2. REVIEW OF APPROACHES TO RESERVOIR SYSTEM OPERATIONS
2.1 Release plan
2.1.3 Explicit stochastic approach
Explicit stochastic optimisation is performed without the perfect foreknowledge of future events and optimal policies are determined without the need for inferring operating rules from results of optimisation. The approach uses probabilistic distributions of random stream flow instead of deterministic hydrologic sequences as shown in Figure 2.4.
A stochastic optimisation tool is a key component of the explicit approach. Several optimisation methods have been applied to reservoir operation and these include Extended Linear Quadratic Gaussian Control (Georgakakos, 1989), Stochastic Dynamic Programming (SDP), and other stochastic control methods (Yakowitz, 1982; Labadie, 2004; Rani and Moreira, 2010). According these literature surveys, Stochastic Dynamic Programming (SDP) methods and their variants or extensions have generally been the most widely applied techniques in the field of stochastic reservoir optimisation.
The Extended Linear Quadratic Gaussian Control Algorithm (ELQGC) algorithm has been applied in real time reservoir operation optimisation studies along the White Nile (WREM Inc
& Norplan (U) Ltd, 2004). The framework of the algorithm offers significant computational advantages over SDP methods but it is a real time control scheme. Real-time reservoir operation models often use as input, target parameters established by the planning model (e.g.
desired ending storage targets and/or reliability levels) and are designed to not only conform to directives that were defined at the long term planning level but also uses the resources in an effective way in the short to medium term by taking into account contingent and unforeseeable situations that may occur. This two-level approach, i.e. strategic planning first with later sequential update is widely prevalent in the management of reservoir systems by state and federal agencies in the United States (Loaiciga et al., 1987; Soncini-Sessa et al., 2007a).
Figure 2.4 Explicit stochastic approach (Labadie, 2004).
The objective of this study is to define long guide curves or storage targets and for this purpose DP is more attractive due to its sequential nature and non- restrictive handling of objective functions. The formulation of SDP algorithms usually requires dense discretization of the reservoir storage space, inflows and release variables to guarantee solution accuracy. Where correlated inflows are taken into account, a probabilistic definition of reservoir inflows is also necessary as illustrated under Figure 2.4. For a complex three cascade lake – reservoir optimisation problem such as the White Nile case study, the computational burden associated with a discrete SDP formulation would be expected to be significantly greater than the implicit approach (Lee and Labadie, 2007) and would ultimately overwhelm even the fastest CPU processors (Malinowski and Sadecki, 1990; Piccardi and Soncini-Sessa, 1991; Sadecki 2002,
Dynamic Programming (DDDP) algorithm with multi-core computer processors to enhance computing efficiency.
Various methods to overcome or cope with the dimensionality problem so as to produce computationally tractable solutions have been designed by reservoir research scientists. The remedies, or combinations thereof, belong to the field of approximate dynamic programming (Powell, 2011) and can be identified as follows:
(i) Decomposition of a multiple reservoir system into single reservoir units and subsequent use of iterative procedures to optimise the single reservoir systems one at a time by SDP or its extensions. (Turgeon, 1981; Nandalal and Sakthivadivel, 2002; Nandalal and Borgardi, 2007; Barty et al., 2010).
(ii) Aggregation of the system of reservoirs or parts thereof, in into an composite reservoir thus allowing a straight forward application of the optimisation procedure and the subsequent disaggregation of the derived composite operating strategy into control policies of individual reservoir elements (Saad et al., 1994; Archibald et al., 1997;
Turgeon and Charbonneau, 1998; Nandalal and Borgardi, 2007).
(iii) Variants of SDP or its extensions that do not require discretization of the storage space.
Examples of such applications with Stochastic Dual Dynamic Programming (Tilmant and Kelman, 2007; Tilmant et al., 2007; Goor et al., 2010) do not provide the traditional release policy tables but rather, a set of piecewise linear cost to go functions which are used to manage mixed hydrothermal systems. Another distinction between SDP and SDDP is that the former can handle a large number of stages while the latter is best suited to handle a larger state space. Hence the claim that SDDP actually mitigates the curse of dimensionality should be viewed within this context since its computational complexity can increase with increase in the number of state variables (Shapiro et al., 2012). For these is reasons, Lee and Labadie ( 2007) contend that SDDP is incapable of finding optimal closed loop or feedback operating policies over long operational horizons and requires special care when applied in the context of highly non-convex problems such as hydropower system optimisation.
(iv) Solving the problem with another variation of SDP known as Neuro Stochastic Dynamic Programming (NSDP), as proposed by Bertsekas and Tsitsiklis (1996). NSDP reduces algorithm complexity by approximation of the Bellman functions via Artificial Neural Networks (ANNs). An ANN is a nonlinear mathematical structure which is capable of representing complex nonlinear processes that relate the inputs and outputs of any system. Castelleti et al. (2007) applied the NSDP algorithm to a multi-objective problem in the Paive catchment in Italy which had 3 three reservoirs. They found NSDP to be 450% faster than an equivalent SDP.
(v) Adoption of a heuristic approach or algorithm that exploits knowledge of the approximate form of the optimal policy of the multi-reservoir systems to simplify the calculations and obtain efficient descriptions of the optimal release rules (Archibald et al., 2001).
Disaggregation/Aggregation techniques are attractive but pose certain difficulties related to loss of interconnectivity modelling detail between reservoirs. It is also cumbersome to disaggregate the composite policies and the results may not meet the accuracy requirements for real-world case study applications (Nandalal and Borgadi, 2007). Almost all the approximate dynamic programming approaches only alleviate the dimensionality problem to some degree and none actually removes it. The challenge in this study will therefore be to devise creative methods of working with a suitable approximate dynamic programming method to yield acceptable results.
In a review of optimal operation of multi-reservoir systems, Labadie (2004) indicates that implicit and explicit stochastic approaches have been useful in determining long-range guide curves and operating policies over weekly, monthly or seasonal time increments but have proven to be of limited use because the release plans they define are unique to the assumed time series (long historical record or stochastic stream flow generation). Consequently, many schemes operated with long-range guide curves continue to be function below their potential.
To remedy such short-comings, long range guide curves are often regarded as planning models that must be complemented by real-time operation models designed to track long-term guidelines over shorter time horizons.
The release plan/rule curve can be categorized as an off-line policy determined a priori for all possible occurrences of the state resulting from analyses of a long historical period or stochastically generated inflows.