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In this paper, we study the existence of equilibrium in a single-leader-multiple-follower game with decision-dependent chance constraints (DDCCs), where decision-dependent uncertainties (DDUs) exist in the constraints of followers. DDUs refer to the uncertainties impacted by the leader's strategy, while the leader cannot capture their exact probability distributions. To address such problems, we first use decision-dependent ambiguity sets under moment information and Cantelli's inequality to transform DDCCs into second-order cone constraints. This simplifies the game model by eliminating the probability distributions. We further prove that there exists at least one equilibrium point for this game by applying Kakutani's fixed-point theorem. Finally, a numerical example is provided to show the impact of DDUs on the equilibrium of such game models.

This two-part tutorial dives into the field of data-driven power flow linearization (DPFL), a domain gaining increased attention. DPFL stands out for its higher approximation accuracy, wide adaptability, and better ability to implicitly incorporate the latest system attributes. This renders DPFL a potentially superior option for managing the significant fluctuations from renewable energy sources, a step towards realizing a more sustainable energy future, by translating the higher model accuracy into increased economic efficiency and less energy losses. To conduct a deep and rigorous reexamination, this tutorial first classifies existing DPFL methods into DPFL training algorithms and supportive techniques. Their mathematical models, analytical solutions, capabilities, limitations, and generalizability are systematically examined, discussed, and summarized. In addition, this tutorial reviews existing DPFL experiments, examining the settings of test systems, the fidelity of datasets, and the comparison made among a limited number of DPFL methods. Further, this tutorial implements extensive numerical comparisons of all existing DPFL methods (40 methods in total) and four classic physics-driven approaches, focusing on their generalizability, applicability, accuracy, and computational efficiency. Through these simulationmethodss, this tutorial aims to reveal the actual performance of all the methods (including the performances exposed to data noise or outliers), guiding the selection of appropriate linearization methods. Furthermore, this tutorial discusses future directions based on the theoretical and numerical insights gained. As the first part, this paper reexamines DPFL theories, covering all the training algorithms and supportive techniques. Capabilities, limitations, and aspects of generalizability, which were previously unmentioned in the literature, have been identified., Comment: 20 pages

Building on the theoretical insights of Part I, this paper, as the second part of the tutorial, dives deeper into data-driven power flow linearization (DPFL), focusing on comprehensive numerical testing. The necessity of these simulations stems from the theoretical analysis's inherent limitations, particularly the challenge of identifying the differences in real-world performance among DPFL methods with overlapping theoretical capabilities and/or limitations. The absence of a comprehensive numerical comparison of DPFL approaches in the literature also motivates this paper, especially given the fact that over 95% of existing DPFL studies have not provided any open-source codes. To bridge the gap, this paper first reviews existing DPFL experiments, examining the adopted test scenarios, load fluctuation settings, data sources, considerations for data noise/outliers, and the comparison made so far. Subsequently, this paper evaluates a total of 44 methods, containing over 30 existing DPFL approaches, some innovative DPFL techniques, and several classic physics-driven power flow linearization methods for benchmarking. The evaluation spans various dimensions, including generalizability, applicability, accuracy, and computational efficiency, using various different test cases scaling from 9-bus to 1354-bus systems. The numerical analysis identifies and examines significant trends and consistent findings across all methods under various test cases. It also offers theoretical insights into phenomena like under-performance, failure, excessive computation times, etc. Overall, this paper identifies the differences in the performances of the wide range of DPFL methods, reveals gaps not evident from theoretical discussions, assists in method selection for real-world applications, and provides thorough discussions on open questions within DPFL research, indicating ten potential future directions., Comment: 26 pages

The popularization of distributed energy resources transforms end-users from consumers into prosumers. Inspired by the sharing economy principle, energy sharing markets for prosumers are proposed to facilitate the utilization of renewable energy. This paper proposes a novel two-layer energy sharing market for massive prosumers, which can promote social efficiency by wider-area sharing. In this market, there is an upper-level wide-area market (WAM) in the distribution system and numerous lower-level local-area markets (LAMs) in communities. Prosumers in the same community share energy with each other in the LAM, which can be uncleared. The energy surplus and shortage of LAMs are cleared in the WAM. Thanks to the wide-area two-layer structure, the market outcome is near-social-optimal in large-scale systems. However, the proposed market forms a complex mathematical program with equilibrium constraints (MPEC). To solve the problem, we propose an efficient and hierarchically distributed bidding algorithm. The proposed two-layer market and bidding algorithm are verified on the IEEE 123-bus system with 11250 prosumers, which demonstrates the practicality and efficiency for large-scale markets.

Traditionally, offline optimization of power systems is acceptable due to the largely predictable loads and reliable generation. The increasing penetration of fluctuating renewable generation and Internet-of-Things devices allowing for fine-grained controllability of loads have led to the diminishing applicability of offline optimization in the power systems domain, and have redirected attention to online optimization methods. However, online optimization is a broad topic that can be applied in and motivated by different settings, operated on different time scales, and built on different theoretical foundations. This paper reviews the various types of online optimization techniques used in the power systems domain and aims to make clear the distinction between the most common techniques used. In particular, we introduce and compare four distinct techniques used covering the breadth of online optimization techniques used in the power systems domain, i.e., optimization-guided dynamic control, feedback optimization for single-period problems, Lyapunov-based optimization, and online convex optimization techniques for multi-period problems. Lastly, we recommend some potential future directions for online optimization in the power systems domain.

Due to possible emergency faults and frequency regulation reserve shortage in the multi-infeed hybrid AC-DC (MIDC) system, the emergency frequency control (EFC) with LCC-HVDC systems participating is important for system frequency stability. Nevertheless, the existing decentralized EFC strategies cannot guarantee the transient stability constraints of lines and cannot meet the frequency restoration requirement. To fill this gap, this paper proposes a complementary distributed EFC strategy. By selecting the semi-distributed or fully-distributed control law according to the state of AC system control center, the proposed distributed EFC can improve the control performance with a normal control center and enhance the control reliability in case of control center failure. Then, to derive the semi-distributed and fully-distributed control laws, a general design method is proposed by formulating an optimal EFC problem. Both of these two control laws can guarantee transient stability constraints, restore system frequency and achieve the defined optimal control objective. Moreover, the optimality of the closed-loop equilibrium is proven, and a Lyapunov analysis shows the asymptotic stability of the closed-loop system with the discontinuous control dynamics. A case study verifies the effectiveness of the proposed distributed EFC strategy.

With the proliferation of distributed generations, traditional passive consumers in distribution networks are evolving into "prosumers", which can both produce and consume energy. Energy trading with the main grid or between prosumers is inevitable if the energy surplus and shortage exist. To this end, this paper investigates the peer-to-peer (P2P) energy trading market, which is formulated as a generalized Nash game. We first prove the existence and uniqueness of the generalized Nash equilibrium (GNE). Then, an distributed online algorithm is proposed to track the GNE in the time-varying environment. Its regret is proved to be bounded by a sublinear function of learning time, which indicates that the online algorithm has an acceptable accuracy in practice. Finally, numerical results with six microgrids validate the performance of the algorithm.

In multi-infeed hybrid AC-DC (MIDC) systems, the emergency frequency control (EFC) with LCC-HVDC systems participating is of vital importance for system frequency stability. Nevertheless, when regional power systems are operated by different decision-makers, the LCC-HVDC systems and their connected AC systems might be unwilling to participate in the EFC due to the costs and losses. In this paper, to incentivize the LCC-HVDC systems and their connected adjacent AC systems to participate in the droop-based EFC, a novel control-parameter-based incentive mechanism is proposed, which can deal with various possible emergency frequency faults. Then, a non-cooperative-based incentive game model is formulated to implement the incentive mechanism in the MIDC system. An algorithm for seeking the Nash equilibrium is designed, and the uniqueness of Nash equilibrium is proven. Moreover, the individual rationality, incentive compatibility and social optimality of the proposed mechanism are analyzed and proven. The effectiveness of the proposed incentive mechanism is verified through a case study.

Contemporary industrial parks are challenged by the growing concerns about high cost and low efficiency of energy supply. Moreover, in the case of uncertain supply/demand, how to mobilize delay-tolerant elastic loads and compensate real-time inelastic loads to match multi-energy generation/storage and minimize energy cost is a key issue. Since energy management is hardly to be implemented offline without knowing statistical information of random variables, this paper presents a systematic online energy cost minimization framework to fulfill the complementary utilization of multi-energy with time-varying generation, demand and price. Specifically to achieve charging/discharging constraints due to storage and short-term energy balancing, a fast distributed algorithm based on stochastic gradient with two-timescale implementation is proposed to ensure online implementation. To reduce the peak loads, an incentive mechanism is implemented by estimating users' willingness to shift. Analytical results on parameter setting are also given to guarantee feasibility and optimality of the proposed design. Numerical results show that when the bid-ask spread of electricity is small enough, the proposed algorithm can achieve the close-to-optimal cost asymptotically., Comment: Accepted by Applied Energy

Due to the different losses caused by various photovoltaic (PV) array faults, accurate diagnosis of fault types is becoming increasingly important. Compared with a single one, multiple PV stations collect sufficient fault samples, but their data is not allowed to be shared directly due to potential conflicts of interest. Therefore, federated learning can be exploited to train a collaborative fault diagnosis model. However, the modeling efficiency is seriously affected by the model update mechanism since each PV station has a different computing capability and amount of data. Moreover, for the safe and stable operation of the PV system, the robustness of collaborative modeling must be guaranteed rather than simply being processed on a central server. To address these challenges, a novel asynchronous decentralized federated learning (ADFL) framework is proposed. Each PV station not only trains its local model but also participates in collaborative fault diagnosis by exchanging model parameters to improve the generalization without losing accuracy. The global model is aggregated distributedly to avoid central node failure. By designing the asynchronous update scheme, the communication overhead and training time are greatly reduced. Both the experiments and numerical simulations are carried out to verify the effectiveness of the proposed method.

In this paper, we discuss a class of two-stage hierarchical games with multiple leaders and followers, which is called Nash-Stackelberg-Nash (N-S-N) games. Particularly, we consider N-S-N games under decision-dependent uncertainties (DDUs). DDUs refer to the uncertainties that are affected by the strategies of decision-makers and have been rarely addressed in game equilibrium analysis. In this paper, we first formulate the N-S-N games with DDUs of complete ignorance, where the interactions between the players and DDUs are characterized by uncertainty sets that depend parametrically on the players' strategies. Then, a rigorous definition for the equilibrium of the game is established by consolidating generalized Nash equilibrium and Pareto-Nash equilibrium. Afterward, we prove the existence of the equilibrium of N-S-N games under DDUs by applying Kakutani's fixed-point theorem. Finally, an illustrative example is provided to show the impact of DDUs on the equilibrium of N-S-N games.

Owing to large industrial energy consumption, industrial production has brought a huge burden to the grid in terms of renewable energy access and power supply. Due to the coupling of multiple energy sources and the uncertainty of renewable energy and demand, centralized methods require large calculation and coordination overhead. Thus, this paper proposes a multi-energy management framework achieved by decentralized execution and centralized training for an industrial park. The energy management problem is formulated as a partially-observable Markov decision process, which is intractable by dynamic programming due to the lack of the prior knowledge of the underlying stochastic process. The objective is to minimize long-term energy costs while ensuring the demand of users. To solve this issue and improve the calculation speed, a novel multi-agent deep reinforcement learning algorithm is proposed, which contains the following key points: counterfactual baseline for facilitating contributing agents to learn better policies, soft actor-critic for improving robustness and exploring optimal solutions. A novel reward is designed by Lagrange multiplier method to ensure the capacity constraints of energy storage. In addition, considering that the increase in the number of agents leads to performance degradation due to large observation spaces, an attention mechanism is introduced to enhance the stability of policy and enable agents to focus on important energy-related information, which improves the exploration efficiency of soft actor-critic. Numerical results based on actual data verify the performance of the proposed algorithm with high scalability, indicating that the industrial park can minimize energy costs under different demands., Comment: Accepted by Applied Energy

The multi-energy management framework of industrial parks advocates energy conversion and scheduling, which takes full advantage of the compensation and temporal availability of multiple energy. However, how to exploit elastic loads and compensate inelastic loads to match multiple generators and storage is still a key problem under the uncertainty of demand and supply. To solve the issue, the energy management problem is constructed as a stochastic optimization problem. The optimization aims are to minimize the time-averaged energy cost and improve the energy efficiency while respecting the energy constraints. To achieve the distributed implementation in real time without knowing any priori knowledge of underlying stochastic process, a distributed stochastic gradient algorithm based on dual decomposition and a fast scheme are proposed. The numerical results based on real data show that the industrial park, by adopting the proposed algorithm, can achieve social welfare maximization asymptotically.

Estimating the stability boundary is a fundamental and challenging problem in transient stability studies. It is known that a proper level set of a Lyapunov function or an energy function can provide an inner approximation of the stability boundary, and the estimation can be expanded by trajectory reversing methods. In this paper, we streamline the theoretical foundation of the expansion methodology, and generalize it by relaxing the request that the initial guess should be a subset of the stability region. We investigate topological characteristics of the expanded boundary, showing how an initial guess can approach the exact stability boundary locally or globally. We apply the theory to transient stability assessment, and propose expansion algorithms to improve the well-known Potential Energy Boundary Surface (PEBS) and Boundary of stability region based Controlling Unstable equilibrium point (BCU) methods. Case studies on the IEEE 39-bus system well verify our results and demonstrate that estimations of the stability boundary and the critical clearing time can be significantly improved with modest computational cost.

Dual decomposition is widely utilized in distributed optimization of multi-agent systems. In practice, the dual decomposition algorithm is desired to admit an asynchronous implementation due to imperfect communication, such as time delay and packet drop. In addition, computational errors also exist when individual agents solve their own subproblems. In this paper, we analyze the convergence of the dual decomposition algorithm in distributed optimization when both the asynchrony in communication and the inexactness in solving subproblems exist. We find that the interaction between asynchrony and inexactness slows down the convergence rate from $\mathcal{O} ( 1 / k )$ to $\mathcal{O} ( 1 / \sqrt{k} )$. Specifically, with a constant step size, the value of objective function converges to a neighborhood of the optimal value, and the solution converges to a neighborhood of the exact optimal solution. Moreover, the violation of the constraints diminishes in $\mathcal{O} ( 1 / \sqrt{k} )$. Our result generalizes and unifies the existing ones that only consider either asynchrony or inexactness. Finally, numerical simulations validate the theoretical results.

Virtual power plant (VPP) provides a flexible solution to distributed energy resources integration by aggregating renewable generation units, conventional power plants, energy storages, and flexible demands. This paper proposes a novel model for determining the optimal offering strategy in the day-ahead energy-reserve market and the optimal self-scheduling plan. It considers exogenous uncertainties (or called decision-independent uncertainties, DIUs) associated with market clearing prices and available wind power generation, as well as the endogenous uncertainties (or called decision-dependent uncertainties, DDUs) pertaining to real-time reserve deployment requests. A tractable solution method based on strong duality theory, McCormick relaxation, and the Benders' decomposition to solve the proposed stochastic adaptive robust optimization with DDUs formulation is developed. Simulation results demonstrate the applicability of the proposed approach.

The proliferation of plug-in electric vehicles (PEVs) advocates a distributed paradigm for the coordination of PEV charging. Distinct from existing primal-dual decomposition or consensus methods, this paper proposes a cutting-plane based distributed algorithm, which enables an asynchronous coordination while well preserving individual's private information. To this end, an equivalent surrogate model is first constructed by exploiting the duality of the original optimization problem, which masks the private information of individual users by a transformation. Then, a cutting-plane based algorithm is derived to solve the surrogate problem in a distributed manner with intrinsic superiority to cope with various asynchrony. Critical implementation issues, such as the distributed initialization, cutting-plane generation and localized stopping criteria, are discussed in detail. Numerical tests on IEEE 37- and 123-node feeders with real data show that the proposed method is resilient to a variety of asynchrony and admits the plug-and-play operation mode. It is expected the proposed methodology provides an alternative path toward a more practical protocol for PEV charging.

In this paper, we investigate the continuous time partial primal-dual gradient dynamics (P-PDGD) for solving convex optimization problems with the form $ \min\limits_{x\in X,y\in\Omega}\ f({x})+h(y),\ \textit{s.t.}\ A{x}+By=C $, where $ f({x}) $ is strongly convex and smooth, but $ h(y) $ is strongly convex and non-smooth. Affine equality and set constraints are included. We prove the exponential stability of P-PDGD, and bounds on decaying rates are provided. Moreover, it is also shown that the decaying rates can be regulated by setting the stepsize.

With the proliferation of distributed generators and energy storage systems, traditional passive consumers in power systems have been gradually evolving into the so-called "prosumers", i.e., proactive consumers, which can both produce and consume power. To encourage energy exchange among prosumers, energy sharing is increasingly adopted, which is usually formulated as a generalized Nash game (GNG). In this paper, a distributed approach is proposed to seek the Generalized Nash equilibrium (GNE) of the energy sharing game. To this end, we convert the GNG into an equivalent optimization problem. A Krasnosel'ski{\v{i}}-Mann iteration type algorithm is thereby devised to solve the problem and consequently find the GNE in a distributed manner. The convergence of the proposed algorithm is proved rigorously based on the nonexpansive operator theory. The performance of the algorithm is validated by experiments with three prosumers, and the scalability is tested by simulations using 123 prosumers.

With the explosion of distributed energy resources (DERs), voltage regulation in distribution networks has been facing a great challenge. This paper derives an asynchronous distributed voltage control strategy based on the partial primal-dual gradient algorithm, where both active and reactive controllable power of DERs are considered. Different types of asynchrony due to imperfect communication or practical limits, such as random time delays and non-identical sampling/control rates, are fitted into a unified analytic framework. The asynchronous algorithm is then converted into a fixed-point problem by employing the operator splitting method, which leads to a convergence proof with mild conditions. Moreover, an online implementation method is provided to make the controller adjustable to time-varying environments. Finally, numerical experiments are carried out on a rudimentary 8-bus system and the IEEE-123 distribution network to verify the effectiveness of the proposed method.

This paper derives distributed conditions that guarantee the system-wide stability for power systems with nonlinear and heterogeneous bus dynamics interconnected via power network. Our conditions require each bus dynamics should satisfy certain passivity-like conditions with a large enough passivity index, a sufficient requirement of which is dictated by the steady-state power flow. The passivity indices uniformly quantify the impacts on the system-wide stability of individual bus dynamics and the coupling strength from the power network. Furthermore, taking three typical bus dynamics as examples, we show that these conditions can be easily fulfilled via proper control design. Simulations on a rudimentary 3-bus example and the IEEE 39-bus system well verify our results under both small and large disturbances., Comment: Accepted by IEEE Transactions on Power Systems

The stability issue emerges as a growing number of diverse power apparatus connecting to the power system. The stability analysis for such power systems is required to adapt to heterogeneity and scalability. This paper derives a local passivity index condition that guarantees the system-wide stability for lossless power systems with interconnected, nonlinear, heterogeneous bus dynamics. Our condition requires each bus dynamics to be output feedback passive with a large enough index w.r.t. a special supply rate. This condition fits for numerous existing models since it only constrains the input-output property rather than the detailed dynamics. Furthermore, for three typical examples of bus dynamics in power systems, we show that this condition can be reached via proper control designs. Simulations on a 3-bus heterogeneous power system verify our results in both lossless and lossy cases. The conservativeness of our condition is also demonstrated, as well as the impact on transient stability. It shows that our condition is quite tight and a larger index benefits transient stability.

With the rapid development of renewable and distributed energies, the underlying dynamics of power systems are no longer dominated by large synchronous generators, but by numerous dynamic components with heterogeneous characteristics. In such a situation, the traditional stability analysis method may fail due to the challenges of heterogeneity and scalability. In this paper, we handle this issue by fundamental circuit theory. Inspired by the work of Brayton and Moser in the nonlinear RLC circuit, we extend the concept of the voltage potential to phasor circuits and offer new results into the distributed stability analytics in power systems. We show that under certain distributed passivity-like conditions the system-wide stability can be ensured. The simulation of a 3-bus system is also provided to verify our results.

This work addresses the distributed frequency control problem in power systems considering controllable load with a nonsmooth cost. The nonsmoothness exists widely in power systems, such as tiered price, greatly challenging the design of distributed optimal controllers. In this regard, we first formulate an optimization problem that minimizes the nonsmooth regulation cost, where both capacity limits of controllable load and tie-line flow are considered. Then, a distributed controller is derived using the Clark generalized gradient. We also prove the optimality of the equilibrium of the closed-loop system as well as its asymptotic stability. Simulations carried out on the IEEE 68-bus system verifies the effectiveness of the proposed method.

Due to the uncertainty of distributed wind generations (DWGs), a better understanding of the probability distributions (PD) of their wind power forecast errors (WPFEs) can help market participants (MPs) who own DWGs perform better during trading. Under the premise of an accurate PD model, considering the correlation among DWGs and absorbing the new information carried by the latest data are two ways to maintain an accurate PD. These two ways both require the historical and latest wind power and forecast data of all DWGs. Each MP, however, only has access to the data of its own DWGs and may refuse to share these data with MPs belonging to other stakeholders. Besides, because of the endless generation of new data, the PD updating burden increases sharply. Therefore, we use the distributed strategy to deal with the data collection problem. In addition, we further apply the incremental learning strategy to reduce the updating burden. Finally, we propose a distributed incremental update scheme to make each MP continually acquire the latest conditional PD of its DWGs' WPFE. Specifically, we first use the Gaussian-mixture-model-based (GMM-based) joint PD to characterize the correlation among DWGs. Then, we propose a distributed modified incremental GMM algorithm to enable MPs to update the parameters of the joint PD in a distributed and incremental manner. After that, we further propose a distributed derivation algorithm to make MPs derive their conditional PD of WPFE from the joint one in a distributed way. Combining the two original algorithms, we finally achieve the complete distributed incremental update scheme, by which each MP can continually obtain its latest conditional PD of its DWGs' WPFE via neighborhood communication and local calculation with its own data. The effectiveness, correctness, and efficiency of the proposed scheme are verified using the dataset from the NREL.

Forming (hybrid) AC/DC microgrids (MGs) has become a promising manner for the interconnection of various kinds of distributed generators that are inherently AC or DC electric sources. This paper addresses the distributed asynchronous power control problem of hybrid microgrids, considering imperfect communication due to non-identical sampling rates and communication delays. To this end, we first formulate the optimal power control problem of MGs and devise a synchronous algorithm. Then, we analyze the impact of asynchrony on optimal power control and propose an asynchronous iteration algorithm based on the synchronous version. By introducing a random clock at each iteration, different types of asynchrony are fitted into a unified framework, where the asynchronous algorithm is converted into a fixed-point problem based on the operator splitting method, leading to a convergence proof. We further provide an upper bound estimation of the time delay in the communication. Moreover, the real-time implementation of the proposed algorithm in both AC and DC MGs is introduced. By taking the power system as a solver, the controller is simplified by reducing one order and the power loss can be considered. Finally, a benchmark MG is utilized to verify the effectiveness and advantages of the proposed algorithm.

This paper addresses the distributed frequency control problem in a multi-area power system taking into account of unknown time-varying power imbalance. Particularly, fast controllable loads are utilized to restore system frequency under changing power imbalance in an optimal manner. The imbalanced power causing frequency deviation is decomposed into three parts: a known constant part, an unknown low-frequency variation and a high-frequency residual. The known steady part is usually the prediction of power imbalance. The variation may result from the fluctuation of renewable resources, electric vehicle charging, etc., which is usually unknown to operators. The high-frequency residual is usually unknown and treated as an external disturbance. Correspondingly, in this paper, we resolve the following three problems in different timescales: 1) allocate the steady part of power imbalance economically; 2) mitigate the effect of unknown low-frequency power variation locally; 3) attenuate unknown high-frequency disturbances. To this end, a distributed controller combining consensus method with adaptive internal model control is proposed. We first prove that the closed-loop system is asymptotically stable and converges to the optimal solution of an optimization problem if the external disturbance is not included. We then prove that the power variation can be mitigated accurately. Furthermore, we show that the closed-loop system is robust against both parameter uncertainty and external disturbances. The New England system is used to verify the efficacy of our design.

This paper addresses the distributed optimal frequency control of power systems considering a network-preserving model with nonlinear power flows and excitation voltage dynamics. Salient features of the proposed distributed control strategy are fourfold: i) nonlinearity is considered to cope with large disturbances; ii) only a part of generators are controllable; iii) no load measurement is required; iv) communication connectivity is required only for the controllable generators. To this end, benefiting from the concept of 'virtual load demand', we first design the distributed controller for the controllable generators by leveraging the primal-dual decomposition technique. We then propose a method to estimate the virtual load demand of each controllable generator based on local frequencies. We derive incremental passivity conditions for the uncontrollable generators. Finally, we prove that the closed-loop system is asymptotically stable and its equilibrium attains the optimal solution to the associated economic dispatch problem. Simulations, including small and large-disturbance scenarios, are carried on the New England system, demonstrating the effectiveness of our design.

Direct-current microgrids (DC-MGs) can operate in either grid-connected or stand-alone mode. In particular, stand-alone DC-MG has many distinct applications. However, the optimal power flow problem of a stand-alone DC-MG is inherently non-convex. In this paper, the optimal power flow (OPF) problem of DC-MG is investigated considering convex relaxation based on second-order cone programming (SOCP). Mild assumptions are proposed to guarantee the exactness of relaxation, which only require uniform nodal voltage upper bounds and positive network loss. Furthermore, it is revealed that the exactness of SOCP relaxation of DC-MGs does not rely on either topology or operating mode of DC-MGs, and an optimal solution must be unique if it exists. If line constraints are considered, the exactness of SOCP relaxation may not hold. In this regard, two heuristic methods are proposed to give approximate solutions. Simulations are conducted to confirm the theoretic results.

Stand-alone direct current (DC) microgrids may belong to different owners and adopt various control strategies. This brings great challenge to its optimal operation due to the difficulty of implementing a unified control. This paper addresses the distributed optimal control of DC microgrids, which intends to break the restriction of diversity to some extent. Firstly, we formulate the optimal power flow (OPF) problem of stand-alone DC microgrids as an exact second order cone program (SOCP) and prove the uniqueness of the optimal solution. Then a dynamic solving algorithm based on primal-dual decomposition method is proposed, the convergence of which is proved theoretically as well as the optimality of its equilibrium point. It should be stressed that the algorithm can provide control commands for the three types of microgrids: (i) power control, (ii) voltage control and (iii) droop control. This implies that each microgrid does not need to change its original control strategy in practice, which is less influenced by the diversity of microgrids. Moreover, the control commands for power controlled and voltage controlled microgrids satisfy generation limits and voltage limits in both transient process and steady state. Finally, a six-microgrid DC system based on the microgrid benchmark is adopted to validate the effectiveness and plug-n-play property of our designs.

In Part I of this paper we propose a decentralized optimal frequency control of multi-area power system with operational constraints, where the tie-line powers remain unchanged in the steady state and the power mismatch is balanced within individual control areas. In Part II of the paper, we propose a distributed controller for optimal frequency control in the network power balance case, where the power mismatch is balanced over the whole system. With the proposed controller, the tie-line powers remain within the acceptable range at equilibrium, while the regulation capacity constraints are satisfied both at equilibrium and during transient. It is revealed that the closed-loop system with the proposed controller carries out primal-dual updates with saturation for solving an associated optimization problem. To cope with discontinuous dynamics of the closed-loop system, we deploy the invariance principle for nonpathological Lyapunov function to prove its asymptotic stability. Simulation results are provided to show the effectiveness of our controller.

This paper addresses the distributed optimal frequency control of multi-area power system with operational constraints, including the regulation capacity of individual control area and the power limits on tie-lines. Both generators and controllable loads are utilized to recover nominal frequencies while minimizing regulation cost. We study two control modes:the per-node balance mode and the network balance mode. In Part I of the paper, we only consider the per-node balance case, where we derive a completely decentralized strategy without the need for communication between control areas. It can adapt to unknown load disturbance. The tie-line powers are restored after load disturbance, while the regulation capacity constraints are satisfied both at equilibrium and during transient. We show that the closed-loop systems with the proposed control strategies carry out primal-dual updates for solving the associated centralized frequency optimization problems. We further prove the closed-loop systems are asymptotically stable and converge to the unique optimal solution of the centralized frequency optimization problems and their duals. Finally, we present simulation results to demonstrate the effectiveness of our design. In Part II of the paper, we address the network power balance case, where transmission congestions are managed continuously., Comment: 12 pages, 5 figures