Linear Power Flow Research Articles

Multi-objective planning of distribution network based on distributionally robust model predictive control

The uncoordinated integration of numerous distributed resources poses significant challenges to the safe and stable operation of distribution networks. To address the uncertainties associated with the intermittent output of distributed power sources, we propose a multi-objective planning strategy for distribution networks based on distributionally robust model predictive control (MPC). Initially, an error fuzzy set is established on a Wasserstein sphere using historical data to enhance out-of-sample performance. Next, a multi-objective optimization framework is constructed, balancing returns and risks, and is subsequently converted into a single-objective solution using value-at-risk conditions. This is followed by the implementation of multi-step rolling optimization within the model predictive control framework. We have linearized the proposed model using the linearized power flow method and conducted a thorough validation on an enhanced IEEE 37-node test system. Distributionally robust optimization (DRO) has been shown to reduce costs by a significant 29.16% when compared to an RO method. Moreover, the energy storage capacity required has been notably reduced by 33.33% on the 29-node system and by 20% on the 35-node system. These quantified results not only demonstrate the substantial economic efficiency gains but also the enhanced robustness of our proposed planning under the uncertainties associated with renewable energy integration.

Synchronized States of Power Grids and Oscillator Networks by Convex Optimization

Synchronization is essential for the operation of ac power systems: all generators in the power grid must rotate with fixed relative phases to enable a steady flow of electric power. Understanding the conditions for and the limitations of synchronization is of utmost practical importance. In this article, we propose a novel approach to computing and analyzing the stable stationary states of a power grid or a network of Kuramoto oscillators in terms of a convex optimization problem. This approach allows us to systematically compute stable states where the phase difference across an edge does not exceed π/2. Furthermore, the optimization formulation allows us to rigorously establish certain properties of synchronized states and to bound the error in the widely used linear power flow approximation. Published by the American Physical Society 2024

Evaluation Method for Voltage Regulation Range of Medium-Voltage Substations Based on OLTC Pre-Dispatch

A new energy industry represented by photovoltaic and wind power has been developing rapidly in recent years, and its randomness and volatility will impact the stable operation of the power system. At present, it is proposed to enrich the regulation of the power grid by tapping the regulation potential of load-side resources. This paper evaluates the overall voltage regulation capability of substations under the premise of considering the impact on network voltage security and providing a theoretical basis for the participation of load-side resources of distribution networks in the regulation of the power grid. This paper proposes a Zbus linear power flow model based on Fixed-Point Power Iteration (FFPI) to enhance power flow analysis efficiency and resolve voltage sensitivity expression. Establishing the linear relationship between the voltages of PQ nodes, the voltage of the reference node, and the load power, this paper clarifies the impact of reactive power compensation devices and OLTC (on-load tap changer) tap changes on the voltages of various nodes along the feeder. It provides theoretical support for evaluating the voltage regulation range for substations. The day-ahead focus is on minimizing network losses by pre-optimizing OLTC tap positions, calculating the substation voltage regulation boundaries within the day, and simultaneously optimizing the total reactive power compensation across the entire network. By analyzing the calculated examples, it was found that a pre-scheduled OLTC (on-load tap changer) can effectively reduce network losses in the distribution grid. Compared with traditional methods, the voltage regulation range assessment method proposed in this paper can optimize the adjustment of reactive power compensation devices while ensuring the voltage safety of all nodes in the network.

A Stochastic Model Predictive Control Method for Tie-Line Power Smoothing under Uncertainty

With the high proportion of distributed energy resource (DER) access in the distributed network, the tie-line power should be controlled and smoothed to minimize power flow fluctuations due to the uncertainty of DER. In this paper, a stochastic model predictive control (SMPC) method is proposed for tie-line power smoothing using a novel data-driven linear power flow (LPF) model that enhances efficiency by updating parameters online instead of retraining. The scenario method is then employed to simplify the objective function and chance constraints. The stability of the proposed model is demonstrated theoretically, and the performance analysis indicates positive results. In the one-day case study, the mean relative error is only 1.1%, with upper and lower quartiles of 1.4% and 0.2%, respectively, which demonstrates the superiority of the proposed method.

Linear optimal power flow model for modern distribution systems: Management of normally closed loop grids and on-load tap changers

Modern distribution systems face new challenges due to the rise of small-scale renewable energy resources and the electrification of energy demand. Additionally, the complexity of network management increases with the exchange of flexibility services between distribution and higher voltage levels. The system operator's role involves optimizing the planning and operation of the distribution network. In current scenarios, this includes managing flexibility resources (such as generators, demand response, and on-load tap changers) and implementing new operating strategies (such as closed-loop topologies and dynamic reconfiguration). This paper proposes a reformulation of one of the most used linear approximations of the Optimal Power Flow model for distribution networks. The novelty of this reformulation lies in integrating equations to handle tap-changing transformers with closed-loop/meshed network operation, while preserving the original linear formulation and computational tractability.

Long solution times or low solution quality: On trade-offs in choosing a power flow formulation for the Optimal Power Shutoff problem

The Optimal Power Shutoff (OPS) problem is an optimization problem that makes power line de-energization decisions in order to reduce the risk of igniting a wildfire, while minimizing the load shed of customers. This problem, with DC linear power flow equations, has been used in many studies in recent years. However, using linear approximations for power flow when making decisions on the network topology is known to cause challenges with AC feasibility of the resulting network, as studied in the related contexts of optimal transmission switching or grid restoration planning. This paper explores the accuracy of the DC OPS formulation and the ability to recover an AC-feasible power flow solution after de-energization decisions are made. We also extend the OPS problem to include variants with the AC, Second-Order-Cone, and Network-Flow power flow equations, and compare them to the DC approximation with respect to solution quality and time. The results highlight that the DC approximation overestimates the amount of load that can be served, leading to poor de-energization decisions. The AC and SOC-based formulations are better, but prohibitively slow to solve for even modestly sized networks thus demonstrating the need for new solution methods with better trade-offs between computational time and solution quality.

Collaborative optimization of reference powers and droop coefficients in DC distribution networks considering flexibility enhancement

In droop-controlled DC distribution networks (DCDNs), the uncertainties of loads and renewable energy powers worsen the system performances. This paper proposes a collaborative optimization method of reference powers and droop coefficients to optimize more objectives that including reducing line losses, improving the voltage level, and enhancing the flexibility of DCDNs. Firstly, a fast and accurate flexibility margin evaluation method is presented for DCDNs, in which the allowable power variation intervals under operational constraints are used to quantify the flexibility margins. The analytic relations of allowable power variation intervals and operational constraints are established by the linear power flow model, and the flexibility margins can be calculated quickly and directly. Secondly, the bi-level collaborative optimization method of reference powers and droop coefficients is proposed to optimize more objectives simultaneously. The reference powers are optimized to improve the voltage level and reduce line power losses in the upper-level optimization, and the droop coefficients are optimized to enhance the system flexibility in the lower-level optimization. Finally, multiple test systems with different topologies are simulated in detail. The case study demonstrates that: 1) The flexibility evaluation method can quantify the margins of operational constraints quickly and accurately. Compared with the bisection method, the calculation time of proposed flexibility evaluation method reduce 99.52% with the error of 1.68%. 2) The proposed collaborative optimization of reference powers and droop coefficients can effectively reduce line losses and the risks of voltage off-limit and VSC overload of DCDN under uncertainties.

A coordinated voltage control scheme based on linearized power flow functions for non-optimal and optimal problems

This paper proposes a coordinated voltage control scheme for transmission systems based on linearized power flow functions to determine the set points of local control voltage devices. The scheme considers models and interactions between different devices to attain the appropriate performance (non-optimal or optimal) of the whole power system. The sensitivity analysis is used to formulate a partly linearized model that relates the voltage variations of buses to the reactive power variation of synchronous machines, settings of static VAr compensators (SVCs), shunt capacitors susceptance, and tap position of load tap changers (LTCs) to determine the set of the settings of the voltage control devices. Two formulations are proposed by considering the susceptance of SVCs or the slope and reference voltage of characteristics of SVCs as control variables. The developed formulation based on the linearized power flow functions provides the possibility of the practical implementation of the coordinated voltage control scheme, especially the optimal one, by reducing the model complexity and solving time. The performance of the proposed formulations is investigated by some studies of voltage control and an optimally coordinated problem on the IEEE 30-bus and IEEE 118-bus test systems.

An online updated linear power flow model based on regression learning

AbstractThe linear power flow (LPF) model is widely used in the optimization, operation, and control of distribution networks. These applications require the LPF model to be accurate, fast, and simple in order to simplify calculations as well as to efficiently perform operations and scheduling. In addition, it is difficult to realize the online update of parameters in the existing LPF models. The model retraining brings serious data burden and inefficiency. To serve these applications and comply with requirements, a brand new LPF model is proposed in this paper. A quadratic power flow model is trained by regression learning first, and then the proposed LPF model is derived by Taylor expansion. After only one initial regression learning, the proposed LPF model no longer needs retraining when updated. The refreshed parameter is simply updated online according to the real‐time measurement data, which improves the generalization ability. In conclusion, the proposed LPF model is accurate, generalizable, and greatly minimizes the data consumption and running time. Performance analysis verifies these superiorities.

Linear Power Flow Method for Radial Distribution Systems Including Voltage Control Devices

Linear Power Flow Method for Radial Distribution Systems Including Voltage Control Devices

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.

Energy optimal scheduling strategy considering V2G characteristics of electric vehicle

in recent years, large-scale electric vehicles connected to the power grid has a significant impact on demand-side resources. In addition, the vehicle-to-grid technology of electric vehicles introduces more uncertainties, which brings great challenges to the optimal scheduling of integrated energy systems. To solve this problem, this paper fully considers the uncertainty and fluctuation of wind power. Firstly, the objective function of joint configuration is constructed to maximize the utilization of wind power and minimize the planned generation cost of the system. Secondly, the linearized power flow equations are used to represent the relations among the state variables, and the second-order cone relaxation method is applied to deal with the branch power flow constraints, which is transformed into a solvable mixed-integer second-order cone programming model. Finally, the validity of the model is verified by an example. The system achieves obvious peak-shaving and valley-filling effect and improves the economic benefit of the system.

Identification of Distribution Network Topology and Line Parameter Based on Smart Meter Measurements

Accurate line parameters are the basis for the optimal control and safety analysis of distribution networks. The lack of real-time monitoring equipment in grids has meant that data-driven identification methods have become the main tool to estimate line parameters. However, frequent network reconfigurations increase the uncertainty of distribution network topologies, creating challenges in the data-driven identification of line parameters. In this paper, a line parameter identification method compatible with an uncertain topology is proposed, which simplifies the model complexity of the joint identification of topology and line parameters by removing the unconnected branches through noise reduction. In order to improve the solving accuracy and efficiency of the identification model, a two-stage identification method is proposed. First, the initial values of the topology and line parameters are quickly obtained using a linear power flow model. Then, the identification results are modified iteratively based on the classical power flow model to achieve a more accurate estimation of the grid topology and line parameters. Finally, a simulation analysis based on IEEE 33- and 118-bus distribution systems demonstrated that the proposed method can effectively realize the estimation of topology and line parameters, and is robust with regard to both measurement errors and grid structures.

Admissible Control Laws for Constrained Linear Power Flow: The General Case

Linearized power flow with line flow and voltage constraints can be modeled as a system of linear inequalities depending on the power injections. When some injections are controlled by the grid operator while others are determined exogenously, robust control aims at determining grid operator's actions under imperfect system observability such that the grid state is feasible for all possible realizations of the exogenous actions. It was shown how to design and analyze such admissible control policies efficiently for the subclass of affine control laws, but this paper shows that there are important cases that require more general control policies. We show that, for the constrained linear power flow setting, general admissible control policies can always be chosen as piecewise affine (PWA) mappings. The PWA mapping can be explicitly characterized offline or control actions can be computed online by solving an optimization problem given a current observation. For the latter setup, we provide an algorithm that verifies offline that the online control scheme is admissible. This verification step is a crucial precondition to apply the online control scheme in real, safety-critical power grids. The proposed framework is demonstrated with applications to voltage regulation in active distribution grids subject to uncertain prosumers and low observability.

Voltage Stability Constrained Operation Optimization: An Ensemble Sparse Oblique Regression Tree Method

Voltage stability remains a main challenge for high renewable penetrated power systems. In future power systems, the dispatch needs to consider the voltage stability margin. However, considering static and transient voltage stability is challenging because a complex relationship exists between voltage stability and operation state variables. This paper proposes an ensemble sparse oblique regression tree method for voltage stability constrained operation optimization. First, we train multiple oblique regression trees on voltage stability simulation datasets. The trained trees are then grouped into ensemble using a boosting method to extract accurate and understandable voltage stability rules. Finally, the rules are embedded as mixed-integer linear programming constraints in a linear optimal power flow model. Validation on the IEEE-14 case shows the interpretability and efficiency of the proposed algorithm. Case studies on the Qinghai power grid and IEEE-118 further show that the proposed strategy can effectively improve the voltage stability margin of optimized operation states.

A Game-Theoretic Approach for Electric Vehicle Aggregators Participating in Phase Balancing Considering Network Topology

With the increasing popularity of electric vehicles (EVs), uncontrolled charging of single-phase plug-in EVs will result in severe three-phase unbalance problem in distribution networks. To solve the problem, a phase balancing (PB) scheme performed by EV aggregators (EVAs) based on the linear multiphase power flow model is proposed in this paper. In the scheme, the PB compensations are provided to encourage EVAs to help the distribution system operator reduce the phase unbalance of the entire grid. Considering all EVAs are selfish and rational, the PB scheme is formulated as a game problem, in which each EVA competes with other EVAs to determine its charging strategy to minimize the charging cost minus the PB compensation. The consideration of coupling voltage constraints leads this game model to a challenging generalized Nash equilibrium problem (GNEP). By employing the theory of variational inequality, the properties of solutions for the GNEP are investigated. Furthermore, a two-level distributed algorithm is designed to find the unique variational solution, a fair and stable solution. Finally, comprehensive case studies are conducted on the IEEE-13 and IEEE-123 test systems to corroborate that the proposed PB game model can effectively mitigate voltage unbalance and cut EV users’ costs.

Event-Triggered Distribution System State Estimation: Sparse Kalman Filtering With Reinforced Coupling

A novel distribution system state estimation (DSSE) method is proposed for power distribution networks with low-observablity, where the measurements come from only a few distribution-level phasor measurement units (D-PMUs). The proposed DSSE method is event-triggered, which means the state variables are updated based on the information that is extracted from the events in the power distribution system. In this regard, the estimations of the state variables during the previous events are used as priori information to predict the state variables at the current event. Accordingly, a novel data-driven method based on elastic net regression analysis is proposed to learn the event-triggered state transition matrix. The DSSE problem is formulated as a generalized group Lasso problem, which is augmented based on the knowledge on the sparsity patterns of the state variables that are extracted from the analysis of the events. Here, in the absence of direct power measurements, we enhance our ability in sparse recovery by developing a new reinforced physics-based coupling method among the state variables, in which we add a novel set of linear differential power flow equations to the DSSE problem formulation in forms of virtual measurements. Finally, two different approaches are proposed to solve the formulated sparse event-triggered DSSE problem. The first approach is exact but computationally expensive, as it requires conducting a batch alternating direction method of multipliers (ADMM) analysis. The second approach is approximate, but it is much faster as it works based on a novel modified Kalman filter/smoother in the presence of ℓ1-norm.

Data-Physical Hybrid Driven Distribution Network Linear Power Flow Considering Non-smooth Constraints

Data-Physical Hybrid Driven Distribution Network Linear Power Flow Considering Non-smooth Constraints

AC Network-Constrained Unit Commitment Based on Adaptive Linear Power Flow Model

AC Network-Constrained Unit Commitment Based on Adaptive Linear Power Flow Model

Robust and Chance-Constrained Dispatch Policies for Linear Power Systems

Several methods have recently been proposed to determine optimal feedback control policies for generator dispatch under uncertain grid in-feeds. This work presents and explains two such approaches based on a constrained linear power flow model in a unified notation, a robust method based on linear programming and a chance-constrained one based on polynomial chaos expansions and second order cone programming. This work compares both methods on the same small test example and discusses consequences for long-term grid expansion planning. While the robust method is easier to implement, it is also not more restrictive. For the chance-constrained method, we highlight a serious drawback in reformulations common in the literature. As an aside, we extend the existing formulation to cases where not all disturbances can be measured, and combine it with a robust pre-processing scheme.