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Energy justice, at the intersection of energy and societal ethics, studies the origins, quantification, and resolution of persistent and potential inequities within the energy sector, serving as a foundational pillar for societal harmony. In this review, we overview the historical and modern definitions of energy equity and frameworks of energy justice. We highlight the tools adopted to measure equity in the energy context, unveiling multifaceted inequities that permeate global energy landscapes. We discuss the limitations of prevalent metrics such as the Gini coefficient and Generalized Entropy Indices in the evaluation of energy justice concerns. Finally, we analyze publications that examined current practices and proposed improving methods towards a more equitable energy market for the society from policy, planning, and operation perspectives.

To increase market transparency, independent system operators (ISOs) have been working on minimizing uplift payments based on convex hull pricing theorems. However, the large-scale complex systems for ISOs bring computational challenges to the existing convex hull pricing algorithms. In this paper, based on the analysis of specific generator features in the Midcontinent ISO (MISO) system, besides reviewing integral formulations for several special cases, we develop two integral formulations of a single generator that can capture these features. We then build a compact convex hull pricing formulation based on these integral formulations. Meanwhile, to improve the computational efficiency, we propose innovative iterative algorithms with convergence properties, plus a complementary algorithm, to obtain a convex hull price. The computational results indicate that our approach leads to an exact convex hull price on MISO instances with and without transmission constraints and the solutions can be obtained within 20 minutes., Comment: Accepted by IEEE Transactions on Power Systems

Reducing uplift payments has been a challenging problem for most wholesale markets in US. The main difficulty comes from the unit commitment discrete decision makings. Recently convex hull pricing has shown promises to reduce the uplift payments. However, it has been intractable to obtain the optimal convex hull price. In this paper, we describe an innovative approach to decide the optimal convex hull price by simply solving a linear program. We also provide an example to illustrate the calculation process. The final computational experiments on a revised IEEE-118 bus system verify the cost effectiveness by utilizing our proposed approach.

In this letter, we describe and develop edge-based and node-based formulations for combined-cycle units (CCUs) with different levels of accuracy in respecting physical restrictions of each turbine, from no restriction requirement of each turbine, to include min-up/-down time restrictions of each turbine, to finally include most restrictions such as min-up/-down time, capacity, and ramping restrictions of each turbine. We develop corresponding models for Independent System Operators (ISOs) and Vertically Integrated Utilities (VIUs), respectively. These models demonstrate the complexity in terms of the size of decision variables and show the tradeoff between accuracy and computational complexity. In general, to the best of our knowledge, this letter provides one of the first models that can capture both configuration-based model needs and all physical restrictions for each turbine within CCUs., Comment: This version has some errors which will mislead the readers

In this paper, we focus on a data-driven risk-averse multistage stochastic programming (RMSP) model considering distributional robustness. We optimize the RMSP over the worst-case distribution within an ambiguity set of probability distributions constructed directly from historical data samples. The proposed RMSP is intractable due to the multistage nested minimax structure in its objective function, so we reformulate it into a deterministic equivalent that contains a series of convex combination of expectation and conditional value at risk (CVaR), which can be solved by a customized stochastic dual dynamic programming (SDDP) algorithm in this paper. As the size of collected data samples increases to infinity, we show the consistency of the RMSP with distributional robustness to the traditional multistage stochastic programming. In addition, to test the computational performance of our proposed model and algorithm, we conduct numerical experiments for a risk-averse hydrothermal scheduling problem, the results of which demonstrate the effectiveness of our RMSP framework.

In this paper, we consider the polyhedral structure of the unit commitment polytope. In particular, we provide the convex hull results for the problem under the following different settings: 1) the convex hulls for the integrated minimum-up/-down time and ramping polytope under the general $T$ time period setting in which the ramping rate equals to the gap between the generation upper and lower bounds and equals to half of the gap between the generation upper and lower bounds, respectively, 2) the convex hull for the integrated minimum-up/-down time and ramping-up polytope for the problem under the general $T$ time period setting, and 3) the convex hull for the integrated minimum-up/-down time and ramping-down polytope for the problem under the general $T$ time period setting.

Recently increasing penetration of renewable energy generation brings challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, a unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this paper, we focus on deriving polynomial time algorithms for the unit commitment problems with general convex cost function and piecewise linear cost function respectively. We refine an $\mathcal{O}(T^3)$ time, where $ T $ represents the number of time periods, algorithm for the deterministic single-generator unit commitment problem with general convex cost function and accordingly develop an extended formulation in a higher dimensional space that can provide an integral solution, in which the physical meanings of the decision variables are described. It means the original problem can be solved as a convex program instead of a mixed-integer convex program. Furthermore, for the case in which the cost function is piecewise linear, by exploring the optimality conditions, we derive more efficient algorithms for both deterministic (i.e., $\mathcal{O}(T)$ time) and stochastic (i.e., $\mathcal{O}(N)$ time, where $N$ represents the number of nodes in the stochastic scenario tree) \rred{single-generator} unit commitment problems. We also develop the corresponding extended formulations for both deterministic and stochastic single-generator unit commitment problems that solve the original mixed-integer linear programs as linear programs. Similarly, physical meanings of the decision variables are explored to show the insights of the new modeling approach.

In this paper, we study the polyhedral structure of an integrated minimum-up/-down time and ramping polytope, which has broad applications in variant industries. The polytope we studied includes minimum-up/-down time, generation ramp-up/-down rate, logical, and generation upper/lower bound constraints. By exploring its specialized structures, we derive strong valid inequalities and explore a new proof technique to prove these inequalities are sufficient to provide convex hull descriptions for variant two-period and three-period polytopes, under different parameter settings. For multi-period cases, we derive generalized strong valid inequalities (including one, two, and three continuous variables, respectively) and further prove that these inequalities are facet-defining under mild conditions. Moreover, we discover efficient polynomial time separation algorithms for these inequalities to improve the computational efficiency. Finally, extensive computational experiments are conducted to verify the effectiveness of our proposed strong valid inequalities by testing the applications of these inequalities to solve both self-scheduling and network-constrained unit commitment problems, for which our derived approach outperforms the default CPLEX significantly., Comment: Earlier versions are available online at http://www.optimization-online.org/DB_HTML/2015/06/4942.html and http://www.optimization-online.org/DB_HTML/2015/08/5070.html