To realize adaptive operation planning with MILP unit commitment, piecewise-linear approximations of the functions that describe the operating behavior of devices in the energy system have to be computed. We present an algorithm to compute a piecewise-linear approximation of a multi-variate non-linear function. The algorithm splits the domain into two regions and approximates each region with a set of hyperplanes that can be translated to a convex set of constraints in MILP. The main advantage of this “piecewise-convex approximation” (PwCA) compared to more general piecewise-linear approximation with simplices is that the MILP representation of PwCA requires only one auxiliary binary variable. For this reason, PwCA yields significantly faster solving times in large MILP problems where the MILP representation of certain functions has to be replicated many times, such as in unit commitment. To quantify the impact on solving time, we compare the performance using PwCA with the performance of simplex approximation with logarithmic formulation and show that PwCA outperforms the latter by a big margin. For this reason, we conclude that PwCA will be a useful tool to set up and solve large MILP problems such as arise in unit commitment and similar engineering optimization problems.

为了利用 MILP 单元实现自适应运行规划，必须计算描述能源系统中设备运行行为的函数的分段线性近似。我们提出了一种计算多元非线性函数的分段线性近似的算法。该算法将域分成两个区域，并用一组超平面来近似每个区域，这些超平面可以转换为 MILP 中的凸约束集。与更一般的带单纯形的分段线性近似相比，这种“分段凸近似”(PwCA) 的主要优点是 PwCA 的 MILP 表示仅需要一个辅助二元变量。因此，PwCA 在大型 MILP 问题中的求解速度显着加快，其中某些函数的 MILP 表示必须重复多次，例如在单元承诺中。为了量化对求解时间的影响，我们将使用 PwCA 的性能与采用对数公式的单纯形近似的性能进行了比较，结果表明 PwCA 的性能大幅优于后者。因此，我们得出结论，PwCA 将成为建立和解决大型 MILP 问题（例如机组组合和类似工程优化问题）的有用工具。