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We study the relation between torsion tensors of principal connections on G-structures and characteristic conic connections on associated cone structures. We formulate sufficient conditions under which the existence of a characteristic conic connection implies the existence of a torsion-free principal connection. We verify these conditions for adjoint varieties of simple Lie algebras, excluding those of type $\textsf{A}_{\ell \neq 2}$ or $\textsf{C}_{\ell}$. As an application, we give a complete classification of the germs of minimal rational curves whose VMRT at a general point is such an adjoint variety: nontrivial ones come from lines on hyperplane sections of certain Grassmannians or minimal rational curves on wonderful group compactifications., Comment: To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

Instruction tuning as an effective technique aligns the outputs of large language models (LLMs) with human preference. But how to generate the seasonal multi-turn dialogues from raw documents for instruction tuning still requires further exploration. In this paper, we present a novel framework named R2S that leverages the CoD-Chain of Dialogue logic to guide large language models (LLMs) in generating knowledge-intensive multi-turn dialogues for instruction tuning. By integrating raw documents from both open-source datasets and domain-specific web-crawled documents into a benchmark K-BENCH, we cover diverse areas such as Wikipedia (English), Science (Chinese), and Artifacts (Chinese). Our approach first decides the logic flow of the current dialogue and then prompts LLMs to produce key phrases for sourcing relevant response content. This methodology enables the creation of the G I NSTRUCT instruction dataset, retaining raw document knowledge within dialoguestyle interactions. Utilizing this dataset, we fine-tune GLLM, a model designed to transform raw documents into structured multi-turn dialogues, thereby injecting comprehensive domain knowledge into the SFT model for enhanced instruction tuning. This work signifies a stride towards refining the adaptability and effectiveness of LLMs in processing and generating more accurate, contextually nuanced responses across various fields., Comment: 11 pages, 3 figures

In an era marked by the rapid scaling of foundation models, autonomous driving technologies are approaching a transformative threshold where end-to-end autonomous driving (E2E-AD) emerges due to its potential of scaling up in the data-driven manner. However, existing E2E-AD methods are mostly evaluated under the open-loop log-replay manner with L2 errors and collision rate as metrics (e.g., in nuScenes), which could not fully reflect the driving performance of algorithms as recently acknowledged in the community. For those E2E-AD methods evaluated under the closed-loop protocol, they are tested in fixed routes (e.g., Town05Long and Longest6 in CARLA) with the driving score as metrics, which is known for high variance due to the unsmoothed metric function and large randomness in the long route. Besides, these methods usually collect their own data for training, which makes algorithm-level fair comparison infeasible. To fulfill the paramount need of comprehensive, realistic, and fair testing environments for Full Self-Driving (FSD), we present Bench2Drive, the first benchmark for evaluating E2E-AD systems' multiple abilities in a closed-loop manner. Bench2Drive's official training data consists of 2 million fully annotated frames, collected from 10000 short clips uniformly distributed under 44 interactive scenarios (cut-in, overtaking, detour, etc), 23 weathers (sunny, foggy, rainy, etc), and 12 towns (urban, village, university, etc) in CARLA v2. Its evaluation protocol requires E2E-AD models to pass 44 interactive scenarios under different locations and weathers which sums up to 220 routes and thus provides a comprehensive and disentangled assessment about their driving capability under different situations. We implement state-of-the-art E2E-AD models and evaluate them in Bench2Drive, providing insights regarding current status and future directions., Comment: Fix typos in text and Table 4. More reference

Real-world autonomous driving (AD) especially urban driving involves many corner cases. The lately released AD simulator CARLA v2 adds 39 common events in the driving scene, and provide more quasi-realistic testbed compared to CARLA v1. It poses new challenge to the community and so far no literature has reported any success on the new scenarios in V2 as existing works mostly have to rely on specific rules for planning yet they cannot cover the more complex cases in CARLA v2. In this work, we take the initiative of directly training a planner and the hope is to handle the corner cases flexibly and effectively, which we believe is also the future of AD. To our best knowledge, we develop the first model-based RL method named Think2Drive for AD, with a world model to learn the transitions of the environment, and then it acts as a neural simulator to train the planner. This paradigm significantly boosts the training efficiency due to the low dimensional state space and parallel computing of tensors in the world model. As a result, Think2Drive is able to run in an expert-level proficiency in CARLA v2 within 3 days of training on a single A6000 GPU, and to our best knowledge, so far there is no reported success (100\% route completion)on CARLA v2. We also propose CornerCase-Repository, a benchmark that supports the evaluation of driving models by scenarios. Additionally, we propose a new and balanced metric to evaluate the performance by route completion, infraction number, and scenario density, so that the driving score could give more information about the actual driving performance., Comment: Accepted by ECCV 2024

This paper extends a new concept of energy-water-hydrogen (EWH) nexus, which was recently developed as a solution for reducing carbon emissions from the generation side of power systems, to the distribution side. Under the concept of distribution-level EWH (micro EWH) nexus, renewable energy sources (RES) are utilized to meet the energy needs of a small community. To avoid the uncertainty caused by RESs, this paper aims to investigate the real-time optimal operation of the micro EWH nexus which is however a challenging optimization problem. First, such a large-scale mixed-integer nonlinear programming problem is relaxed into a mixed-integer convex program (MICP) by leveraging the effective convex-hull relaxation technique. Second, a fast data-driven solution method based on active constraint and integer variable prediction is presented, which can solve the MICP problem very fast since it utilizes historical optimization data to quickly predict binary variable values and a limited set of active constraints., Comment: arXiv admin note: text overlap with arXiv:2306.17395

This paper aims to evaluate the economic viability of the energy-water-hydrogen (EWH) nexus as a new solution for reducing carbon emissions from power systems. The urgency around climate change emphasizes the pressing need to mitigate carbon emissions, especially from the electricity sector, which accounts for a significant portion of total emissions in the US. In response, incorporating more renewable energy sources (RESs) and green hydrogen, created through water electrolysis and RES, stands out as a crucial strategy to combat climate challenges. We delve into various aspects of the EWH nexus, including carbon emissions from different power plants, capturing these emissions, and potential options for their reuse or storage. This paper involves modeling different sections of the EWH nexus and conducting an economic analysis across scenarios in power plants to determine optimal water supply methods, suitable chemical products for carbon reuse, and an appropriate carbon emission penalty to encourage emission reduction through the EWH nexus. The results indicate that reusing captured carbon emissions emerges as the most beneficial option across all power plant types. This finding underscores the potential of carbon reuse as a pivotal strategy within the EWH nexus framework for addressing carbon emissions., Comment: 5 pages, 3 figures, 2 tables, conference

The streaming temporal action segmentation (STAS) task, a supplementary task of temporal action segmentation (TAS), has not received adequate attention in the field of video understanding. Existing TAS methods are constrained to offline scenarios due to their heavy reliance on multimodal features and complete contextual information. The STAS task requires the model to classify each frame of the entire untrimmed video sequence clip by clip in time, thereby extending the applicability of TAS methods to online scenarios. However, directly applying existing TAS methods to SATS tasks results in significantly poor segmentation outcomes. In this paper, we thoroughly analyze the fundamental differences between STAS tasks and TAS tasks, attributing the severe performance degradation when transferring models to model bias and optimization dilemmas. We introduce an end-to-end streaming video temporal action segmentation model with reinforcement learning (SVTAS-RL). The end-to-end modeling method mitigates the modeling bias introduced by the change in task nature and enhances the feasibility of online solutions. Reinforcement learning is utilized to alleviate the optimization dilemma. Through extensive experiments, the SVTAS-RL model significantly outperforms existing STAS models and achieves competitive performance to the state-of-the-art TAS model on multiple datasets under the same evaluation criteria, demonstrating notable advantages on the ultra-long video dataset EGTEA. Code is available at https://github.com/Thinksky5124/SVTAS., Comment: submit to TNNLS

This paper focuses on developing a real-time optimal operation model for a new engineering system, wind-to-hydrogen-driven low-carbon critical infrastructure (W2H-LCCI), that utilizes wind power to generate hydrogen through electrolysis and combines it with carbon capture to reduce carbon emissions from the power sector. First, a convex mathematical model for W2H-LCCI is proposed, and then optimization models for its real-time decision-making are developed, which are mixed-integer convex programs (MICPs). Furthermore, since this large-scale MICP problem must be solved in real-time, a fast solution method based on active constraint and integer variable prediction (ACIVP) is presented. ACIVP method predicts the binary variable values and the set of limited-number constraints, which most likely contain all of the active constraints, based on historical optimization data. It results in only a small-scale continuous convex optimization problem needing to be solved by optimization solvers for W2H-LCCI real-time optimal operation. To increase the accuracy of the ACIVP method, feature space expansion (FSE) is employed, and a multi-stage ACIVP-FSE method is proposed. The effects of stage design and stage ordering on ACIVP-FSE performance are also discussed. We validate the effectiveness of the developed system and solution method using two water-energy nexus case studies.

The electric power distribution network (PDN) and the transportation network (TN) are generally operated/coordinated by different entities. However, they are coupled with each other due to electric vehicle charging stations (EVCSs). This paper proposes to coordinate the operation of the two systems via a fully decentralized framework where the PDN and TN operators solve their own operation problems by sharing only limited information. Nevertheless, the operation problems generally are in mixed-integer programming (MIP) form. To the best of our knowledge, the most existing decentralized/distributed optimization algorithms, such as the alternating direction method of multipliers (ADMM), are not always guaranteed to converge for such MIP problems. Therefore, a novel fully decentralized optimization algorithm is proposed, whose contributions include: 1) it is applicable to MIP problems with convergence and optimality guaranteed for mild assumptions, 2) it only requires limited information exchange between PDN and TN operators, which will help preserve the privacy of the two systems and reduce the investment in building communication channels, and 3) it is fully decentralized so that all the computations are carried out by PDN operator and TN coordinator only. Simulations on the test cases show the efficiency and efficacy of the proposed framework and algorithm.

Since the celebrated work by Cartan, distributions with \nobreak{small} growth vector $(2,3,5)$ have been studied extensively. In the holomorphic setting, there is a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5. We generalize this correspondence to higher dimensions by studying nondegenerate lines on holomorphic contact manifolds and the corresponding class of distributions of small growth vector $(2m, 3m, 3m+2)$ for any positive integer $m$., Comment: To appear in Nagoya Mathematical Journal

Pasquier and Perrin discovered that the ${\rm G}_2$-horospherical manifold ${\bf X}$ of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric, in other words, it can be deformed nontrivially to the rational homogeneous space. We show that ${\bf X}$ is the only smooth projective variety with this property. This is obtained as a consequence of our main result that ${\bf X}$ can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to ${\bf X}$ if and only if its VMRT at a general point is projectively isomorphic to that of ${\bf X}$. We employ the method the authors developed to solve the corresponding problem for symplectic Grassmannians, which constructs a flat Cartan connection in a neighborhood of a general minimal rational curve. In adapting this method to ${\bf X}$, we need an intricate study of the positivity/negativity of vector bundles with respect to a family of rational curves, which is subtler than the case of symplectic Grassmannians because of the nature of the differential geometric structure on ${\bf X}$ arising from VMRT., Comment: To appear in Transformation Groups

This paper considers endogenous uncertainty (EnU) in the stochastic economic dispatch (SED) problem, where the endogenous uncertainty means decision dependent uncertainty. In this problem, demand response (DR) commitment is the source of the EnU. Nevertheless, EnU is not well considered in existing literature. Our first contribution is to build up an optimization model of DR-involved SED under EnU (SED-DR-EnU). This is a computational challenging problem due to the EnU. Our second contribution is introducing a coupled learning enabled optimization algorithm which can effectively solve the proposed SED-DR-EnU problem. This strategy is tested on the IEEE 14 bus, and IEEE 39 bus systems, and the results showed the importance of considering EnU in the DR-involved SED problem.

To each complex composition algebra $\mathbb{A}$, there associates a projective symmetric manifold $X(\mathbb{A})$ of Picard number one, which is just a smooth hyperplane section of the following varieties ${\rm Lag}(3,6), {\rm Gr}(3,6), \mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to $X(\mathbb{A})$, then every fiber is isomorphic to $X(\mathbb{A})$.

This paper solves a new class of optimization problems under uncertainty which optimizes an objective function of decision variables and subjects to a set of probability contour constraints (PCC). The proposed PCC logically means that an optimal solution should satisfy a set of algebraic constraints for all possible high-probability realizations of the uncertain parameters. The PCC is an alternative to the conventional chance constraint while the latter cannot guarantee the solution's feasibility to high-probability realizations of uncertainty. Given that the existing solution methods of the conventional chance-constrained optimization are not suitable for solving the proposed probability contour constrained optimization (PCCO), we develop a novel data-based solution paradigm that uses historical measurements of the uncertain parameters as input samples. This solution paradigm is conceptually simple and allows us to develop effective data-reduction schemes which reduces computational burden while reserves high accuracy., Comment: 26 pages

1-flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachh\"ofer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves., Comment: 36 pages, to appear in Journal of Geometric Analysis in the collection in memory of Nessim Sibony

In this paper, we propose a novel gait recognition method based on a bag-of-words feature representation method. The algorithm is trained, tested and evaluated on a unique human gait data consisting of 93 individuals who walked with comfortable pace between two end points during two different sessions. To evaluate the effectiveness of the proposed model, the results are compared with the outputs of the classification using extracted features. As it is presented, the proposed method results in significant improvement accuracy compared to using common statistical features, in all the used classifiers., Comment: 14 pages, 7 figures, submitted to AHFE conference

Water and power systems are increasingly interdependent due to the growing number of electricity-driven water facilities. The security of one system can be affected by a contingency in the other system. This paper investigates a security-constrained operation problem of the energy-water nexus (EWN), which is a computationally challenging optimization problem due to the nonlinearity, nonconvexity, and size. We propose a two-step iterative contingency filtering method based on the feasibility and rating of the contingencies to decrease the size of the problem. The optimal power and water flow are obtained in a normal situation by considering the set of contingencies that can not be controlled with corrective actions. The feasibility check of the contingencies is performed in the second step, followed by a rating of the uncontrollable contingencies. Finally, the critical contingencies are obtained and added to the first step for the next iteration. We also employ convex technologies to reduce the computation burden. The proposed method is validated via two case studies. Results indicate that this approach can efficiently attain optimal values.

The multi-period dynamics of energy storage (ES), intermittent renewable generation and uncontrollable power loads, make the optimization of power system operation (PSO) challenging. A multi-period optimal PSO under uncertainty is formulated using the chance-constrained optimization (CCO) modeling paradigm, where the constraints include the nonlinear energy storage and AC power flow models. Based on the emerging scenario optimization method which does not rely on pre-known probability distribution functions, this paper develops a novel solution method for this challenging CCO problem. The proposed meth-od is computationally effective for mainly two reasons. First, the original AC power flow constraints are approximated by a set of learning-assisted quadratic convex inequalities based on a generalized least absolute shrinkage and selection operator. Second, considering the physical patterns of data and motived by learning-based sampling, the strategic sampling method is developed to significantly reduce the required number of scenarios through different sampling strategies. The simulation results on IEEE standard systems indicate that 1) the proposed strategic sampling significantly improves the computational efficiency of the scenario-based approach for solving the chance-constrained optimal PSO problem, 2) the data-driven convex approximation of power flow can be promising alternatives of nonlinear and nonconvex AC power flow.

The uncertainty of multiple power loads and renewable energy generations (PLREG) in power systems increases the complexity of power flow analysis for decision-makers. The chance-constrained method can be applied to model the optimization problems of power flow under uncertainty. This paper develops a novel solution approach for chance-constrained AC optimal power flow (CCACOPF) problem based on the data-driven convexification of power flow and a fast algorithm for scenario technique (FAST). This method is computationally effective for mainly two reasons. First, the original nonconvex AC power flow (ACPF) constraints are approximated by a set of learning-based quadratic convex ones. Second, FAST is an advanced scenario-based solution method (SSM) that doesn't rely on the pre-assumed probability distribution, using far less scenarios than the conventional SSM. Eventually, the CCACOPF is converted into a computationally tractable convex optimization problem. The simulation results on IEEE test cases indicate that 1) the proposed solution method can outperform the conventional SSM in computational efficiency, 2) the data-driven convexification of power flow is effective in approximating original complex AC power flow.

This paper presents a novel solution paradigm of general optimization under both exogenous and endogenous uncertainties. This solution paradigm consists of a probability distribution (PD)-free method of obtaining deterministic equivalents and an innovative approach of scenario reduction. First, dislike the existing methods that use scenarios sampled from pre-known PD functions, the PD-free method uses historical measurements of uncertain variables as input to convert the logical models into a type of deterministic equivalents called General Scenario Program (GSP). Our contributions to the PD-free deterministic equivalent construction reside in generalization (making it applicable to general optimization under uncertainty rather than just chance-constrained optimization) and extension (enabling it to the problems under endogenous uncertainty via developing an iterative and a non-iterative frameworks). Second, this paper reveals some unknown properties of the PD-free deterministic equivalent construction, such as the characteristics of active scenarios and repeated scenarios. Base on this discoveries, we propose a concept and methods of strategic scenario selection which can effectively reduce the required number of scenarios as demonstrated in both mathematical analysis and numerical experiments. Numerical experiments are conducted on two typical smart grid optimization problems under exogenous and endogenous uncertainties., Comment: Subtitle: Applications in Smart Grid Optimization under Both Exogenous and Endogenous Uncertainty

This paper defines a security injection region (SIR) to guarantee reliable operation of water distribution systems (WDS) under extreme conditions. The model of WDSs is highly nonlinear and nonconvex. Understanding the accurate SIRs of WDSs involves the analysis of nonlinear constraints, which is computationally expensive. To reduce the computational burden, this paper first investigates the convexity of the SIR of WDSs under certain conditions. Then, an algorithm based on a monotone inner polytope sequence is proposed to effectively and accurately determine these SIRs. The proposed algorithm estimates a sequence of inner polytopes that converge to the whole convex region. Each polytope adds a new area to the SIR. The algorithm is validated on two different WDSs, and the conclusion is drawn. The computational study shows this method is applicable and fast for both systems.

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension $n$ with normal bundles isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus (n-1-p)}$ for some nonnegative integer $p$. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of $p=1$ and $n \leq 5$, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role., Comment: To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

This paper presents a practical approach to utilizing emergency response resources (ERRs) and post-disaster available distributed energy resources (PDA-DERs) to improve the resilience of power distribution systems against natural disasters. The proposed approach consists of two sequential steps: first, the minimum amount of ERRs is determined in a pre-disaster planning model; second, a post-disaster restoration model is proposed to co-optimize the dispatch of pre-planned ERRs and PDA-DERs to minimize the impact of disasters on customers, i.e., unserved energy for the entire restoration window. Compared with existing restoration strategies using ERRs, the proposed approach is more tractable since 1) in the pre-disaster stage, the needed EERs are determined based on the prediction of energy shortage and disaster-induced damages using machine learning-based algorithms (i.e., cost-sensitive-RFQRF for prediction of outage customers, random forest for prediction of outage duration, and CART for prediction of disaster-induced damages); 2) in the post-disaster stage, the super-node approximation (SNA) and the convex hull relaxation (CHR) of distribution networks are introduced to achieve the best trade-off between computational burden and accuracy. Tests of the proposed approach on IEEE test feeders demonstrated that a combination of SNA and CHR remarkably reduces the solution time of the post-disaster restoration model., Comment: 8 pages, 7 figures!

For a complex connected semisimple linear algebraic group $G$ of adjoint type and of rank $n$, De Concini and Procesi constructed its wonderful compactification $\bar{G}$, which is a smooth Fano $G \times G$-variety of Picard number $n$ enjoying many interesting properties. In this paper, it is shown that the wonderful compactification $\bar{G}$ is rigid under Fano deformation. Namely, for any regular family of Fano manifolds over a connected base, if one fiber is isomorphic to $\bar{G}$, then so are all other fibers. This answers a question raised by Bien and Brion in their work on the local rigidity of wonderful varieties., Comment: 42 pages, to appear in Journal of Differential Geometry

This paper presents a novel optimization framework of formulating the three-phase optimal power flow that involves uncertainty. The proposed uncertainty-aware optimization (UaO) framework is: 1) a deterministic framework that is less complex than the existing optimization frameworks involving uncertainty, and 2) convex such that it admits polynomial-time algorithms and mature distributed optimization methods. To construct this UaO framework, a methodology of learning-aided uncertainty-aware modeling, with prediction errors of stochastic variables as the measurement of uncertainty, and a theory of data-driven convexification are proposed. Theoretically, the UaO framework is applicable for modeling general optimization problems under uncertainty., Comment: Accepted for pubication in the IEEE Transactions on Power Systems

This paper develops an ensemble learning-based linearization approach for power flow, which differs from the network-parameter based direct current (DC) power flow or other extended versions of linearization. As a novel data-driven linearization through data mining, it firstly applies the polynomial regression (PR) as a basic learner to capture the linear relationships between the bus voltage as the independent variable and the active or reactive power as the dependent variable in rectangular coordinates. Then, gradient boosting (GB) and bagging as ensemble learning methods are introduced to combine all basic learners to boost the model performance. The fitted linear power flow model is also relaxed to compute the optimal power flow (OPF). The simulating results of standard IEEE cases indicate that (1) ensemble learning methods outperform PR and GB works better than bagging; (2) as for solving OPF, the data-driven model excels the DC model and the SDP relaxation in the computational accuracy, and works faster than ACOPF and SDPOPF., Comment: 5 pages, 9 figures, 3 tables

Though the convex optimization has been widely used in power systems, it still cannot guarantee to yield a tight (accurate) solution to some problems. To mitigate this issue, this paper proposes an ensemble learning based convex approximation for AC power flow equations that differs from the existing convex relaxations. The proposed approach is based on quadratic power flow equations in rectangular coordinates and it can be used in both balanced and unbalanced three-phase power networks. To develop this data-driven convex approximation of power flows, the polynomial regression (PR) is first deployed as a basic learner to fit convex relationships between the independent and dependent variables. Then, ensemble learning algorithms such as gradient boosting (GB) and bagging are introduced to combine learners to boost model performance. Based on the learned convex approximation of power flows, optimal power flow (OPF) is formulated as a convex quadratic programming problem. The simulation results on IEEE standard cases show that, in the context of solving OPF, the proposed data-driven convex approximation outperforms the conventional SDP relaxation in both accuracy and computational efficiency, especially in the cases that the conventional SDP relaxation fails., Comment: 8 pages, 8 figures, 4 tables

Energy storage has been proven to yield positive effects on planning, operation and control of electric grids. It has become a crucial task to properly model the energy storage systems (ESS) under the framework of grid optimization on transmission and distribution networks including microgrids. This paper presents a review on mathematical models and test cases of ESSs used for grid optimization studies, where the network constraints of power systems are included. The existing ESS models are mainly classified into two categories; linear and nonlinear models. The two main categories are further divided into several subcategories respectively; such as mixed integer linear and convex nonlinear subcategories. Based on the review and discussions, this paper aims at providing suggestions for choosing proper ESS models for specific grid optimization studies considering the chosen power network model., Comment: Accepted for publication ahead of the PES General Meeting 2019