This paper deals with the modeling of stochastic processes in long-term multi-stage energy planning problems when limited information is available about the distributions of such processes. Starting from simple estimates of variation intervals for uncertain parameters, such as energy demands and costs, we model the temporal correlation of these parameters through carefully constructed autoregressive (AR) models that respect the intervals defined in each period. We introduce a coefficient for “zigzag” effects in the evolution of uncertain processes, which controls the correlation across periods and also the likelihood of extreme scenarios. To preserve the convexity of the stochastic problem, we discretize the AR models associated with the cost parameters involved in the objective function by Markov chains. The resulting formulation is then solved using a Stochastic Dual Dynamic Programming (SDDP) algorithm available in the literature that handles finite-state Markov chains. Our numerical experiments, carried out on the Swiss energy system, show a very desirable adaptation strategy of investment decisions to uncertainty scenarios, a behavior that is not observed when the temporal correlation is ignored. Moreover, the solutions lead to better out-of-sample cost performances, especially on extreme scenario realizations, than the non-correlated ones which usually result in overcapacities to protect against high, but unlikely, parameter variations over time.

中文翻译：

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能源规划模型中的不确定性动态：自回归和马尔可夫链建模方法

本文研究了长期多阶段能源规划问题中随机过程的建模，当有关过程分布的信息有限时。从对不确定参数（例如能源需求和成本）的变化区间的简单估计开始，我们通过精心构建的自回归（AR）模型对这些参数的时间相关性进行建模，该模型尊重每个时期定义的区间。我们引入了不确定过程演化中“锯齿形”效应的系数，该系数控制跨时期的相关性以及极端情景的可能性。为了保持随机问题的凸性，我们通过马尔可夫链离散化与目标函数中涉及的成本参数相关的 AR 模型。然后使用处理有限状态马尔可夫链的文献中提供的随机对偶动态规划 (SDDP) 算法来求解所得公式。我们在瑞士能源系统上进行的数值实验表明，投资决策对不确定性情景有一种非常理想的适应策略，这种行为在忽略时间相关性时不会被观察到。此外，与不相关的解决方案相比，这些解决方案可以带来更好的样本外成本性能，尤其是在极端场景的实现上，而非相关的解决方案通常会导致容量过剩，以防止随着时间的推移出现高但不太可能的参数变化。