towards first

Dalam dokumen Development and application of decision support systems for improved planning and operation of large dams along the White Nile. (Halaman 165-170)