Traditional Optimal Power Flow Research Articles

Distributed sequential optimal power flow under uncertainty in power distribution systems: A data-driven approach

Modern distribution systems with high penetration of distributed energy resources face multiple sources of uncertainty. This transforms the traditional Optimal Power Flow problem into a problem of sequential decision-making under uncertainty. In this framework, the solution concept takes the form of a policy, i.e., a method of making dispatch decisions when presented with a real-time system state. Reasoning over the future uncertainty realization and the optimal online dispatch decisions is especially challenging when the number of resources increases and only a small dataset is available for the system’s random variables. In this paper, we present a data-driven distributed policy for making dispatch decisions online and under uncertainty. The policy is assisted by a Graph Neural Network but is constructed in such a way that the resulting dispatch is guaranteed to satisfy the system’s constraints. The proposed policy is experimentally shown to achieve a performance close to the optimal-in-hindsight solution, significantly outperforming state-of-the-art policies based on stochastic programming and plain machine-learning approaches.

Robust optimal power flow considering uncertainty in wind power probability distribution

This paper proposes an optimal power flow model that takes into account the uncertainty in the probability distribution of wind power. The model can schedule controllable generators under any possible distribution of wind power to ensure the safe and economic operation of the system. Firstly, considering the incompleteness of historical wind power data, the paper models the uncertainty of wind power using second-order moments of probability distribution and their fluctuation intervals. Subsequently, a robust optimal power flow model based on probability distribution model and joint chance constraints is established. The Lagrangian duality theorem is then employed to eliminate random variables from the optimization model, transforming the uncertainty model into a deterministic linear matrix inequality problem. Finally, a convex optimization algorithm is used to solve the deterministic problem, and the results are compared with traditional chance-constrained optimal power flow model. The feasibility and effectiveness of the proposed method are validated through case study simulations.

Decentralized DC optimal power flow model based on improved Lagrangian and consensus algorithm

Renewable energy sources such as wind power and photovoltaic are integrated into the power system in a high proportion and a decentralized manner. Driven by factors such as the privacy of the interests of various power sources, the efficiency of communication and computing and the reliability of transaction security, various forms of power sources are managed and controlled in a decentralized manner. It makes the traditional centralized DC optimal power flow difficult to realize. So, based on Lagrangian algorithm and improving traditional consistency algorithms, a decentralized DC optimal power flow model is proposed. Firstly, the decentralized calculation methods of active power, phase and line active power flow at nodes based on the consensus are presented. Secondly, based on Lagrangian algorithm, an improved DC optimal power flow model considering line constraints is derived. Thirdly, the calculation principle of decentralized DC optimal power flow model is proposed. It consists of two parts: the main iteration and the sub iteration. The main iteration is an iterative process with phase angle as a variable. Sub iteration is a process with Lagrange multipliers as iteration variables and using the Kaczmarz algorithm to prevent non convergence of iterations. Finally, a real system and IEEE14 bus system are taken as an example to demonstrate the accuracy of the proposed method compared to the Matpower and Injection Shift Factors(ISF) method.

Peer-to-peer energy trading among multiple microgrids considering risks over uncertainty and distribution network reconfiguration: A fully distributed optimization method

This paper proposes a two-layer optimization framework to co-optimize the P2P energy trading among multiple microgrids (MMGs) under uncertainty and optimal topology planning of the distribution networks (DNs). At the upper layer, the traditional verification optimal power flow model of DNs is transformed into a prosumer-focused and transaction-oriented dynamic network reconfiguration model. At the lower layer, uncertainty from wind power generations is integrated into the operating model of individual MGs and addressed by the stochastic programming (SP) method. Meanwhile, the conditional value at risk technique is introduced to find a trade-off between cost minimization and risk aversion flexibly. To establish the global negotiation mechanism among all participants (not only between distribution system operators and MGs, but also among MMGs), a fully distributed method is developed by combining an analytical target cascading algorithm and an alternating direction multiplier method. Furthermore, a diagonal quadratic approximation method is utilized to linearize the quadratic penalty term so that achieving parallel computing for all independent optimization subproblems. Simulations of different strategies, models, and distributed algorithms are implemented to verify the rationality and validity of the proposed method. The results of these case studies demonstrate that the proposed risk-averse SP approach can avoid over-optimistic solutions, the obtained P2P trading strategies are immune to uncertainty and P2P trading behaviors among MMGs can help reduce network losses of DNs. In addition, comparisons with other distributed algorithms verify the high performance of the proposed fully distributed method.

Distributed bi-level optimal power flow calculation for SVSM interval of integrated transmission and distribution networks considering the uncertain renewables

To accurately and efficiently calculate the static voltage stability margin (SVSM) interval of integrated transmission and distribution networks (ITDNs) considering the uncertain fluctuation of renewables, a distributed bi-level optimal power flow (OPF) calculation method based on the alternating direction method of multipliers for the SVSM interval of ITDNs is proposed in this paper. Due to the different structure characteristics of transmission network (TN) and distribution network (DN), a hybrid node-branch OPF method for calculating the SVSM of ITDNs is proposed, which has better convergence than the traditional node OPF method. Different reactive power-voltage control modes of renewable energy stations (RESs) are also considered in the calculation. Considering the uncertain fluctuation of RESs, two distributed bi-level OPF models for calculating the upper and lower bounds of the SVSM interval of ITDNs are established. Based on the alternating direction method of multipliers, the SVSM interval bounds of ITDNs can be obtained by solving the sub-optimization models and transmitting the boundary coupling variables and the load growth parameter between the TN and multiple DNs iteratively until the given convergence criterion is satisfied. Test results on two ITDNs, the modified IEEE 39–33 & 69 bus system and an actual 2285-bus ITDNs, demonstrate the high computation accuracy and efficiency of the proposed distributed bi-level OPF method compared with other calculation methods.

Intra-day optimal power flow considering flexible workload scheduling of IDCs

The online interactive workloads and offline batch computing services of Internet data centers (IDCs) have scheduling flexibility. Batch computing tasks with low latency sensitivity and priority can adjust the processing time and provide flexibility temporally. In contrast, interactive workloads with high latency sensitivity and priority can change the physical server for processing and provide flexibility spatially. Considering the change of power supply cost with time and space, the load flexibility of IDCs brought by Internet workload scheduling can be utilized to respond to the power system for more economical operation. To take full advantage of the flexibility of IDCs’ load demand, IDCs coordinated by an Internet service company (ISC) as a special controllable load with both temporal and spatial flexibility potential are considered. Instead of operating separately in normal optimal power flow (OPF) models, a two-stage interactive framework of the distribution system operator (DSO) and ISC is established. The operating state of the ISC is considered in OPF and is introduced as regularization terms into the objective function, thus fully exploiting its load flexibility. Some operation constraints are also added to guarantee the quality of service (QoS) of the ISC. The effectiveness of the model and the interactive framework are evaluated in a typical 33-bus medium-voltage distribution network. The results validate the further reduction of the power grid operation cost compared with the traditional OPF operation.

POWER SYSTEM UNDER FREE MARKET CONDITIONS

During the past 20 years, we have seen an expanding role for the free market in the electricity sector as well as other sectors. A perfectly competitive power market is efficient but difficult to achieve. The special features of electricity should be taken into account. In this paper, it is shown how traditional Optimal Power Flow (OPF) can be extended to include the demand side, thereby reflecting the supply-demand balance of an open market. The OPF technique can be used for the calculation of a set of nodal prices representing the optimal price at each node in the grid, from which we can derive a set of optimal transmission prices. Depending on the market design, generation planning can be performed by central optimization or in a decentralized manner by each generation company. For an individual generating company, the planning criterion will change from cost minimization to profit maximization. Practical experiences are now available from various parts of the world. It is clearly demonstrated that a power market can be operated under free market conditions. Various solutions concerning ownership and market design have been implemented. However, experiences have been mixed. In the future, it will be important to draw upon the lessons that have been learned.

Many-objective optimal power flow problems based on distributed power flow calculations for hierarchical partition-managed power systems

It is more and more challenging to obtain the global optimal power flow (OPF) information of a whole system with the developing trend of power systems toward hierarchical partition-managed ones. Traditional centralized OPF problems are not adaptive to modern partition-managed power systems because of the emergence of distributed sources and multi-stakeholders and the protection of their private data. This paper extends the traditional OPF to a many-objective optimization problem which is helpful to the independent power flow calculation (PFC) of the various subregions in partition-managed networks. To address the coordination of various regions, the individual distributed PFC problem is transformed into a whole optimization problem through node tearing and line disconnection methods, and then the pattern search algorithm is employed to address its solution. Then, the distributed PFC is integrated into the improved NSGA-III (I-NSGA-III) approach to solve the entire many-objective OPF (Ma-OPF) for partition-managed power systems. The effectiveness and feasibility of the proposed distributed approach are demonstrated in multiple benchmarks of IEEE 30/39/118-bus systems and a practical partitioned power system in Guizhou Province of China. The results show that the average voltage difference between distributed PFC and centralized PFC is within 1.12% while the phase angle difference is within 0.44%. In the application of OPF problems, the proposed distributed approach shows the high ability to yield approximate solutions of the original I-NSGA-III with centralized PFC. It is also found from the simulation that the distributed approach shows competitive performance compared to other considered centralized PFC-based evolutionary algorithms and weighted interior point method while it does not require complete information about various regions and is beneficial to the privacy protection of multi-stakeholders.

Large Scale Multi-Period Optimal Power Flow With Energy Storage Systems Using Differential Dynamic Programming

With theincreased penetration of renewable sources, power gridsare becoming stressed due to fluctuating generation. To alleviate stress from inconsistent sources, utilities employ energy storage systems alongside renewable sources and rely on dispatching synchronous generators. However, optimal dispatch of such devices is limited by traditional AC optimal power flow methods that do not account for time-dependent constraints, such as the state of charge of energy storage systems and generator ramping constraints. Multi-period optimal power flow is proposed as a large non-convex non-linear problem to optimally dispatch and control generators and energy storage elements across multiple time periods. In this paper, we introduce a scalable, robust framework to solve multi-period optimal power flow using a differential dynamic programming scheme that makes it capable of scaling to large systems containing energy storage devices. We demonstrate the efficacy of this solution by optimizing the SyntheticUSA testcase over a set of time periods. A robust homotopy method is applied to achieve fast simulation times that can be parallelized for further improvements.

Multi-Objective Optimization Framework for Optimal Power Flow Problem of Hybrid Power Systems Considering Security Constraints

The hybrid model of the power network infrastructure is an essential part of the sophisticated technology of the electrical network. For the traditional Optimal Power Flow (OPF) problem, thermal generators are typically employed to generate electricity. In this case, the amount of fuel required to produce power is constrained, and emissions from the network are frequently neglected. Renewable Energy Sources (RESs) have received increasing attention due to various potential characteristics such as cleanliness, diversity, and renewability. As an outcome, RESs are being integrated into the electrical grid at an increasing rate. The study in this paper proposes a techno-monetary investigation into the single- and multi-objective OPF, coordinating with RESs like wind, PhotoVoltaic (PV), and small hydropower units with hybrid PV. Moreover, the probability density functions of Weibull, Lognormal, and Gumble have been used independently to predict the required power in this paper. This paper considers an equilibrium optimizer for handling the OPF problem. The superior performance of the equilibrium optimizer is further verified with the results of both single- and multi-objective through comparative analysis with state-of-the-art counterparts, and the indications are that the proposed method can find better optimal solutions, use a lower number of generated solutions, and faster convergence with well distributed optimal Pareto front for multi-objective problems. The results are verified by employing a modified IEEE-30 bus hybrid power network, and performance comparisons are made among several well-established algorithms. Simulation findings show that the suggested strategy can achieve a reasonable compromise solution for different objectives.

Multiobjective optimal power flow for static voltage stability margin improvement

Worldwide, utilities are aiming to increase the stability of modern power systems during system disturbances. Optimizing generation scheduling can improve system security in contingency and stressed conditions while lowering losses and generation costs. An efficient operating strategy for maintaining power system stability is proposed in this work. The paper focuses on incorporating a Voltage Collapse Proximity Index (VCPI) in the traditional optimal power flow problem for multiobjective optimization (MO). Different case studies are assessed to evaluate the impact on the control variables. A Preference Selection index (PSI) is utilized to determine the best-case study for optimal system operation. The effectiveness of the proposed approach is tested on standard IEEE 30-bus and IEEE 57-bus during normal, contingency, and stressed conditions using MATPOWER. During normal conditions, the MO voltage stability constrained optimal power flow (VSC-OPF) increases the system stability by 28.13 % higher than the single objective (SO) case. Furthermore, the transmission losses are lowered by 14.69% with the proposed MO approach. During line outage contingency conditions, the voltage stability enhancement and loss reduction are higher in the MO than in the SO case by 13.60% and 23.19%. However, the loss minimization and stability improvement during normal and contingency conditions come at a slightly higher generation cost of 5.05% in both systems. On the other hand, during stressed conditions, the SO performs better in voltage stability improvement (by 8.77%) and loss reduction (by 6.97%) than in the MO voltage stability constrained OPF. Additionally, PV Curve analysis for the two systems indicates that voltage stability in MO OPF problems provides a more significant margin enhancement of 9.00%, 118.95% in normal and contingency, respectively, higher than the SO case. However, the SO case increases the load margin by 12.36% more than the MO case in stressed conditions. Consequently, the PSI ranks the multiobjective optimization of the three objectives as the most optimal way for operating the systems in normal and line outage contingency conditions. However, during increased load conditions, the system performance is better if a singular objective function is considered. This is due to the lack of adequate reactive power generation during stressed conditions, and hence a singular objective focus is sufficient to assure system stability. Therefore, the proposed approach is an effective preventive control measure for possible voltage collapse in typical power systems. The resulting improvement also brings about a sufficient system stability margin, causing the system to become more secure.

Solution approach for optimal power flow considering wind turbine and environmental emissions

The power grid is changing expeditiously with the increasing penetration of renewable energy sources (RES). Optimal utilization of RES reduces the burden on the primary grid and makes the grid more resilient. Traditional optimal power flow (OPF) is a complex problem in power management systems, and the complexity further increases with the integration of RES due to their intermittency. This paper presents the complete formulation of the OPF model incorporating wind turbines (WT) and environmental emissions for proper scheduling, planning, and efficient operation of thermal generating units (TGU) using the Ant Lion Optimization (ALO) algorithm. The formulation of the OPF problem comprises forecasted active power generation of WT, depending on the real-time measurement and probabilistic wind speed models. The results are analyzed from the perspective of operating cost, voltage profile, and transmission power losses in the system. The OPF approach and the solution methodology are tested on the IEEE 30 and IEEE 57-bus systems. The effectiveness of the proposed ALO algorithm is evaluated against well-established algorithms like Particle Swarm Optimization and Teaching-learning-based optimization. The comparison emphasizes the effectiveness of the ALO approach for solving various OPF problems with complex and non-smooth objective functions.

Economic dispatch of power systems with LMP-dependent demands: A non-iterative MILP model

The proliferation of demand response programs in the smart grid provides the system operator unique opportunities to reduce the load peak and alleviate network congestions. This paper considers the economic dispatch problem with elastic demands which flexibly respond to the locational marginal prices (LMPs). However, LMP is the dual variable of optimal power flow (OPF) problem and thus is unknown before the OPF problem is solved. Without LMP, the elastic demand is unclear, and the OPF problem cannot be set up. Given this interactive nature, it is difficult to acquire the dispatch strategy as well as the LMP according to the traditional OPF method. This paper thoroughly addresses this problem. Specifically, the limitation of the traditional LMP scheme in the described situation is analyzed. An equilibrium solution may not exist because the demand function and the discontinuous LMP may not have an intersection. To overcome this difficulty, LMP at the discontinuity point is redefined, so that the dispatch problem always has an equilibrium solution. A mixed-integer linear programming model for the economic dispatch problem with LMP-dependent load is proposed, and the equilibrium solution simultaneously offers the dispatch strategy and LMPs. Case studies demonstrate the difficulties of traditional approaches and the effectiveness of the proposed method.

A Progressive Period Optimal Power Flow for Systems with High Penetration of Variable Renewable Energy Sources

Renewable energy sources including wind farms and solar sites, have been rapidly integrated within power systems for economic and environmental reasons. Unfortunately, many renewable energy sources suffer from variability and uncertainty, which may jeopardize security and stability of the power system. To face this challenge, it is necessary to develop new methods to manage increasing supply-side uncertainty within operational strategies. In modern power system operations, the optimal power flow (OPF) is essential to all stages of the system operational horizon; underlying both day-ahead scheduling and real-time dispatch decisions. The dispatch levels determined are then implemented for the duration of the dispatch interval, with the expectation that frequency response and balancing reserves are sufficient to manage intra-interval deviations. To achieve more accurate generation schedules and better reliability with increasing renewable resources, the OPF must be solved faster and with better accuracy within continuous time intervals, in both day-ahead scheduling and real-time dispatch. To this end, we formulate a multi-period dispatch framework, that is, progressive period optimal power flow (PPOPF), which builds on an interval optimal power flow (IOPF), which leverages median and endpoints on the interval to develop coherent coordinations between day-ahead and real-time period optimal power flow (POPF). Simulation case studies on a practical PEGASE 13,659-bus transmission system in Europe have demonstrated implementation of the proposed PPOPF within multi-stage power system operations, resulting in zero dispatch error and violation compared with traditional OPF.

Surrogate‐assisted optimal re‐dispatch control for risk‐aware regulation of dynamic total transfer capability

To enable reliable power delivery through transmission tie-lines, total transfer capability (TTC) must be calculated and regulated to accommodate the transferred amount. However, the traditional optimal power flow (OPF)-based total transfer capability calculation is computationally expensive for efficient total transfer capability control due to the inclusion of a large set of differential-algebraic equations (DAEs) to verify transient stability constraints. In order to enable practicable total transfer capability regulation, a novel risk-aware deep learning-assisted paradigm is proposed here. First, a deep belief network (DBN) is employed to establish the total transfer capability predictor and surrogate the computation-intensive differential-algebraic equations in original optimal power flow formulas, simplifying the high-dimensional and intractable constraints deep belief networks without loss of nonlinearity. Particularly, in order to be aware of control risk from the predictive error of the deep belief networks, prediction intervals (PIs) are produced improved by using ensemble learning and used to disclose the probability of insufficient actions, further guaranteeing the sufficient and cost-effective control by compromising the tradeoff between cost and risk. Symbiotic organisms search (SOS) is then applied to solve the proposed risk-aware deep belief network-assisted total transfer capability control problem globally. The numerical studies testify that the proposed method enables economical, reliable, and full nonlinearity-retained dynamic total transfer capability regulation control within a risk-free surrogate-assisted and tractable physical model-driven hybrid framework.

Distributed Optimization for Distribution Grids With Stochastic DER Using Multi-Agent Deep Reinforcement Learning

This article develops a special decomposition methodology for the traditional optimal power flow which facilitates optimal integration of stochastic distributed energy resources in power distribution systems. The resulting distributed optimal power flow algorithm reduces the computational complexity of the conventional linear programming approach while avoiding the challenges associated with the stochastic nature of the energy resources and loads. It does so using machine learning algorithms employed for two crucial tasks. First, two proposed algorithms, Dynamic Distributed Multi-Microgrid and Monte Carlo Tree Search based Reinforcement Learning, constitute dynamic microgrids of network nodes to confirm the electric power transaction optimality. Second, the optimal distributed energy resources are obtained by the proposed deep reinforcement learning method named Multi Leader-Follower Actors under Centralized Critic. It accelerates conventional linear programming approach by considering a reduced set of resources and their constraints. The proposed method is demonstrated through a real-time balancing electricity market constructed over the IEEE 123-bus system and enhanced using price signals based on distribution locational marginal prices. This application clearly shows the ability of the new approach to effectively coordinate multiple distribution system entities while maintaining system security constraints.

Integrating Dynamic Line Rating in Security Constrained Optimal Power Flow Considering Wind Power Generation. (Dept. E )

Traditional optimal power flow (OPF) problem is solved considering static line rating (SLR) of the transmission lines which are constant values of power flow limits. This led to underutilization of the network and higher locational marginal prices (LMPs). Dynamic line rating (DLR) is one of the active solutions to enhance ampacity for overhead transmission lines (OHTL), especially with the penetration of high wind power without investing in an additional transmission line network. This paper investigates the effect of integrating the DLR on power dispatch with uncertainty levels due to renewable energy sources (RES) generation. The model propose is a multi-period security-constrained OPF based on particle swarm optimization (SCOPF/PSO) that takes into consideration the steady-state heat-balance equation (SS-HBE) of the OHTL and the ramp-up time of generation units. The problem is a multi-objective optimization one; the main objective is to maximize the social welfare (SW) by minimizing total system operation cost and maximizing the revenue from the energy consumers, whereas the second objective is to minimize the thermal emissions. To demonstrate the effectiveness of the proposed model, a case study is applied to the modified IEEE 30-bus test system. The results expound on the effectiveness of the proposed approach.

Quadratically Constrained Quadratic Programming Formulation of Contingency Constrained Optimal Power Flow with Photovoltaic Generation

Contingency Constrained Optimal Power Flow (CCOPF) differs from traditional Optimal Power Flow (OPF) because its generation dispatch is planned to work with state variables between constraint limits, considering a specific contingency. When it is not desired to have changes in the power dispatch after the contingency occurs, the CCOPF is studied with a preventive perspective, whereas when the contingency occurs and the power dispatch needs to change to operate the system between limits in the post-contingency state, the problem is studied with a corrective perspective. As current power system software tools mainly focus on the traditional OPF problem, having the means to solve CCOPF will benefit power systems planning and operation. This paper presents a Quadratically Constrained Quadratic Programming (QCQP) formulation built within the matpower environment as a solution strategy to the preventive CCOPF. Moreover, an extended OPF model that forces the network to meet all constraints under contingency is proposed as a strategy to find the power dispatch solution for the corrective CCOPF. Validation is made on the IEEE 14-bus test system including photovoltaic generation in one simulation case. It was found that in the QCQP formulation, the power dispatch calculated barely differs in both pre- and post-contingency scenarios while in the OPF extended power network, node voltage values in both pre- and post-contingency scenarios are equal in spite of having different power dispatch for each scenario. This suggests that both the QCQP and the extended OPF formulations proposed, could be implemented in power system software tools in order to solve CCOPF problems from a preventive or corrective perspective.

Preventive Security-Constrained Optimal Power Flow with Probabilistic Guarantees

The traditional security-constrained optimal power flow (SCOPF) model under the classical N-1 criterion is implemented in the power industry to ensure the secure operation of a power system. However, with increasing uncertainties from renewable energy sources (RES) and loads, the existing SCOPF model has difficulty meeting the practical requirements of the industry. This paper proposed a novel chance-constrained preventive SCOPF model that considers the uncertainty of power injections, including RES and load, and contingency probability. The chance constraint is used to constrain the overall line flow within the limits with high probabilistic guarantees and to significantly reduce the constraint scales. The cumulant and Johnson systems were combined to accurately approximate the cumulative distribution functions, which is important in solving chance-constrained optimization problems. The simulation results show that the model proposed in this paper can achieve better performance than traditional SCOPF.

Distributed multi-period three-phase optimal power flow using temporal neighbors

The penetration of distributed generation in medium (MV) and low (LV) voltage distribution grids has been steadily increasing every year in multiple countries, thus creating new technical challenges in grid operation and motivating developments in distributed optimization for flexibility management. The traditional centralized optimal power flow (OPF) algorithm can solve technical constraints violation. However, computational efficiency, new technologies (e.g., edge computing) and control architectures (e.g., web-of-cells) are demanding for distributed approaches. This work formulates a novel distributed multi-period OPF for three-phase unbalanced grids that is essential when integrating energy storage units in operational planning (e.g., day-ahead) of LV or local energy community grids. The decentralized constrained optimization problem is solved with the alternating direction method of multipliers (ADMM) adapted for unbalanced LV grids and multi-period optimization problems. A 33-bus LV distribution grid is used as a case-study in order to define optimal battery storage scheduling along a finite time horizon that minimizes overall grid operational costs, while complying with technical constraints of the grid (e.g., voltage and current limits) and battery state-of-charge constraints.