Linearized AC Power Flow Research Articles

Dynamic hierarchical partitioning method of flexible resource for virtual power plant based on event triggering

Abstract With the advantage of efficient aggregation of flexible resources on the demand side, virtual power plants have been widely considered as one of the effective solutions to solve the lack of large-scale access systems and regulation capacity of new energy. Aiming at the problem of how the flexible resources of the virtual power plant can effectively participate in the optimal operation of hierarchical zoning of the power grid, this paper takes the aggregation of flexible resources of the virtual power plant to alleviate multiple line overload as an example. By deducing the transfer factor matrix under the linearized AC power flow model, an event-triggered hierarchical zoning method of flexible resources of virtual power plants considering resource effectiveness is proposed. The effectiveness of the algorithm is verified in several network examples, which provides a new idea for the virtual power plant to participate in the optimal operation of various levels and regions of the power grid.

A four-stage fast reliability assessment framework for renewables-dominated strong power systems with large-scale energy storage by temporal decoupling and contingencies filtering

Reliability assessment for renewables-dominated strong power systems presents significant challenges, primarily due to the overwhelming computational demands posed by large-scale renewable generation and energy storage. In response to the limitations and lack of scalability in traditional methods, this study introduces a novel four-stage fast reliability assessment framework tailored for renewables-dominated strong power systems. First and foremost, the pre-dispatch and temporal decoupling of energy storage serve as vital foundations for the fast assessment framework. Following the pre-dispatch model, an adaptive scenario clustering technique is proposed to eliminate redundant scenarios while retaining extreme ones. Consequently, a quick verification process for contingencies is established, employing DC optimal power flow in the worst-case scenario to identify potential load shedding. Then, all contingencies are filtered to only effective ones for reliability indices computation. Finally, a modified reliability calculation model with linear AC power flow is built using the filtered contingency set and reduced scenarios. The efficacy of this approach is validated through four numerical experiments, demonstrating impressive efficiency and computation feasibility. For accuracy, the proposed method shows a 3.7% average error in a modified IEEE RTS system, and for efficiency, the proposed method finishes N-2 analysis in 424 s for a 62-bus-212-line case as well as 4924 s for a 200-bus-490-line case.

Hierarchical Stochastic Frequency Constrained Micro-Market Model for Isolated Microgrids

With the developments of isolated microgrids (IMGs) and prosumers in remote areas, energy trading has emerged as a critical aspect of IMGs. However, the lack of an upstream network and the low inertia of the system may threaten the secure operation of these networks. This paper proposes a Micro-Market (lM) model for IMGs that includes a precise hierarchical control structure. To address the IMGs low inertia and high intermittency of renewable energy sources (RES), the proposed lM manages the active-reactive power and schedules primary and secondary active reserves to maintain the frequency within in a predefined range. Additionally, a bidirectional linearized AC power flow is established to schedule the reactive reserve and the proposed model is formulated as a two-stage stochastic mixed-integer linear problem (MILP) to maximize social welfare (SW) over the next 24 hours. To validate the effectiveness of the proposed model, the lM is tested on an IMG based on a CIGRE medium-voltage benchmark system, and different operational cases are simulated. The results demonstrate that the proposed model, which takes into account hierarchical control levels and technical issues of the IMG, is a cost-effective way to maximize social welfare while ensuring the secure operation of the IMG.

Linearized AC power flow model based interval total transfer capability evaluation with uncertain renewable energy integration

Total transfer capability (TTC) evaluation plays an essential role in guaranteeing reliable power transactions in a deregulated electricity market. However, the large-scale integration of wind power brings significant challenges to TTC calculations. The conventional probabilistic TTC evaluation methods are highly dependent on the pre-defined probabilistic distributions of the input random variables, which could not be accurately predicted in practical situations. This paper proposes an interval TTC evaluation method based on the linearized AC power flow (LACPF) model, which can provide intuitive TTC boundary information without any prior distributions. The adoption of LACPF also contributes to yielding more accurate TTC values than those DC power flow (DCPF) based methods. The proposed interval optimization model is equivalently translated into an optimistic model and a pessimistic model. The optimistic model is formulated as a max-max problem and can be directly converted to a single-level linear programming model. The pessimistic model is formulated as a min-max problem that cannot be solved directly. Hence, the strong duality theory and the big-M method are introduced to transfer the original bilevel model into a single-level mixed integer linear programming (MILP) model for an efficient solution. The proposed method is validated on the IEEE 118-bus system and a modified actual power grid in Northwest China. Numerical results demonstrate its computational efficiency.

Segmentation of interconnected power systems considering microgrids and the uncertainty of renewable energy sources

AbstractThis paper proposes an approach for segmentation of power systems to prevent uncontrolled islanding. With the development of microgrids (MGs) and increased penetration of renewable energy sources (RESs) in the power system, it is necessary to consider the effect of MGs and the stochastic behaviour of RESs in the controlled islanding process. First, the enormous search space for finding the boundary points is reduced by the clustering of the coherent generators and the buses which are placed in sections that cannot be separated from the coherent groups. Then, linear AC power flow is defined as a sub‐problem and solved by linear programming to satisfy the power system and MG constraints. In this step, an optimization model is defined, and a mixed‐integer linear programming (MILP)‐based model is suggested to find cut‐sets and islands arrangement with reduced load shedding. In view of the fact that the fluctuation of RESs may have the significant effects on the power imbalance, to cover the uncertainties a scenario based stochastic simulation is considered. Finally, the effect of uncertainties on cut‐sets and load curtailment is evaluated. The effectiveness of the proposed methodology is demonstrated by the simulation on the test system.

Robust Planning of Distributed Generators in Active Distribution Network Considering Network Reconfiguration

The energy crisis and environmental concerns have accelerated the development of the active distribution network (ADN) with a high proportion of renewable energy, which poses a challenge to the operation of the power system. Moreover, using active management means to promote the consumption of renewable energy is an important task of ADN. Therefore, as an important operation means, the network reconfiguration is used to enhance the adjustable capacity of the power system at the planning stage. Firstly, a “wind–light–load” uncertain scenario set is constructed to address the uncertainty of wind speed, lighting, and load. On this basis, a robust optimization model for distributed power generation taking into account network reconstruction and in ADN is proposed. In addition, the distributed generator (DG) permeability indicator is introduced in the planning model to improve the ADN ability of absorbing renewable energy. A linearized AC power flow model is utilized to calculate the power flow. Finally, via simulation in an IEEE 33-bus system and IEEE 69-bus system, the influence of network reconfiguration and robustness on distributed generator planning, economy and reliability of ADN is analyzed, and the validity of the model is verified.

A multi-energy multi-microgrid system planning model for decarbonisation and decontamination of isolated systems

Decarbonising and decontaminating remote regions in the world presents several challenges. Many of these regions feature isolation, dispersed demand in large areas, and a lack of economic resources that impede the development of robust and sustainable networks. Furthermore, isolated systems in the developing world are mostly based on diesel generation for electricity, and firewood and liquefied petroleum gas for heating, as these options do not require a significant infrastructure cost. In this context, we present a stochastic multi-energy multi-microgrid system planning model that integrates electricity, heat and hydrogen networks in isolated systems. The model is stochastic to capture uncertainty in renewable generation outputs, particularly hydro and wind, and thus design a multi-energy system proved secured against such uncertainty. The model also features two distinct constraints to limit the emissions of CO2 (for decarbonisation) and particulate matter (for decontamination), and incorporates firewood as a heating source. Moreover, given that the focus is on low-voltage networks, we introduce a fully linear AC power flow equations set, allowing the planning model to remain tractable. The model is applied to a real-world case study to design a multi-energy multi-microgrid system in an isolated region in Chilean Patagonia. In a case with a zero limit over direct CO2 emissions, the total system’s cost increases by 34% with respect to an unconstrained case. In a case with a zero limit over particulate matter emissions, the total system’s cost increases by 189%. Finally, although an absolute zero limit over both, particulate matter and direct CO2 emissions, leads to a total system’s cost increase of 650%, important benefits in terms of decarbonisation and decontamination can be achieved at marginal cost increments.

Optimal controlled-islanding with AC power flow constraints: A benders-decomposition approach

The controlled-islanding process aims to prevent large devastating blackouts through an optimal splitting of a power network into smaller islands followed by corrective control actions. Generally, the controlled-islanding problem is cast as a mixed-integer non-linear programming (MINLP) model, which can be prohibitively cumbersome, particularly when it accommodates AC power flow constraints. This paper exploits the Benders decomposition to enhance the mathematical tractability of the problem. The AC power flow constraints are converted to equality constraints using proper auxiliary variables. These new constraints are applied to form a modified Jacobian matrix. The modified equations are embedded in the Benders subproblem involving slack variables in a linear iterative-based optimization structure. Hence, the obtained solution is totally free of any approximation or linearization error. The objective function of the controlled-islanding problem is a combination of total load shedding and generation curtailment of the network. This objective function implicitly mitigates the stability concerns in the islands, which can threaten their steady operation. As opposed to a unique value, adaptive lower bound are considered for the steady-state frequency deviation of islands. This adaptive setting prevents an incomplete frequency recovery after the islanding, considering the reserve capacity of each island. The proposed method yields more practically sound results than a standard mixed-integer linear programming (MILP) model with DC or linearized AC power flow constraints as voltage and apparent power quantities are retained in the permissible ranges. The effectiveness of the proposed algorithm is examined via the IEEE 30-bus test system and a 76-bus case as a part of an actual power system. The time-domain simulation results are also provided to support the accuracy of the frequency stability model captured in the optimization problem.

Three Solution Approaches to Stochastic Multi-Period AC Optimal Power Flow in Active Distribution Systems

This paper proposes three novel solution approaches (A1, A2, A3) to solve stochastic multi-period AC optimal power flow (S-MP-OPF) for day-ahead flexibility procurement from distributed energy resources (DER) in active distribution systems (ADSs). The S-MP-OPF is a mixed-integer nonlinear programming (MINLP) problem due to the binary variables modeling the operation of storage devices and flexible loads. The proposed three approaches have a shared common first step, which resorts to a new mixed-integer linear programming (MILP) model approximation of the S-MP-OPF problem. The MILP model employs second-order Taylor series expansion of trigonometric terms and formulates the linear approximations relying on variables such as square of voltage magnitude and voltage angle difference. This first step serves also the purpose of fixing the binary variables to the values computed by the MILP problem. Then, the approach A1 only checks the AC feasibility of MILP solution while approaches A2 and A3 further optimize continuous variables. Specifically, the sophisticated heuristic approach A2 employs sequential linear programming and AC power flow while the approach A3 models and solves directly the remaining nonlinear programming (NLP) problem. The performances of these approaches, a benchmark MINLP solver, and a state-of-the-art method are thoroughly compared in three radial or weakly meshed ADSs of 34, 31, and 191 nodes, respectively. The numerical results indicate that, albeit the approach A2 performs best overall, these approaches present distinct accuracy vs speed trade-offs, which make them suitable to problems of different sizes and diverse accuracy or speed requirements.

Optimal Planning and Bidding Strategy for Wind Farms in Joint Balancing and Day‐Ahead Energy Markets

Due to the fact that wind farms have a meagre operating cost, it is expected that their optimum utilization in the transmission network would improve the economic and technical condition of the network. Henceforth, this work aims to allocate wind farms' locations in the transmission network while persuading the farm owners to participate in the balancing and day-ahead energy markets. The proposed scheme is formulated in the form of a bi-level optimization model. The upper-level problem aims to maximize wind farms' profit in joint day-ahead and balancing markets bounded by the location and operational constraints. On the other hand, the lower-level problem provides the operating model of the transmission network. Herein, the network operator is in authority to minimize the operational cost of non-renewable resources, considering the linearized AC power flow and technical constraints of resources. Then, the Karush—Kuhn–Tucker is adopted to extract the proposed scheme single-level model. One of the eminent novelties of this work is to propose a linear model for wind farms and to introduce an incentive plan for farm owners in the energy market by considering bidding errors. In the end, by implementing this scheme on IEEE 6-bus and 24-bus transmission networks, the numerical outcomes indicate the proposed scheme's potential to improve the economic and operational status of wind farms and the transmission network. With linear programming based on optimum utilization of wind farms, the utilization of traditional units, energy prices, energy plans, and the maximum voltage drop are reduced by 35%, 15%, 20%, and 21%, respectively, compared to the cases without wind farms penetrations.

Coordination of Preventive, Emergency and Restorative Dispatch in Extreme Weather Events

The increase of extreme weather events with wide-area enduring adverse impact has caused enormous loss to power systems. Coordination of various emergency measures can effectively reduce the overall loss over the time-horizon of these events. However, coordination of preventive dispatch, emergency dispatch, and restorative dispatch in transmission systems under extreme weather conditions has not been thoroughly addressed. To fill this gap, we first propose a methodology to coordinate the above dispatch stages based on a novel hierarchical defender-attacker model (HDAM). In this model, various dispatch stages are modelled as defenders, whereas the extreme weather event is modelled as the attacker. Second, frequency security constraints, load shedding, and transmission line switching are modelled with the piecewise linear AC power flow in the HDAM. Particularly, an adaptive approximation of cosine function in the piecewise linear power flow equation is applied to greatly reduce the computational complexity. Third, the HDAM is recast as a two-stage robust optimization model with binary decision variables in the recourse stage and solved by the nested column-and-constraint generation algorithm. Then, the robust resilience trapezoid and coordinated strategy during extreme weather events are obtained by the proposed methodology. Moreover, by inference-based sensitivity analysis of the discrete line outage scenarios, the critical transmission lines and transformers are identified for pre-event hardening. Comprehensive case studies using IEEE benchmark systems and a Chinese 132-bus system validate that coordination of multiple dispatch stages is effective to enhance power system resilience against extreme weather events by reducing the overall loss.

Line outage distribution factors of a linearized AC model with reactive power and voltage magnitude for resilience-constrained economic dispatch

Research on preventive generation dispatch schemes for resilient power system operation is often based on the DC power flow model, ignoring the influence of reactive power and voltage magnitude. This paper presents an in-depth study of a linearized AC power flow (LAC) model with reactive power and voltage magnitude to derive sensitivity factors, including shift factors and line outage distribution factors, for pre-contingency and post-contingency power flow calculations for N-k contingencies. Based on the derived sensitivity factors, a resilience-constrained economic dispatch (LAC-RCED) strategy is developed, which considers the security constraints of N-1 contingency for all lines and N-2 contingency for the affected lines, as well as optimization objectives to improve the power flow distribution in the transmission system. To deal with the computational difficulties associated with the N-k contingency constraints, an iterative contingency filtering algorithm based on the derived line outage distribution factor is proposed for contingency screening and creating security constraints for violated contingency scenarios. In the case study, the accuracy of the power flow solution obtained from the derived sensitivity factors is investigated by comparison with the AC model. The proposed LAC-RCED model and the iterative contingency filtering algorithm are tested on the IEEE 30-bus and 118-bus systems.

Distributed Scheme for Line Overload Mitigation With Linearized AC Power Flow

The transmission overload after a disturbance poses significant security risk to the power system. Once it happens, an efficient remedial action must be taken to relieve the line overloads as quickly as possible. The line overload mitigation problem becomes more urgent and challenging in the future power grid that integrates distributed and fluctuating renewable energies. In this brief, we propose a distributed corrective control scheme for alleviating the overloads in transmission lines. A linearized AC power flow model is introduced, and the error of linearizion is compensated through a close loop of power flow control. The proposed distributed control framework not only satisfies the need of accommodating distributed resources in future grids, but also guarantees the system security during the line overload mitigation process. The performance of our corrective control scheme is demonstrated in the simulations using the IEEE 14-bus test system.

Minimal Controlled Islanding With Similarity-Based Coherency Identification Using Phasor Measurement Data

Controlled islanding is the last protective scheme to prevent passive islanding or total system blackout caused by cascading failures. In this article, a novel coherency detection approach is proposed, which is based on a density and distance clustering algorithm. The coherency detection approach is parameter free and requires only the frequency measurement data since the approach utilizes the cosine similarity combined by the Gaussian principle. Following the coherency detection, the proposed island reduction algorithm utilizes frequency deviation at center-of-inertia reference and locates the fault-affected generators plus the faulted area to construct the islanding scenarios. To find a practical splitting strategy with better network reconnection, an island reduction algorithm is presented to reduce the number of islands and isolate the faulted area from the rest of the system considering the coherency and system operational constraints. Additionally, an mixed integer liner programming (MILP) optimization model based on the linearized ac power flow is employed to find the optimal island boundaries and guarantee the efficient isolation of critical or most affected islands. The efficiency and performance of the proposed islanding approach are tested on the IEEE 118-bus test system.

Bidding Strategy of a Wind-Thermal GENCO considering Piecewise Linear AC Power Flow and Correlated Uncertainties

In deregulated electricity markets, generation companies (GENCO) try to maximize their economic benefits considering the electricity demand, transmission network condition, and other participants’ behaviors. The increasing penetration of renewable sources such as wind power generation with intermittent nature poses several challenges to the participation of GENCOs in the electricity market. Thus, this paper presents a stochastic bilevel optimization model to determine the coordinated bidding strategy of a wind-thermal GENCO with the aim of maximizing its profit in the day-ahead and real-time balancing market. Herein, the model aims to maximize the profit of GENCO in the day-ahead and the balancing market in the upper-level problem while minimizing the operation cost of the system in the lower-level problem. The uncertainties of wind power generation and electricity demand are modeled by defining a set of scenarios considering their mutual correlation using the copula technique. Additionally, incorporating AC power flow constraints in the proposed optimization model offers a better solution to the coordinated bidding strategy of the wind-thermal GENCO. Further, the nonlinear AC power flow equations are linearized using the piecewise approximation technique to reduce the computational complexity and enhance the accuracy of the optimal solution. In the end, the developed algorithm is implemented on the IEEE 24-bus RTS, and the simulation results are provided to validate the efficiency and applicability of the proposed coordinated bidding strategy model. The results advocate that the participation of the thermal unit along with the wind farm might mitigate the risk of uncertainties, but it causes an intense increase in the locational marginal price of the system. Importantly, the simulation results indicate the computational efficiency of the model by developing an exact AC power flow model without compromising the results. Notably, it has been found that the profit of the wind-thermal GENCO would be increased by 35.2% employing the copula technique to model the mutual correlation of uncertain parameters.

Optimal Power Flow in Distribution Networks Under N – 1 Disruptions: A Multistage Stochastic Programming Approach

Contingency research to find optimal operations and postcontingency recovery plans in distribution networks has gained major attention in recent years. To this end, we consider a multiperiod optimal power flow problem in distribution networks, subject to the N – 1 contingency in which a line or distributed energy resource fails. The contingency can be modeled as a stochastic disruption, an event with random magnitude and timing. Assuming a specific recovery time, we formulate a multistage stochastic convex program and develop a decomposition algorithm based on stochastic dual dynamic programming. Realistic modeling features, such as linearized AC power flow physics, engineering limits, and battery devices with realistic efficiency curves, are incorporated. We present extensive computational tests to show the efficiency of our decomposition algorithm and out-of-samplex performance of our solution compared with its deterministic counterpart. Operational insights on battery utilization, component hardening, and length of recovery phase are obtained by performing analyses from stochastic disruption-aware solutions. Summary of Contribution: Stochastic disruptions are random in time and can significantly alter the operating status of a distribution power network. Most of the previous research focuses on the magnitude aspect with a fixed set of time points in which randomness is observed. Our paper provides a novel multistage stochastic programming model for stochastic disruptions, considering both the uncertainty in timing and magnitude. We propose a computationally efficient cutting-plane method to solve this large-scale model and prove the theoretical convergence of such a decomposition algorithm. We present computational results to substantiate and demonstrate the theoretical convergence and provide operational insights into how making infrastructure investments can hedge against stochastic disruptions via sensitivity analyses.

Optimal market-based operation of microgrid with the integration of wind turbines, energy storage system and demand response resources

This paper deals with an optimal operation of a microgrid in the electricity market and presents the communication between the distribution market operator and microgrid operator. The distribution market operator controls and manages the electricity market established in the distribution level, determining the amount of both electricity price and power exchange between market participants. The microgrid operator is able to purchase active and reactive power from the local distribution market. An effective short-term scheduling of the microgrid is implemented to ensure optimal operation. A risk-based stochastic model is used to model the prevailing uncertainties such as loads, wind generation, and main-grid availability in a market-based operation framework. Moreover, in this model, a linearized AC power flow is added to the mathematical formulations to offer a comprehensive solution to the security-constraint operation of the microgrid. The stochastic operation strategy is formulated as a mixed integer linear programming problem. Regarding the uncertainty modeling, the substation equipment failure is modeled with Monte-Carlo algorithm. The effectiveness of the risk-based stochastic method is demonstrated using a microgrid test bed in the presence of demand response resources, dispatchable and wind generation units as well as energy storage system. The results demonstrate demand response program can significantly reduce the operation cost in worst scenarios. Also, it is indicated that the risk-averse decisions reduce the risk of experiencing costly scenario. The proposed framework incorporating the distribution market constraints reduces the uncertainty in real-time operation as it can specified the required energy before running the problem. The deviations from assigned energy to the microgrid are penalized through the distribution market operator.

A Linear AC Power Flow Model for Unbalanced Multi-Phase Distribution Networks Based on Current Injection Equations

A realistic and practicable modeling of unbalanced distribution networks (DNs) is the missing link in most studies undertaken in these networks, e.g., optimal power flow (OPF), energy markets, etc. To address the issue, this letter proposes an unbalanced linear three-phase four-wire power flow (PF) model based on current injection equations at nodes. Also, the ZIP representation is considered for loads. The model can be easily generalized for unbalanced three-wire networks with minor changes and assumptions. Hence, the scope of the proposed method can be extended to cover both low-voltage and medium-voltage DNs. The performance of the proposed model is verified through simulations on two IEEE test feeders.

Optimal coordination of thyristor controlled series compensation and transmission expansion planning: Distributionally robust optimization approach

This paper proposes a distributionally robust optimization method for assessing economic benefits of transmission expansions with thyristor controlled series compensation devices. To accommodate the uncertainty of wind generations and electrical vehicle loads, a distributionally robust optimization method is proposed to use historical data instead of predefined distribution probabilities. In the proposed framework, firstly, a confidence set for unknown distributions are constructed based on the historical data. Then, based on worst-case distributions within a constructed confidence set, the optimal results of transmission expansion planning and allocation of thyristor controlled series compensation device are obtained. In order to take advantage of the flexible alternative current transmission devices (FACTS) in both economic and technical considerations, the optimal allocation of thyristor controlled series compensation is integrated into the transmission expansion planning. This requires AC-based transmission expansion planning model. AC power flow suffers from high computational complexity. To tackle this issue, the linearized AC power flow model is utilized. Furthermore, to solve the optimization problem, the combined technique including Bender's decomposition and cut-and-column are utilized. The numerical results on the Garver 6-bus and IEEE 118-bus systems demonstrate the effectiveness of the proposed method in comparison with scenario-based stochastic and robust optimization methods. Besides, the impacts of the thyristor controlled series compensation on total cost reduction has been investigated.

Network-constrained unit commitment with piecewise linear AC power flow constraints

This paper presents a new network-constrained unit commitment (UC) model, which is formulated as a mixed-integer linear programming (MILP) optimization problem with embedded ac power flow (AC-PF) constraints. Piecewise linear AC-PF equations are formulated, which constrain the UC dispatch schedule within permissible transmission capacity and voltage deviation limits, solely by executing the main optimization problem. The proposed model is tested on a six-bus and the IEEE RTS-79 test systems, using GAMS software, whereas the obtained AC-PF results are validated against DIgSILENT/Powerfactory package. The UC schedules are comparatively assessed with detailed results presented in the relevant literature for the same test systems, employing different UC+AC-PF models.