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A unifying energy-based approach to stability of power grids with market dynamics
Tjerk Stegink, Claudio De Persis, Arjan van der Schaft
TL;DR
Power-system market dynamics require stability analysis that accounts for their coupling with physical network dynamics. The paper combines third-order voltage-aware modeling with primal-dual pricing in a port-Hamiltonian closed loop, proving convergence results and covering constrained and acyclic congestion settings.
Problem
Dynamic pricing couples market updates with physical network dynamics, motivating stability analysis that considers both simultaneously.
Method
The paper applies a primal-dual gradient method to social welfare optimization and interconnects the resulting port-Hamiltonian pricing controller with a third-order physical network model.
Results
The closed-loop system converges to optimal points; the result extends to nodal power constraints, while line congestion and transmission costs require acyclic networks.
Takeaways & Limitations
The energy-based framework provides a unified stability analysis for nonlinear power networks coupled with market dynamics and supports several operational extensions.
Takeaways & Limitations
For nonlinear networks, the results provide only local asymptotic stability, and line congestion with transmission costs remains unresolved for cyclic networks.
Abstract
from arXiv · showhide
In this paper a unifying energy-based approach is provided to the modeling and stability analysis of power systems coupled with market dynamics. We consider a standard model of the power network with a third-order model for the synchronous generators involving voltage dynamics. By applying the primal-dual gradient method to a social welfare optimization, a distributed dynamic pricing algorithm is obtained, which can be naturally formulated in port-Hamiltonian form. By interconnection with the physical model a closed-loop port-Hamiltonian system is obtained, whose properties are exploited to prove asymptotic stability to the set of optimal points. This result is extended to the case that also general nodal power constraints are included into the social welfare problem. Additionally, the case of line congestion and power transmission costs in acyclic networks is covered. Finally, a dynamic pricing algorithm is proposed that does not require knowledge about the power supply and demand.
I. INTRODUCTION
The paper develops a passivity-based, port-Hamiltonian framework for coupling dynamic pricing with a higher-order nonlinear power-network model. It targets optimal dispatch and stability while accommodating realistic voltage dynamics, nonlinear flows, constraints, and selected network extensions.
- Motivation: Increasing renewable penetration places generators nearer capacity limits and makes transmission congestion more frequent.
- Motivation: The social welfare problem optimizes supply-demand matching while sharing generation and consumption utilities and costs across control areas.
- Motivation: Market dispatch dynamically couples market updates with physical electromechanical responses, so stability must be analyzed for both processes simultaneously.
- Main contributions: The analysis uses a third-order synchronous-generator model with voltage dynamics, considering market, frequency, and voltage dynamics together.
- Main contributions: The approach represents both the physical network and gradient-based pricing controller in port-Hamiltonian form, then interconnects them through passivity.
- Main contributions: Unlike reverse-engineering approaches, the method addresses nonlinear power flows and time-varying voltages without relying on linearization.
- Main contributions: State transformation removes the requirement that the pricing controller know demand to determine the market price.
4) Port-Hamiltonian framework:
The framework supports controller extensions for constraints, transmission costs, transient behavior, and more flexible objective functions. Barrier functions can keep trajectories feasible during transients.
- 4) Port-Hamiltonian framework:: Transmission costs can be included to reduce energy losses or the risk of breakdown on selected transmission lines.
- 4) Port-Hamiltonian framework:: Nonstrict convex cost and concave utility functions can be handled by relaxing objective-function conditions.
- 4) Port-Hamiltonian framework:: Damping can be added to the gradient-based controller, potentially improving the closed-loop convergence rate.
- 4) Port-Hamiltonian framework:: Barrier functions enforce feasibility throughout transients by keeping trajectories within the capacity-constrained region.
- 4) Port-Hamiltonian framework:: The controller extensions include nodal power constraints, line congestion, and transmission costs for acyclic power networks.
B. Power network model
The paper models a connected power network with third-order synchronous-generator dynamics that include voltage behavior. It represents the physical system using energy functions and port-Hamiltonian structure under standard operating assumptions.
- Network structure: The network consists of buses connected by transmission lines represented as a connected undirected graph with an incidence matrix.Each bus is treated as a control area with controllable power supply and demand.
- Generator dynamics: The model uses a third-order flux-decay representation for synchronous generators, including frequency and voltage dynamics.The model notation and parameters are summarized in Table I.
- Model assumptions: The analysis assumes purely inductive lines, operation near synchronous frequency, constant excitation voltage, and angle differences within a security range.Additional sufficient conditions include small generator reactances relative to line reactances and small voltage-angle differences.
- Energy representation: The physical Hamiltonian combines shifted rotor kinetic energy with magnetic energy stored in generator circuits and inductive transmission lines.Voltage angle differences and angular momenta provide key state variables for the energy representation.
- Port-Hamiltonian formulation: The physical network can be written in port-Hamiltonian form, supporting an energy-based analysis of its stability and equilibria.The formulation relies on a positive-definite Hessian near equilibria satisfying the stated security-angle condition.
C. Social welfare problem
The social welfare problem optimizes controllable generation and consumption while enforcing power balance and zero frequency deviation. Convexity and concavity assumptions allow its optimality conditions to characterize the solutions.
- Objective: Social welfare is defined as consumer utility minus producer cost, S(Pg, Pd) := U(Pd) − C(Pg).The cost is strictly convex and the utility is strictly concave.
- Optimization goal: The optimization seeks maximum social welfare while achieving zero frequency deviation through total supply-demand balance.The necessary balance condition is 1T Pd = 1T Pg.
- Network constraint: Supply-demand balance is equivalently represented using virtual flows on a connected communication graph with incidence matrix Dc.This equivalence motivates the convex minimization formulation used in the paper.
- Optimality conditions: The Lagrangian and KKT conditions provide the first-order optimality characterization for the constrained welfare problem.The multipliers λ represent the dual variables associated with the balance constraints.
- Optimality conditions: Because the minimization problem is convex, strong duality makes the KKT conditions necessary and sufficient for optimality.An optimal solution exists exactly when an associated multiplier satisfies the stated conditions.
III. BASIC PRIMAL-DUAL GRADIENT CONTROLLER
The basic controller applies a primal-dual gradient method to the welfare problem and expresses the resulting dynamic pricing mechanism in port-Hamiltonian form. Interconnection with the physical network yields local convergence to optimal zero-frequency equilibria.
- Controller design: Applying the primal-dual gradient method to the welfare problem produces the basic distributed dynamic pricing controller.The controller is the starting point for the later design variations.
- Economic interpretation: Producer and consumer decisions respond to local prices, while communication-graph variables implement distributed price updates through virtual power flows.The virtual flow need not match the physical power flow because the communication and physical topologies may differ.
- Port-Hamiltonian controller: The controller admits a port-Hamiltonian representation because the welfare gradient is incrementally passive under the concavity assumption.The relevant inequality is (z1 − z2)T(∇S(z1) − ∇S(z2)) ≤ 0.
- Closed-loop system: Interconnecting controller and physical-network ports produces a closed-loop port-Hamiltonian system with total Hamiltonian H = Hp + Hc.The interconnection uses uc = −yp and up = yc.
- Equilibrium: The desired equilibrium set simultaneously satisfies the welfare optimality conditions and zero frequency deviation.The set is therefore the target set for the closed-loop convergence result.
- Stability result: Every trajectory initialized in a neighborhood of an equilibrium satisfying Assumption 2 converges to the optimal equilibrium set and, additionally, to a point.The proof uses a shifted Hamiltonian, dissipation inequality, and LaSalle’s invariance principle.
IV. VARIATIONS IN THE CONTROLLER DESIGN
The controller design is extended to incorporate additional network and market constraints. These variations include nodal power constraints and line congestion with transmission costs.
- Controller extensions: The extended controller design incorporates nodal power constraints into the social welfare problem.These constraints are presented as a variation of the basic controller.
- Controller extensions: For acyclic networks, the design also incorporates line congestion together with power transmission costs.The extensions address constraints and costs associated with transmission lines.
A. Including nodal power constraints
The framework extends the social welfare formulation and closed-loop stability analysis to general nodal power constraints. Under the stated assumptions, trajectories locally converge to optimal operation points.
- A. Including nodal power constraints: General convex nodal constraints can be incorporated into the social welfare optimization problem.
- A. Including nodal power constraints: The inequality constraints are characterized through KKT conditions and decentralized multiplier subsystems with complementary-slackness equilibria.
- A. Including nodal power constraints: The controller and physical network are interconnected in a power-preserving way to form the constrained closed-loop system.
- A. Including nodal power constraints: The closed-loop equilibrium set satisfies the optimization conditions together with zero frequency deviation and the physical power-balance equations.
- A. Including nodal power constraints: For every equilibrium satisfying the Hessian assumption, trajectories initialized in a neighborhood converge to the equilibrium set, with each trajectory converging to a point.
- A. Including nodal power constraints: Barrier functions provide an alternative that preserves feasibility but converges to a ν-dependent suboptimal welfare value, approaching optimal welfare as ν → 0 under Slater’s condition.
B. Including line congestion and transmission costs
For acyclic networks, the framework incorporates transmission costs and line-congestion constraints into the social welfare problem. The resulting closed loop converges locally asymptotically to an isolated equilibrium under the stated assumptions.
- B. Including line congestion and transmission costs: In acyclic networks, line congestion and power transmission costs can be added to the social welfare optimization.
- B. Including line congestion and transmission costs: The line constraints bound virtual transmission flows componentwise, while the communication graph is chosen to match the physical network topology.
- B. Including line congestion and transmission costs: Applying the gradient method and interconnecting its controller with the physical model yields a closed-loop system with a partial port-Hamiltonian representation.
- B. Including line congestion and transmission costs: Because the network is a tree, the steady-state controller variable corresponds to physical power flow, so its constraints and costs represent physical-flow constraints and costs.
- B. Including line congestion and transmission costs: All trajectories initialized sufficiently near an isolated equilibrium converge asymptotically to that equilibrium.
- B. Including line congestion and transmission costs: The combined inclusion of nodal constraints, line congestion, and transmission costs remains restricted to tree physical networks.
C. State transformation
A state transformation removes uncertain demand from the dynamic pricing controller. The transformed system retains local convergence to the set of optimal points, while the controller requires no supply or demand information.
- C. State transformation: The demand appears in the original λ-dynamics and is often uncertain, motivating a state transformation to eliminate it from the controller dynamics.
- C. State transformation: The transformation introduces a new state related to the original pricing state and power-balance variable, yielding an equivalent transformed port-Hamiltonian system.
- C. State transformation: For every equilibrium in the transformed optimal set satisfying the Hessian assumption, nearby trajectories converge to that set and each trajectory converges to a point.
- C. State transformation: Choosing τλ = τθ = M simplifies the controller dynamics without changing the stated convergence result.
- C. State transformation: The resulting dynamic pricing algorithm requires no information about power supply, demand, or ω̇; θ − 2ω acts as the electricity price.
- C. State transformation: The stability analysis still requires knowledge of physical power flows and parameters M and A, whose uncertainty radius preserving asymptotic stability remains open.
D. Relaxing the strict convexity assumption
The paper relaxes strict convexity and concavity requirements while preserving local convergence to optimal points. An augmented formulation also adds tunable damping that may improve closed-loop convergence.
- D. Relaxing the strict convexity assumption: A quadratic augmentation preserves the optimal-point set while allowing convex costs and concave utilities that need not be strict.
- D. Relaxing the strict convexity assumption: Applying the primal-dual gradient method to the augmented problem produces distributed dynamics that retain the port-Hamiltonian form.
- D. Relaxing the strict convexity assumption: For every equilibrium satisfying the Hessian assumption, trajectories initialized in a neighborhood converge to a point in the optimal set.
- D. Relaxing the strict convexity assumption: The quadratic term introduces additional damping into the gradient-based controller, with its amount freely selected through ρ and potentially improving convergence properties.
V. CONCLUSIONS AND FUTURE RESEARCH
The paper establishes an energy-based framework proving convergence to optimal points, extends it to nodal constraints, and covers congestion and transmission costs under an acyclic-network condition. Future work includes higher-dimensional models, voltage regulation, cyclic networks, and the region of attraction.
- Passivity-based arguments prove convergence of gradient-method controllers to the set of optimal points.The result is established within a unifying, systematic energy-based framework for modeling and stability analysis.
- The stability result extends to social-welfare problems with general nodal power constraints.
- Line congestion and power transmission costs admit asymptotic-stability results when the power network is acyclic.The acyclicity requirement applies to proving asymptotic stability to the set of optimal points in this case.
- A possible extension is a passive controller for voltage regulation or optimal reactive-power sharing.The paper suggests continuing along the lines of [28] to pursue these objectives.
- The port-Hamiltonian framework may support higher-dimensional synchronous-generator models, while current work considers network-preserving generator-load distinctions.
- Open questions include cyclic networks with nonlinear flows and the region of attraction, since nonlinear-network results provide only local asymptotic stability.