Multi-period Optimal Power Flow Research Articles

A Fully-Decentralized Transactive Energy Management Under Distribution Network Constraints via Peer-to-Peer Energy Trading

Peer-to-peer (P2P) energy trading is seen as a promising method for distributed transactive energy (TE) management. Since distribution network is used for electricity transfer, physical constraints should be imposed during the P2P trading. In this paper, we propose a P2P-based distributed TE management approach considering distribution network constraints and product differentiation for P2P energy trading between different buses. A multi-period AC optimal power flow (OPF) model is used to monitor power flow in distribution system, and the exactness of second-order cone relaxation is recovered by the convex-concave procedure, making sure that P2P transactions do not violate network constraints. Moreover, we propose a fully-decentralized variant of alternating direction method of multipliers for clearing P2P transactions in a fully-decentralized manner under AC power flow constraints. Our proposed variant does not need any interactions with the central supervisor in comparison with the traditional one. Finally, numerical results on IEEE 33-bus and 141-bus distribution systems verify the effectiveness of our proposed method in ensuring physical feasibility, lowering data communication and protecting data privacy during P2P energy trading.

A Tractable Linearization-Based Approximated Solution Methodology to Stochastic Multi-Period AC Security-Constrained Optimal Power Flow

The growing presence of renewable energy sources (RES), energy storage systems (ESSs) and flexible loads (FLs) in power systems necessitates a new approach to N-1 security in day-ahead operation planning considering stochasticity and time coupling. To this end, a comprehensive approach based on stochastic multi-period AC security-constrained optimal power flow (S-MP-AC-SCOPF) was proposed recently in a form of a large-scale non-linear programming problem, which is not scalable. This paper proposes the first-time new tractable solution methodology for the most complete AC-SCOPF problem to date: S-MP-AC-SCOPF. As other main novelty, the proposed methodology achieves tractability by solving sequentially a limited number of different linear approximations of the full S-MP-AC-SCOPF problem. These linear approximations differ in terms of: state dependency regarding losses approximation, carefully reduced sets of constraints or tightening of critical constraints. The performance of the methodology is demonstrated in power system models of Sweden and Portugal. Extensive numerical experiments have shown that the methodology is able to reduce progressively until complete removal the number of violated thermal or voltage constraints. It leads to reasonably accurate solution with small optimality gap within few iterations and is at least 85% faster than the competitor IPOPT solver.

A nonlinear multi-period hydrothermal optimal power flow model for hydropower systems

Despite their social and climate benefits, renewable energy sources also present challenges such as intermittency, volatility and randomness, which may considerably reduce security and reliability of the system. Hydro units have the ability to provide flexible and rapid energy reserves, serving as an effective way of handling these challenges. Yet, many Hydrothermal Optimal Power Flow (HOPF) models in the literature often provide a simplified and sometimes inaccurate representation of the hydraulic aspects. In most HPOF models proposed in the literature, the representation of hydraulic aspects disregards important physical and operational hydro constraints. Such simplifications can curtail the potential benefits of hydro generation, especially concerning system security and reliability. In this paper, we investigate the impact of hydro constraints in the operation of hydrothermal systems by means of HOPF models. For such a purpose, we propose a nonlinear multi-period active–reactive HOPF model where hydro constraints are represented in detail, by means of a nonlinear framework. This model presents innovative formulations for hydro units, specifically designed to address pivotal non-differentiable constraints. The proposed multi-period HOPF model is used as a benchmark to assess inaccuracies present in linear or linearized HOPF model variants. The results using the IEEE 24-bus and 300-bus systems, adapted so they become hydropower systems, clearly identify significant impact of some hydro constraints which have been neglected in most models described in the literature, in the hydrothermal operational policy and scheduling.

Multi-Period Optimal Power Flow with Photovoltaic Generation Considering Optimized Power Factor Control

This paper presents a Multi-Period Optimal Power Flow (MOPF) modeling applied to the minimization of energy losses in Distribution Networks (DNs) considering the reactive power control of Photovoltaic Generation (PVG) that can be applied to both short-term and long-term operation planning. Depending on the PV Power Factor (PVpf) limitations, PVG may provide both active and reactive power. The optimal power factor control on the buses with PVG contributes to an economical and safe operation, minimizing losses and improving the voltage profile of the DN. The proposed MOPF was modeled in order to minimize active energy losses subject to grid constraints and PVpf limitations. The variations of loads and PVG were discretized hour by hour, composing a time horizon of 24 h for day-ahead planning; nonetheless, the methodology can be applied to any other time period, such as a month, year, etc., by simply having generation and load forecasts. To demonstrate the effectiveness and applicability of the proposed approach, various tests were carried out on 33-bus and 69-bus distribution test systems. The analyses considered the DN operating with PVG in four different cases: (a) PVpf fixed at 1.0; (b) PVpf fixed at 0.9 capacitive; (c) hourly PVpf optimization; and (d) optimization of PVpf for a single value. The results show that a single optimal adjustment of PVpf minimizes losses, improves voltage profile, and promotes safe operation, avoiding multiple PVpf adjustments during the operating time horizon. The algorithm is extremely fast, taking around 2 s to reach a solution.

Applied Power Wheeling Concept Considering Site-Specific and Variability of VRE under Contingency Events

Unlike non-variable renewable energy (VRE) generators, VRE generators have site-specific, various, and uncertain characteristics that require different approaches for power wheeling cost calculations. The conventional power wheeling method cannot be used to accommodate the variability and site specificity of VRE. This study aims to develop a power wheeling calculation method that accommodates the variability and site specificity of VRE by comparing the postage stamp and MW mile methods in a multi-period optimal power flow simulation. The impact of line contingencies on the total power wheeling cost is also assessed. The simulation uses a modified IEEE 39 bus system as the test network, with three study cases centered on assessing the impacts of variability, site specificity, and line contingency. Based on the simulation, it is understood that the pivotal power wheeling cost is the line used by power wheeling actors, as there are cheaper and more expensive lines. Therefore, the total power wheeling cost is less affected by variability and more affected by the site specificity of VRE. On the other hand, because of the meshed network being employed as the test case, line contingency results in a lower cost for power wheeling actors to use fewer and cheaper lines.

A Multi-Period Charging Service Pricing Game for Public Charging Network Operators Considering the Dynamics of Coupled Traffic-Power Systems

The proliferation of electric vehicles (EVs) couples the operations of traffic systems and power systems, necessitating the interdependent traffic-power modeling to optimize the on-road EV charging decisions. In this paper, the multi-period charging service pricing interactions between multiple charging network operators (CNOs) are discussed considering the dynamics of traffic-power systems. A multi-period user equilibrium model considering mixed vehicle types is formulated to describe the possible vehicle trip transitions between different periods in urban transportation network. For the power distribution network, a convexified multi-period AC optimal power flow model is adopted for local electricity market clearing. With the dynamics of traffic-power systems, the pricing interaction between multiple CNOs is a non-cooperative game. We analyze the existence of Nash Equilibrium with fixed-point theory, and propose an iterative method based on the best response strategy to find <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon$</tex-math> </inline-formula> -Nash Equilibrium. Numerical studies based on an interdependent power-traffic system consisting of two 18-node power distribution networks and one revised Nguyen-Dupuis transportation network demonstrate that with the proposed CNOs’ charging pricing strategy, the CNOs’ service profits can increase by 1%-3% while the PDNs’ operation costs decrease about 1%-2% for the interdependent traffic-power system.

A DSO-based day-ahead market mechanism for optimal operational planning of active distribution network

Grid-edge distributed energy resources (DERs) present both operational challenges and opportunities to distribution system operators (DSO). This paper presents a day-ahead scheduling strategy for DER-rich active distribution networks to leverage operational flexibilities of various DERs for effective network management. The scheduling strategy is realized based on a DSO-centric day-ahead market model that incorporates a multi-period and two-stage optimal power flow (OPF) model to determine scheduling and pricing for network and non-network DERs. The uncertainties in day-ahead operation stemming from local generation, load consumption, and market prices are robustly considered for hedging the intermittency and forecasting uncertainties. A novel bid price function is proposed for local distributed solar PV generation aiming at maximizing participation in network service. A new reactive power management and pricing scheme is also presented for optimal scheduling of voltage support devices within distribution network. The proposed framework is implemented on IEEE-33 bus balanced test system and operational planning strategy is analyzed for overall cost-benefits of the system at various uncertainty levels. A 7-10 percent saving in cost of procurement for DSO can be accrued at various uncertainty levels, and a 10 percent increase in revenue for the PV generation can be observed.

Robust flexibility needs assessment with bid matching framework for distribution network operators

For market-based procurement of low-voltage flexibility, DSOs identify the amount of flexibility needed for resolving probable distribution network (DN) voltage and thermal congestion. A framework is required to avoid over or under-procurement of flexibility in the presence of uncertainty. To this end, we propose a scenario-based robust chance-constrained (CC) day-ahead flexibility needs assessment (FNA) framework. The CC level is analogous to the risk DSO is willing to take in flexibility planning. Multi-period optimal power flow is performed to calculate the amount of flexibility needed to avoid network issues. Future uncertainties are considered as multiple scenarios, which are utilized to solve the flexibility needs assessment optimal power flow (FNA-OPF) problem. The FNA tool calculates the temporal and locational ramp-up and ramp-down flexibility needs of the DN. We also propose a Pareto optimal mechanism for selecting CC level to reduce flexibility needs while reducing DN congestion. Zonal clustering of a DN feeder is performed using electrical distance as a measure, with spatial partitioning and silhouette coefficient. Further, a zonal and nodal bid matching algorithm is proposed, where alternate flexibility bids in the same zone are identified in case nodal matching is not exact, compared to the calculated FNA. The proposed bid matching algorithm assists the system operator in bid selection by avoiding rerunning power flow evaluations. The zonal and nodal bid matching leads to more than 43% additional reduction of probable network incidents compared to naive nodal bid matching. Furthermore, we also calculate FNA for a real German DN with 646 nodes and 331 consumers.

The promise of EV-aware multi-period optimal power flow problem: Cost and emission benefits

Increased electric vehicle (EV) penetration brings considerable challenges to the daily planning of the power grid operations. A careful coordination of the grid operations and charging schedules is needed to alleviate these challenges, and turn them into opportunities. For this purpose, we study the Multi-Period Optimal Power Flow problem (MOPF) with electric vehicles under emission considerations. We integrate three different real-world datasets: household electricity consumption, marginal emission factors, and EV driving profiles. We present a systematic solution approach based on second-order cone programming to find globally optimal solutions for the resulting nonconvex optimization problem. To the best of our knowledge, our paper is the first to propose such a comprehensive model integrating multiple real datasets and a promising solution method for the EV-aware MOPF Problem. Our computational experiments on various instances with up to 2000 buses demonstrate that our solution approach leads to high-quality feasible solutions with provably small optimality gaps. In addition, we show the importance of coordinated EV charging to achieve significant emission savings and reductions in cost. In turn, our findings can provide quantitative insights to decision-makers on how to incentivize EV drivers depending on the trade-off between cost and emission.

Efficient Reallocation of BESS in Monopolar DC Networks for Annual Operating Costs Minimization: A Combinatorial-Convex Approach

This article deals with the solution of a mixed-integer nonlinear programming (MINLP) problem related to the efficient reallocation of battery energy storage systems (BESS) in monopolar direct current (DC) grids through a master–slave optimization approach. The master stage solves the integer nature of the MINLP model, which is related to the nodes where the BESS will be located. In this stage, the discrete version of the vortex search algorithm is implemented. To determine the objective function value, a recursive convex approximation is implemented to solve the nonlinear component of the MINLP model (multi-period optimal power flow problem) in the slave stage. Two objective functions are considered performance indicators regarding the efficient reallocation of BESS in monopolar DC systems. The first objective function corresponds to the expected costs of the annual energy losses, and the second is associated with the annual expected energy generation costs. Numerical results for the DC version of the IEEE 33 bus grid confirm the effectiveness and robustness of the proposed master–slave optimization approach in comparison with the solution of the exact MINLP model in the General Algebraic Modeling System (GAMS) software. The proposed master–slave optimizer was programmed in the MATLAB software. The recursive convex solution of the multi-period optimal power flow problem was implemented in the convex discipline tool (CVX) with the SDPT3 and SEDUMI solvers. The numerical reductions achieved with respect to the benchmark case in terms of energy loss costs and energy purchasing costs were 7.2091% and 3.2105%, which surpassed the results reached by the GAMS software, with reductions of about 6.0316% and 1.5736%.

Analytical uncertainty propagation for multi-period stochastic optimal power flow

The increase in renewable energy sources (RESs), like wind or solar power, results in growinguncertainty also in transmission grids. This affects grid stability through fluctuating energy supplyand an increased probability of overloaded lines. One key strategy to cope with this uncertainty isthe use of distributed energy storage systems (ESSs). In order to securely operate power systemscontaining renewables and use storage, optimization models are needed that both handle uncertaintyand apply ESSs. This paper introduces a compact dynamic stochastic chance-constrained DC optimalpower flow (CC-OPF) model, that minimizes generation costs and includes distributed ESSs. AssumingGaussian uncertainty, we use affine policies to obtain a tractable, analytically exact reformulation asa second-order cone problem (SOCP). We test the new model on five different IEEE networks withvarying sizes of 5, 39, 57, 118 and 300 nodes and include complexity analysis. The results showthat the model is computationally efficient and robust with respect to constraint violation risk. Thedistributed energy storage system leads to more stable operation with flattened generation profiles.Storage absorbed RES uncertainty, and reduced generation cost.

A Mathematical Programming Approach for the Optimal Operation of Storage Systems, Photovoltaic and Wind Power Generation

The ever-growing participation of Renewable Energy Sources (RES) in modern distribution networks is replacing an important portion of Conventional Generation (CG), which brings along new challenges in the planning and operation of distribution grids. As RES such as Photovoltaic Energy (PV) and Wind Power Generation (WPG) increase in distribution networks, studies regarding their integration and coordination become more important. In this context, the purpose of this paper is to propose a Multi-period Optimal Power Flow (MOPF) model for the optimal coordination of Battery Energy Storage Systems (BESSs) with PV, WPG, and CG in modern distribution networks. The model formulation was developed in A Modeling Language for Mathematical Programming (AMPL) and solved through the Knitro solver within a time horizon of 24 h. A distinctive feature and one of the main contributions of the proposed approach is the fact that BESSs can provide both active and reactive power. The proposed optimization model reduces power losses and improves voltage profiles. To show the applicability and effectiveness of the proposed model, several tests were carried out on the 33-bus distribution test system and a real distribution system of 141 buses located in the metropolitan area of Caracas. Power loss reductions of up to 58.4% and 77% for the test systems of 33 and 141 buses were obtained, respectively, when BESSs provided both active and reactive power. The results allow us to conclude that the proposed model for optimal coordination of BESSs with RES is suitable for real-life applications, resulting in important reductions of power losses and flattening of voltage profiles.

Optimal Power Dispatch of PV Generators in AC Distribution Networks by Considering Solar, Environmental, and Power Demand Conditions from Colombia

This paper deals with the problem regarding the optimal operation of photovoltaic (PV) generation sources in AC distribution networks with a single-phase structure, taking into consideration different objective functions. The problem is formulated as a multi-period optimal power flow applied to AC distribution grids, which generates a nonlinear programming (NLP) model with a non-convex structure. Three different objective functions are considered in the optimization model, each optimized using a single-objective function approach. These objective functions are (i) an operating costs function composed of the energy purchasing costs at the substation bus, added with the PV maintenance costs; (ii) the costs of energy losses; and (iii) the total CO2 emissions at the substation bus. All these functions are minimized while considering a frame of operation of 24 h, i.e., in a day-ahead operation environment. To solve the NLP model representing the studied problem, the General Algebraic Modeling System (GAMS) and its SNOPT solver are used. Two different test feeders are used for all the numerical validations, one of them adapted to the urban operation characteristics in the Metropolitan Area of Medellín, which is composed of 33 nodes, and the other one adapted to isolated rural operating conditions, which has 27 nodes and is located in the department of Chocó, Colombia (municipality of Capurganá). Numerical comparisons with multiple combinatorial optimization methods (particle swarm optimization, the continuous genetic algorithm, the Vortex Search algorithm, and the Ant Lion Optimizer) demonstrate the effectiveness of the GAMS software to reach the optimal day-ahead dispatch of all the PV sources in both distribution grids.

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.

A Data-Driven Optimization Method Considering Data Correlations for Optimal Power Flow Under Uncertainty

This paper introduces a data-driven optimization (DDO) method based on novel strategic sampling (SS) considering data correlations for multiperiod optimal power flow (OPF) considering energy storage devices under uncertainty (OPF-ESDUU) of uncertain renewable energy and power loads (UREPL). This DDO method depends only on the uncertainty samples to yield an optimal solution that satisfies a specific confidence level, which is effective because of two resounding learning algorithms: Bayesian hierarchical modeling (BHM) and determinantal point process (DPP). Considering both the local bus information and spatial correlations over all buses, BHM learns the convex approximation of AC power flow (CAACPF) more accurately than the existing learning methods, converting the originally non-convex OPF-ESDUU to a convex optimization problem. DPP considers the correlations between samples to find a small set of significant samples by measuring the relative weight of each sample using the random matrix theory, significantly decreasing the data samples required by the existing SS. The experimental analysis in IEEE test cases shows that after considering data correlations, 1) BHM learns CAACPF better with 13–90% accuracy improvement on average, compared with the existing learning methods, and 2) the proposed DDO performs more efficiently than the existing DDO as DPP-based SS boosts the sampling efficiency by 50% at least.

A novel two-stage TSO–DSO coordination approach for managing congestion and voltages

This paper proposes a novel approach to TSO–DSO coordination in day-ahead operation planning for the procurement of ancillary services (AS) such as congestion management and voltage control. The approach overcomes two major issues hindering practical applicability of existing TSO–DSO coordination mechanisms, namely the computational complexity due to the large number of times the TSO and DSO problems are resolved as well as the inability to map analytically the cost and amount of flexibility at DSO level into TSO problem. To this end, a two-fold novelty of our approach consists of the decomposition of the overall AS problem into two sequential stages, which seek the procurement of: (i) “active power related” AS (i.e., for congestion management) and then (ii) the “reactive power related” AS (i.e., for voltage support), respectively as well as a fast approximation of the cost of aggregated active and reactive power flexibilities of active distribution systems (ADSs). Unlike existing mechanisms, the proposed approach considers real-world challenging aspects such as the N-1 security at TSO level and operation uncertainties at both TSO and DSO levels. Accordingly, the approach relies on tailored versions of stochastic multi-period AC security-constrained optimal power flow (S-MP-SCOPF) at TSO level and stochastic multi-period AC optimal power flow (S-MP-OPF) at DSO level, which are formulated as non-linear programming (NLP) problems. The importance and performance of the proposed approach are illustrated on a power system connecting a 60-bus transmission system (under 33 N-1 contingencies) with five 34-bus ADSs.

Congestion mitigation in unbalanced residential networks with OPF-based demand management

This paper proposes a novel congestion mitigation strategy for low voltage residential feeders in which the rising power demand due to the electrification of the transport and heating systems leads to congestion problems. The strategy is based on requiring residential customers to limit their demand for a certain amount of time in exchange for economic benefits. The main novelty of the method consists of combining a thorough representation of the network physics with advanced constraints that ensure the comfort of residential users, in a scalable manner that suits real systems. The mitigation strategy is presented from a DSO perspective, and takes the form of contracts between users and system operator. The focus on user comfort aims to make the contracts appealing, encouraging users to voluntarily enroll in the proposed mitigation scheme. The presented solution is implemented as a mixed-integer multi-period optimal power flow problem which relies on a linearized three-phase power flow formulation. Calculations on 100 real-life distribution feeders are performed, to analyze the congestion-relieving potential of several possible system operator-user contracts. From a planning perspective, the results can help the system operator define contractual terms that make a specific congestion mitigation scheme effective and viable. From an operational perspective, the same calculations can be used to optimally schedule power reduction on a day-ahead basis.

BELTISTOS: A robust interior point method for large-scale optimal power flow problems

Optimal power flow (OPF) problems are ubiquitous for daily power grid operations and planning. These optimal control problems are nonlinear, non-convex, and computationally demanding for large power networks especially for OPF problems defined over a large number of time periods, which are commonly intertemporally coupled due to constraints associated with energy storage devices. A robust interior point optimization library BELTISTOS is proposed, which allows fast and accurate solutions to single-period OPF problems and significantly accelerates the solution of multi-period OPF problems via the aid of structure-exploiting algorithms. Adhering to high reporting standards for replicable and reliable analysis, BELTISTOS is compared with interior point optimizers within the software package MATPOWER and evaluated using large scale power networks with up to 193,000 buses and problems spanning up to 4800 time periods. • BELTISTOS optimization library provides structure-exploiting interior point methods. • BELTISTOS accelerates convergence of multi-period optimal power flow problems. • BELTISTOS provides algorithms with significantly lower memory footprint.

Hierarchical Flexible Operation Approach on a VSC-MTDC Interconnected Hybrid Grid With a High Share of Renewable Power

Voltage source converter (VSC) based multi -terminal HVDC (VSC-MTDC) systems have emerged as flexible collector systems for multi-area interconnected AC/DC hybrid grids with high share of volatile renewables energy. However, it remains an open question as how to exploit the flexible and cooperative regulation of multiple VSC stations and enhance secure operation on the premise of information privacy protection of multiple independent dispatch areas. This paper proposed an Analytical Target Cascading (ATC)-based three-layer decentralized optimization approach on flexible operation of VSC-MTDC hybrid systems. The low layer is a chance-constrained stochastic optimal power flow (SOPF) problem for AC grids. To guard against the secure operation risk by a VSC's random unipolar atresia fault, a modified loss of load probability index (MLOLP) is proposed to improve the day-ahead generation-reserve scheme. The middle layer and the high layer are a multiperiod OPF problem for VSC stations and the MTDC grid, respectively, where a detailed power regulation model is built to fully exert the flexible cooperation of multiple VSC stations. Two case systems are tested to verify the effectiveness and superiority of the proposed flexible operation hierarchy.

A systematic analysis of integrating variable wind power into Fujian power grid

To realize the carbon emission peak and carbon neutrality objectives in China, efforts should focus on the electricity system, with an emphasis on creating low-carbon and renewable energy sources. The improvement of installed capacities of variable wind power and photovoltaics in the power system affect the entire power system. With the relatively independent power system of Fujian Province as an example, this study comprehensively considers the startup/shutdown, ramp-up/down, and power generation costs of the generators. Furthermore, the study uses the multi-period direct-current optimal power flow with unit-commitment method to analyze the impacts of variable wind power on the outputs of generators in the power system. The results show that at the current wind power-capacity level, the power system can still achieve the optimal dispatch without wind curtailment, and the marginal cost of wind power is relatively low. With further improvement of the installed capacity of wind power, it is necessary to fully mobilize the flexibility of all generators and PSHs, which will raise the overall operating cost of the whole power grid system. Finally, the study concludes with some policy recommendations for lowering the systematic integration cost of variable renewables.