An increasing number of electric loads, such as hydrogen producers or data centers, can be characterized as carbon-sensitive, meaning that they are willing to adapt the timing and/or location of their electricity usage in order to minimize carbon footprints. However, the emission reduction efforts of these carbon-sensitive loads rely on carbon intensity information such as average carbon emissions, and it is unclear whether load shifting based on these signals effectively reduces carbon emissions. To address this open question, we design a carbon-aware equilibrium model, which expands the commonly used equilibrium model for standard (carbon-agnostic) electricity market clearing to include carbon-sensitive consumers that adapt their consumption based on average carbon emission signals and carbon costs. This analysis represents an idealized situation for carbon-sensitive consumers, where their carbon preferences are reflected directly in the market clearing, and contrasts with current practice, where carbon emission signals only become known to consumers a posteriori (i.e., after the market has already been cleared). Furthermore, we extend our model to consider temporal load shifting and time-varying maximum renewable generations. We employ illustrative three-bus examples and numerical simulations on the IEEE RTS-GMLC system to reveal the limitations of the widely adopted average carbon emission signal for guiding carbon emission reduction. Our model offers a novel perspective for evaluating the effectiveness of different carbon signals and contributes to new carbon signal design.

Power systems modeling and planning has long leveraged mathematical programming for its ability to provide optimality and feasibility guarantees. One feature that has been recognized in the optimization literature since the 1970s is the existence and meaning of multiple exact optimal and near-optimal solutions, which we call alternative solutions. In power systems modeling, the use of alternative solutions has been limited to energy system optimization modeling (ESOM) applications and modeling to generate alternative (MGA) techniques. We present three key results about alternative solutions for power systems modeling. First, we give a perspective, based on sublevel sets and projection, for characterizing alternative solutions as a facet of general optimization theory. Second, we include pointers to alternative solution generation methods and tools beyond MGA-style techniques. Third, we demonstrate the use cases for alternative solutions in power system modeling on the fundamental optimal power flow problem.

Energy production throughout the world is transitioning from fossil fuels to renewable sources such as wind power and solar power. This transition has been gradual—over half of the world’s electricity is still produced by coal, oil, and gas—but must accelerate to meet global emission targets. This paper examines the contributions that mathematical optimization and equilibrium models can make to help accelerate this transition. The models we catalog cover a range of physical scales and timescales. Our focus is on novel model formulations that can help overcome the challenges of the transition by unpicking the complexity inherent in many settings and quantifying the trade-offs that must be made when developing energy policy. Funding: This research was performed while the authors were participating in the Architecture of Green Energy Systems Program hosted by the University of Chicago, which is supported by the National Science Foundation [Grant DMS-1929348]. A. B. Philpott acknowledges support from MBIE Catalyst Fund New Zealand German Platform for Green Hydrogen Integration (HINT) [Grant UOCX2117].

We show how to extract alternative solutions for optimization problems solved by Benders Decom- position. In practice, alternative solutions provide useful insights for complex applications; some solvers do support generation of alternative solutions but none appear to support such generation when using Benders Decomposition. We propose a new post-processing method that extracts multiple optimal and near-optimal solutions using the cut-pool generated during Benders Decomposition. Further, we provide a geometric framework for understanding how the adaptive approximation in Benders Decomposition re- lates to alternative solutions. We demonstrate this technique on stochastic programming and interdiction modeling, and we highlight use cases that require the ability to enumerate all optimal solutions.

375 Background: Prior studies with crefmirlimab, a 89-Zr labeled antibody with specificity for CD8+ cells, have shown a strong correlation with SUV uptake and IHC of the same lesions validating its use (Pal 2023). We predicted that responders to IO therapy would have higher CD8 lymph node avidity than non-responders. Methods: Pts enrolled had a diagnosis of mRCC and must have received IO therapy (CPI/TKI, CPI/CPI, or CPI monotherapy). Pts obtained a crefmirlimab PET/CT within 1 week before CPI infusion as baseline. PET intensity was determined using SUVMax, SUVMean, SUVPeak. Highly avid lymph nodes (SUVMax > 10) were quantified. Best overall response was determined using RECIST 1.1 criteria with SUV values and number of CD8 PET-avid lymph nodes compared using a Wilcoxon Signed Rank test in R. Results: A total of 17 pts (9 M: 8 F) were enrolled across 3 sites with 9 pts locally available for analysis. The median age of pts was 64 years old (54-71). Histology types included 12 clear cell (71%), 2 papillary (12%), and 3 indeterminate (17%). There were 3 responders (18%), 12 non-responders (71%), and 2 non-evaluable (12%). The most common treatment regimens patients received in descending order were cabozantinib + nivolumab (44.44%), nivolumab (33.33%), and ipilimumab + nivolumab (22.22%). Strong correlation was observed between CD8 cell density and SUVMean (p <0.0003 by Spearman’s correlation). The average number of CD8 PET-avid lymph nodes (defined as SUVMax > 10) in responders was 6 versus 1.17 in non-responders (p = 0.025, 95% CI [1.99,7]). SUVMax, SUVMean, SUVPeak across all CD8 PET-avid lymph nodes for responders were 15.58, 12.63, 11.44 and for non-responders were 9.86, 8.53, 7.79 respectively. The difference between SUVMax, SUVMean, SUVPeak values reached statistical significance for all three values (p = 0.023 95% CI [3.42,7.84], p = 0.023 95% CI [2.64,5.84], p = 0.029 95% CI [1.18, 5.41] respectively). Conclusions: Our results show that pts with higher number of CD8 PET-avid lymph nodes at baseline was associated with better response to IO therapy; responders also had overall higher SUV uptake in CD8 PET-avid lymph nodes than non-responders. Our findings suggest that those with a more enriched CD8 phenotype trended towards better outcomes and may be a reliable predictor for IO therapy response. Clinical trial information: NCT03802123 .

We formulate and compare optimization models of investment in renewable generation using a suite of social planning models that compute optimal generation capacity investments for a hydro-dominated electricity system where inflow uncertainty results in a risk of energy shortage. The models optimize the expected cost of capacity expansion and operation allowing for investments in hydro, geothermal, solar, wind, and thermal plant, as well as battery storage for smoothing load profiles. A novel feature is the integration of uncertain seasonal hydroelectric energy supply and short-term variability in renewable supply in a two-stage stochastic programming framework. The models are applied to data from the New Zealand electricity system and used to estimate the costs of moving to a 100% renewable electricity system by 2035. We also explore the outcomes obtained when applying different forms of CO 2 constraint that limit respectively non-renewable capacity, non-renewable generation, and CO 2 emissions on average, almost surely, or in a chance-constrained setting, and show how our models can be used to investigate the merits of a proposed pumped-hydro scheme in New Zealand’s South Island.

Action towards a renewable-centric electricity system needs to occur now; however, energy market incentives are typically aimed at short-term payoffs, leaving questions about optimal ways to achieve United Nations Sustainable Development Goals (SDG) and climate and energy goals established by the European Commission (EC). Current planning models for this purpose are two-stage models. They typically detail generator, transmission and control/policy system investment on a 30to 50-year time horizon, coupled to an operational model that shows that the real-time system is resilient to many different scenarios of future demand for electricity. Models of these effects are plentiful in the literature, but often lack a clear user interface to enable policy analysts to direct their application, leading to a gap between data and policy. Minding this gap, we have developed WEREWOLF (Wisconsin Expansion of Renewable Electricity with Optimization under Long-term Forecasts) as an independent, multi-year planning tool that provides data driven cost and benefit information regarding investments and energy system operations. WEREWOLF engages state of the art computing and data science technology, coupled with strong economic principles of competition and efficiency, to provide cost estimates and scenarios for strategic investments in new energy technologies that will be flexible to the uncertainties of technologies, policies and economics in the rapidly changing energy marketplace. The user-friendly software infrastructure allows policy analysts to directly interface to the model, and in collaboration with the proposer and associates or independently, carry out scenario runs (comparative statics) to explore the design space fully. The open-source WEREWOLF model is available on Github. This paper will briefly describe the rapidly changing energy landscape in the state of Wisconsin USA, the WEREWOLF model and software infrastructure, and policy scenarios within the state of Wisconsin USA currently explored in the early stages of research. The authors discuss potential advantages of the tool, including its opensource transparency and availability, its user-friendly interface, use of primary data sources, and model outputs in minutes. The authors conclude with addressing current limitations for using such tools to inform policy and planning and possible options to transfer the model to help meet SDG and EC energy and climate goals.

We study a competitive partial equilibrium in markets where risk-averse agents solve multistage stochastic optimization problems formulated in scenario trees. The agents trade a commodity that is produced from an uncertain supply of resources. Both resources and the commodity can be stored for later consumption. Several examples of a multistage risked equilibrium are outlined, including aspects of battery and hydroelectric storage in electricity markets, distributed ownership of competing technologies relying on shared resources, and aspects of water control and pricing. The agents are assumed to have nested coherent risk measures based on one-step risk measures with polyhedral risk sets that have a nonempty intersection over agents. Agents can trade risk in a complete market of Arrow-Debreu securities. In this setting, we define a risk-trading competitive market equilibrium and establish two welfare theorems. Competitive equilibrium will yield a social optimum (with a suitably defined social risk measure) when agents have strictly monotone one-step risk measures. Conversely, a social optimum with an appropriately chosen risk measure will yield a risk-trading competitive market equilibrium when all agents have strictly monotone risk measures. The paper also demonstrates versions of these theorems when risk measures are not strictly monotone.

We present a mixed complementarity problem (MCP) formulation of continuous state dynamic programming problems (DP-MCP). We write the solution to projection methods in value function iteration (VFI) as a joint set of optimality conditions that characterize maximization of the Bellman equation; and approximation of the value function. The MCP approach replaces the iterative component of projection based VFI with a one-shot solution to a square system of complementary conditions. We provide three numerical examples to illustrate our approach.

Management decisions can be informed by near-real-time data streams to improve the economics of the farm and to positively benefit the overall health of a dairy herd or the larger environment. Decision support tools can use data management services and analytics to exploit data streams from farm and other economic, health, and agricultural sources. We will describe a decision support tool that couples data analytics tools to underlying cow, herd, and economic data with an application programming interface. This interface allows the user to interact with a collection of dairy applications without fully exposing the intricacies of the underlying system model and understand the effects of different decisions on outputs of interest. The collection of these applications will form the basis of the Dairy Brain decision support system, which will provide management suggestions to farmers at a single animal or farm level. Dairy operations data will be gathered, cleaned, organized, and disseminated through an agricultural data hub, exploiting newly developed ontologies for integration of multiple data sources. Models of feed efficiency, culling, or other dairy operations (such as large capital expenditures, outsourcing opportunities, and interactions with regulators) form the basis of analytical approaches, operationalized via tools that help secure information and control uncertainties. The applications will be independently generated to provide flexibility, and use tools and modeling approaches from the data science, simulation, machine learning, and optimization disciplines to provide specific recommendations to decision makers. The Dairy Brain is a decision support system that couples data analytics tools with a suite of applications that integrate cow, herd, and economic data to inform management, operational, and animal health improving practices. Research challenges that remain include dealing with increased variability as predictions go from herd or pen level down to individual cow level and choosing the appropriate tool or technique to deal with a specific problem.

We study demand-side participation in an electricity market for an industrial consumer of electricity, with some flexibility to reduce demand, and capable of offering interruptible load reserve. Our consumer is a price maker, and the impact of its actions in the market is modelled via a bi-level optimization problem. We have extended a standard model for optimal strategic consumption, to the case where reserve offer curves need to be optimized simultaneously with consumption curves; our models provide intuition into this interaction. Furthermore, we provide tailor-made solution strategies for the resulting problems under uncertainty, and report numerical results of our implementation on instances over the full New Zealand network yielding a realistic and large problem set.

In many applications, conservation organizations depend on one species to indicate the presence of another. While extensive research has gone into methods for selecting these indicator species, few studies have directly measured the performance of indicator species in guiding conservation actions. Here, we evaluated whether a small number of indicator species could be used to efficiently select barrier removal projects to restore breeding habitat access for many other Great Lakes migratory fishes in the highly fragmented tributaries of the North American Great Lakes. First, we used a dataset of the historical distributions of 35 species of native migratory fishes to identify four clusters of co-occurring species, and then selected an indicator species for each cluster based on within-group co-occurrence or range width. We evaluated the utility of these indicator species by using upstream habitat and removal costs for 103,894 dams and road culverts across 1800 tributaries of the Great Lakes. We compared the potential increase in accessible tributary habitat for each species when barrier removals were prioritized to maximize benefits for (1) each species itself, versus (2) possible indicator species. We found that for 80% of the species, habitat gains from indicator-based project selection were at least 75% of the maximum gains possible under species-specific planning. However, a small subset of species would receive few habitat gains under indicator-directed project selection. Overall, our findings suggest that a suite of indicator species could be an efficient basis for planning restoration efforts for a majority of native migratory fishes in the Great Lakes.

In recent years, the power systems research community has seen an explosion of novel methods for formulating the AC power flow equations. Consequently, benchmarking studies using the seminal AC Optimal Power Flow (AC-OPF) problem have emerged as the primary method for evaluating these emerging methods. However, it is often difficult to directly compare these studies due to subtle differences in the AC-OPF problem formulation as well as the network, generation, and loading data that are used for evaluation. To help address these challenges, this IEEE PES Task Force report proposes a standardized AC-OPF mathematical formulation and the PGLib-OPF networks for benchmarking AC-OPF algorithms. A motivating study demonstrates some limitations of the established network datasets in the context of benchmarking AC-OPF algorithms and a validation study demonstrates the efficacy of using the PGLib-OPF networks for this purpose. In the interest of scientific discourse and future additions, the PGLib-OPF benchmark library is open-access and all the of network data is provided under a creative commons license.

Assembly of power-flow equations has traditionally begun from a nodal analysis formulation of the underlying transmission circuit's behavior. Most power-flow formulations encapsulate network constraints in the bus admittance matrix, Y bus . From a circuit perspective, this admittance representation restricts the network elements to be voltage controlled; resulting drawbacks treating zero-impedance branches in such applications as state estimation have long been recognized. This paper explores the advantages of the alternatives to Y bus -based formulations in power flow. It proposes a sparse tableau formulation (STF), and it demonstrates its computational efficiency, robustness, and generality in detailed comparison to traditional Y bus -based solution algorithms. In the examples of power networks ranging from 1888 to 82 000 buses, computational case studies indicate that STF provides comparable computational speed, while allowing simple treatment of zero-impedance branches and more reliably converging to solutions in many cases for which Y bus -based Newton algorithms diverge.

We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages. We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems. Our constructs enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of sophisticated structures such as shared constraints and additional constraints on their solutions. We define shared variables and demonstrate their uses for sparse reformulation, economic equilibrium problems sharing economic states, mixed pricing behavior of agents, and so on. We give some equilibrium and variational examples from the literature and describe how to formulate them using our framework. Experimental results comparing performance of various complementarity formulations for shared variables are provided. Our framework has been implemented and is available within GAMS/EMP.

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Affine variational inequalities (AVI) are an important problem class that subsumes systems of linear equations, linear complementarity problems and optimality conditions for quadratic programs. This paper describes PathAVI, a structure-preserving pivotal approach, that can efficiently process (solve or determine infeasible) large-scale sparse instances of the problem with theoretical guarantees and at high accuracy. PathAVI implements a strategy known to process models with good theoretical properties without reducing the problem to specialized forms, since such reductions may destroy sparsity in the models and can lead to very long computational times. We demonstrate formally that PathAVI implicitly follows the theoretically sound iteration paths, and can be implemented in a large scale setting using existing sparse linear algebra and linear programming techniques without employing a reduction. We also extend the class of problems that PathAVI can process. The paper illustrates the effectiveness of our approach by comparison to the Path solver used on a complementarity reformulation of the AVI in the context of applications in friction contact and Nash Equilibria. PathAVI is a general purpose solver, and freely available under the same conditions as Path.