T. AOUAM, A. SOUFYANE, M. CHACHA
. Here, we consider two realizations of the inflow process at : � t � � t ,
� t � � t . Where �t = mean of at , and �t = standard deviation of at . This corresponds to low or
high realizations in each time period over the planning horizon. The total number of
combinations gives the number of scenarios S. To calculate the WS value we are required to
solve S deterministic problems. For the case of a planning horizon of T = 8, one would be
required to solve S = 256 deterministic problem of 8 periods. Here for the purpose of
illustration, we consider only two extreme scenarios: inflow low in all periods and inflow
high in all periods. Table 2 presents the optimal values for the two scenarios in the
hydrothermal planning problem over the planning horizon of 8 periods.
Therefore we approximate WS by
= $82,333,522.2. While,
= $91,642,658.3. This
= $9,309,136.1 as a rough approximation of EVPI. This is how
leads to:
much roughly the decision maker is willing to pay for perfect information.
Value of the stochastic solution (VSS): In practice several people would find the calculation
of WS cumbersome, furthermore each of the S problems would possibly have a different
solution and further effort would be required to combine these solutions, using some sort of
heuristic, into a solution to implement. Therefore, the majority of people would solve the
deterministic problem by replacing the random variables by their corresponding expected
values. Suppose that one solves the mean value problem or the deterministic problem with
mean values. Let z be the optimal decision of the first stage in the mean value problem and
x be the optimal state vector of the system in the mean problem. Q ( z ) is the optimal value of
the stochastic problem given that the decision z is made in the first stage. Hence,
Q ( z 1 ) � c1 ( z 1 ) � Q 2 ( x 2 ) . The VSS measures how good or bad the decision z is. The VSS is
defined by VSS � Q0* � Q ( z 1 ) . This quantity is positive because a solution that contains z is a
feasible solution to the SP.
For the hydrothermal planning problem VSS = 91,677,050.3 – 91,642,658.3 = $34,392. This
represents the cost of ignoring uncertainty for the first period only.
1
1
1
1
1
1
Table 2. Optimal values for deterministic problems with different scenarios.
S
Q (s)
Low
$ 241279844.6
High
$ 5720722.1
8. SUMMARY
The paper presents a multistage stochastic hydrothermal planning model, which is solved
using algorithms based on stochastic dynamic programming (SDP). One such algorithm is the
stochastic dual dynamic programming (SDDP) algorithm, which was implemented for the
hydrothermal power system of the Pacific Northwest in the U.S. In this work, we
characterized the vanishing property of the first period decision and proposed a modified
algorithm (MSDDP). Computational results showed that the algorithm can speed up
computation by nearly 50% compared to the classical SDDP.
REFERENCES
Arvanitidis, N.V., Rosing, J., 1970. Composite representation of a multireservoir
hydroelectric power system, IEEE Transactions on Power Apparatus and Systems,
PAS-89 2 (1970), pp. 319–326
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