Decision Dependent Uncertainty

In the field of decision making under uncertainty, most of the existing literature assumes that the uncertainty is modeled as an exogenous process, which cannot be influenced by the optimization process (see Assumption 2.1). In the real world, however, the decision making might have a significant influence on the stochastic process. For example, the demand of commodity on the shelf of supermarkets highly depends on the price and inventory level (Dana Jr and Petruzzi, 2001).

From modeling perspective, Pflug (1990) was the pioneer to address this issue of decision dependent uncertainty, in which the parameter of the underlying probability distribution was featured by the decision variable. A measure-valued differentiation method was presented to resolve the challenge. Later on, Goel and Grossmann (2006) accommodated the decision dependent uncertainty in the context of classical stochastic programming model, extending it to a more general framework. They also proposed a Lagrangian duality based branch and bound algorithm to solve the problem. Recently, Lee et al. (2012) utilized the interaction between stochastic process and optimization process to develop an iterative decision process (IDP) algorithm, which is able to learn the structure of the endogenous uncertainty on the decision variable and solve the optimization problem simultaneously.

Chapter 3

Short-Term Natural Gas Supply Management

3.1 Introduction

Natural gas contracts have been widely used to manage financial risks caused by volatile spot prices and stochastic demands, especially after the radical restructure of the industry. With the natural gas spot price becoming increasingly volatile, the profitability of the gas-fired GENCOs heavily relies on their ability to manage natural gas portfolio over bilateral contract and spot trading. In this chapter, we consider an optimization problem of dynamically allocating contracted gas over a finite planning horizon taking into account volatile spot gas prices and stochastic power demands.

For the past decades, there is an emerging practice-based literature on nat-ural gas supply management and energy portfolio optimization (see Chen and Baldick, 2007; Asif and Jirutitijaroen, 2009; Kittithreerapronchai et al., 2010;

Jirutitijaroen et al., 2013). The problems are commonly modeled as a stochastic mixed integer linear programming (SMILP) problem and are solved by commer-cial software such as CPLEX, without looking deep into the structure of the model.

Algorithms based on Lagrangian relaxation method and Bender’s decomposition

have been proposed to efficiently solve the energy portfolio and generation plan-ning problems by exploiting the structure of the optimization models (Cerisola et al., 2009). However, it should be noted that solving the SMILP solely produces an optimal action and corresponding performance in the current stage. But it cannot track optimal decision making strategies for all possible states in the fol-lowing stages. However, in practice, the GENCO bound by a specified contract is more concerned about establishing an optimal contract delivery policy that allows the GENCO to dynamically allocate contracted gas in response to the re-vealed uncertainty. In order to achieve this goal, researchers proposed stochastic dynamic programming to study contract allocation and valuation in energy (see Jaillet et al., 2004; Løland and Lindqvist, 2008; Secomandi, 2010; Edoli et al., 2013).

With the presence of spot trading, whether the GENCO should withdraw nat-ural gas from the contract or not highly depends on the interaction of spot price and remaining contract amount. Barrera-Esteve et al. (2006) proved that under some mild assumptions, the optimal policy is of bang-bang type. That is, the op-timal consumption amount can only take two values, either the minimum or the maximum delivery amount. The trigger event is characterized by a price thresh-old depending on time and inventory such that if the unfthresh-olding spot price is higher than the price threshold, it is optimal to take the maximum consumption amount, otherwise the minimum consumption amount is optimal. Subsequently, Bardou et al. (2010) extended the work by presenting conditions for the exis-tence of bang-bang optimal policy in a more general situation. Compared with the bang-bang type policy, base stock policy appears more common (especially in the literature of inventory management), as it does not need those critical assumptions required for the existence of former. Secomandi (2010) showed that the optimal inventory-trading policy for a natural gas storage facility is fully char-acterized by two base-stock levels which only depend on stage and spot price.

Under some mild assumptions, it is found that these two stage and price

de-pendent target levels monotonically decrease in spot prices. As with Secomandi (2010), we also establish an optimal base stock policy for dynamic contract

alloca-tion and show the price monotonicity of our base stock levels, but in a novel way using the theory of supermodularity (Topkis, 1998). Similarities and differences between Secomandi (2010) and our work are presented in Table 3.1.

Table 3.1: Comparison between Secomandi (2010) and our work

Compare objective Secomandi (2010) our work

Similarity

Problem NG storage valuation or contract valuation Model Stochastic dynamic programming Policy parameters Two base stock levels One base stock level

How to prove

Another motivation for this work lies in the explanation of the strategic role of the bilateral forward contract. It is well known that the hedging role is the reason for the existing of forward market, which has been extensively investigated in the field of supply chain management (Dong and Liu, 2007). Popescu and Seshadri (2013) stated that the forward contract can help the contract holders to hedge against the financial risks and therefore appeals to risk-averse buyers and sellers.

We, on the other hand, argue that the bilateral contract can also allow risk neutral participants to gain profit by strategically allocating the contracted gas. The strategic role of the forward contract here is similar to that of the storage facility (Secomandi, 2010), enabling the buyers to time-shift the natural gas usage for

more favorable prices in the future.

The remainder of this chapter is organized as follows. In Section 3.2, we present a detailed description of the natural gas supply management problem for a GENCO and formulate an MDP model. The convexity of the optimal value function is discovered and further leveraged to establish the optimal base-stock

policy in Section 3.3. Moreover, we show that the optimal base-stock target levels are monotonously decreasing in spot prices using the theory of supermodularity.

Section 3.4 is devoted to the description of our continuous price evolution model and its discretization framework. In Section 3.5, results on the value of stochastic solution are presented. We end this chapter with a brief discussion in Section 3.6.