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Generalized Model of VSC-based Energy Storage Systems for Transient Stability Analysis

Álvaro Ortega, Federico Milano

arXiv:1509.05290v1eess.SY

TL;DR

ESS studies lack a commonly accepted model that is simple yet accurate across storage technologies for voltage and angle stability analysis. The paper introduces a fixed-structure DAE model retaining storage, VSC, dc-link, controller, and hard-limit dynamics, and reports accurate agreement with detailed models across disturbances and operating cycles.

  • Problem

    There is no commonly accepted simple yet accurate general ESS model for voltage and angle stability studies across diverse technologies.

  • Method

    The paper defines a fixed set of linear DAEs combining generalized storage-device dynamics with balanced fundamental-frequency VSC, dc-link, PLL, regulator, and hard-limit models.

  • Results

    The generalized model accurately reproduces detailed transient-stability models for faults and loss-of-load disturbances across the whole operating cycle.

  • Takeaways & Limitations

    The fixed-equation structure supports modeling different storage technologies and comparing control strategies using the same control scheme.

Abstract

from arXiv · show

This paper presents a generalized energy storage system model for voltage and angle stability analysis. The proposed solution allows modeling most common energy storage technologies through a given set of linear differential algebraic equations (DAEs). In particular, the paper considers, but is not limited to, compressed air, superconducting magnetic, electrochemical capacitor and battery energy storage devices. While able to cope with a variety of different technologies, the proposed generalized model proves to be accurate for angle and voltage stability analysis, as it includes a balanced, fundamental-frequency model of the voltage source converter (VSC) and the dynamics of the dc link. Regulators with inclusion of hard limits are also taken into account. The transient behavior of the generalized model is compared with detailed fundamental-frequency balanced models as well as commonly-used simplified models of energy storage devices. A comprehensive case study based on the WSCC 9-bus test system is presented and discussed.

I. INTRODUCTION

The paper addresses the lack of a commonly accepted ESS model that is both simple and accurate for voltage and angle stability studies. It develops a fixed-structure model retaining storage, VSC, dc-link, controller, and limiter dynamics.

  • ESS modeling is complex because many existing and future technologies require different device representations.
  • The VSC representation includes an average dq-axis converter model, dc-circuit and PLL dynamics, and equivalent switching-loss modeling.
  • The proposed generalized model uses a fixed structure and parameter set of linear differential algebraic equations independent of storage technology.
  • The generalized ESS comprises balanced fundamental-frequency models of the VSC, dc link, regulators, and storage-device dynamics.
  • The paper validates the generalized approach against detailed technology-specific and simplified ESS models, including controllers that retain only active and reactive power dynamics.
  • Storage control regulates a measured quantity through charging or discharging, with dead-band and energy-limit handling to reduce unnecessary operations and smooth saturation transients.

B. Simplified ESS scheme

The simplified ESS scheme represents storage dynamics through active- and reactive-power controllers while omitting the storage device itself and dc/VSC circuitry.

  • Simplified ESS models represent ESS dynamics using only active and reactive power controllers.

III. PROPOSED GENERALIZED MODEL OF ENERGY STORAGE DEVICES

The generalized ESS model linearizes storage-device DAEs, separates energy-related states from other variables, and reduces fast or immaterial dynamics while retaining a common VSC-based structure.

  • General formulation: The storage-device behavior is initially represented by nonlinear differential-algebraic equations, then linearized around an equilibrium point.The linear time-invariant formulation specifies state, input, output, and equilibrium-point terms.
  • General formulation: Energy-related potential and flow variables are separated from other states, with VSC dc voltage as input and dc current as output.The storage-control output and VSC dc voltage form the input vector, while dc current is the output.
  • Model reduction: Fast or immaterial dynamics are neglected to reduce the model to a technology-independent set of linear differential-algebraic equations.The reduction assumes the transient response of the eliminated variables is much faster than, or immaterial to, the retained states.
  • Model components: The generalized ESS model combines the VSC block, controllers, and storage-device equations, with stored-energy limits incorporated through the storage-input limiter.The limiter uses stored energy to smooth transients caused by energy saturation.
  • Model components: The framework is used to derive generalized models for multiple ESS technologies.The paper illustrates the procedure for several storage technologies.

A. Electrochemical Capacitor Energy Storage Model

The ECES is connected to the VSC through a bidirectional buck-boost converter, whose averaged switching behavior is incorporated into the generalized model.

  • ECES configuration: The ECES connects to the VSC through a bidirectional dc/dc buck-boost converter.Buck operation delivers energy from storage to the grid, while boost operation stores energy in the ECES.
  • Converter model: The same model represents both buck and boost operation modes using the converter switch logic.With averaged variables, the switch state is continuous and represents the converter duty cycle.
  • Energy representation: The ECES stored-energy expression includes the dominant capacitance term and the inductance contribution.The capacitance Csc term is identified as the main contribution, while inductive energy is retained for completeness.
  • Generalized mapping: In the generalized formulation, ECES states are capacitor voltage and current, the converter duty cycle is the control input, and no additional z states are used.The parameter mapping applies the proposed general notation to the ECES equations.

B. Superconducting Magnetic Energy Storage Model

The SMES model represents magnetic-energy storage through a superconducting coil connected to the VSC by a boost converter, with averaged circuit dynamics mapped into the generalized formulation.

  • SMES configuration: The SMES is connected in parallel to the VSC through a dc/dc boost converter.The converter duty cycle controls magnetic-energy injection into the network.
  • Circuit dynamics: The model describes the superconducting coil and dc/dc converter using averaged circuit equations.The coil inductance and converter duty cycle are explicit model quantities.
  • Energy representation: The stored-energy formulation accounts for the magnetic energy of the SMES.The energy expression is then written using the generalized model notation.
  • Generalized mapping: In the generalized mapping, the SMES state vector contains coil current and voltage, the duty cycle is the control input, and no additional z states are used.The notation identifies x, z, and u for the SMES representation.

C. Compressed Air Energy Storage Model

The CAES model represents pressure, mechanical, electrical-machine, and converter dynamics within the generalized ESS framework, reducing the original nonlinear description to three linear DAEs.

  • CAES configuration: CAES injects air into a tank through a compressor and extracts it through a turbine, with both machines interfaced to the VSC dc link by ac/dc converters.The compressor and turbine are driven by asynchronous motor and generator machines, respectively.
  • Storage and machine dynamics: Tank-pressure dynamics use an ideal-gas model with pressure, density, temperature, molecular weight, tank volume, and air flow variables.The air-flow quantity Q is used for compressor or turbine operation depending on its sign.
  • Storage and machine dynamics: Compressor and turbine behavior is modeled with polytropic air-process equations and fifth-order dq electrical-machine models.The electrical machines and their ac/dc converters are represented separately from the displayed CAES equations.
  • Energy representation: The CAES energy expression includes tank-pressure and mechanical-rotor-speed terms, although the rotor-speed contribution is expected to be much smaller.Both energy terms are retained in the model.
  • Generalized mapping: The generalized CAES mapping uses x = [Π2 Ω]T, air flow Q as the input, and compressor-plus-turbine dc current as the output.The remaining machine and VSC variables are collected in z, which contains 25 state and algebraic variables.
  • Generalized mapping: The nonlinear CAES DAE system is reduced to three linear DAEs because the air valve is not modeled and the generalized-model input is air flow Q.The reduction includes the storage, machine, and ac/dc-converter equations.

D. Battery Energy Storage Model

The battery energy storage model uses the Shepherd battery representation and maps its nonlinear dynamics into the generalized ESS framework. Its equations account for charge/discharge-dependent behavior and a converter connection to the VSC.

  • The Shepherd model represents rechargeable battery-cell dynamics within the battery energy storage model.
  • The polarization-voltage relation is specified with separate charge and discharge cases through the battery state of charge.The cited expression distinguishes the condition im > 0, corresponding to discharge.
  • Because polarization voltage is nonlinear, the generalized model switches between two equation sets according to whether the battery is charging or discharging.
  • The BES is connected to the VSC similarly to the SMES, using a dc/dc converter whose duty cycle and cell-series/parallel counts parameterize the connection.The paper notes that other converter configurations are possible but are not considered.
  • The model defines extracted capacity, filtered current, voltage components, resistance, and battery voltage as state and parameter quantities.Qe is extracted capacity; im is filtered battery current; voc, vp, and ve are open-circuit, polarization, and exponential voltages; Ri is internal resistance; vb is battery voltage.

IV. CASE STUDY

The case study validates the generalized ESS model through WSCC 9-bus time-domain simulations involving SMES, CAES, and BES devices. It compares proposed, detailed, and simplified representations across contingencies and stochastic-load scenarios.

  • The validation uses the WSCC 9-bus system, which includes three synchronous machines, transformers, transmission lines, loads, and primary frequency and voltage regulators.An ESS is connected to bus 8.
  • The study considers SMES, CAES, and BES devices across fault, loss-of-load, and stochastic-load scenarios.
  • Results are randomly selected from several hundred simulations conducted to assess the generalized model’s validity and accuracy.
  • The SMES comparisons include detailed, proposed, and commonly used simplified models.The corresponding figures compare the WSCC system with SMES using these model variants.

A. SMES

The SMES case study evaluates a 15 MW, 60 MJ device connected to bus 8 under a three-phase fault and stochastic load variations. The scenarios compare its detailed, proposed, and simplified model responses.

  • The SMES connected to bus 8 is rated at 15 MW and 60 MJ.The paper contrasts this case with the largest installed SMES cited, rated at 100 MJ and capable of 100 MW peak and ±50 MW oscillatory power.
  • The SMES study compares dynamic responses after a three-phase fault and under stochastic load variations with different initial states of charge.
  • The fault scenario applies a three-phase fault at bus 7, cleared after 70 ms by disconnecting the line between buses 7 and 5.

1) Contingency:

The simulations show that the proposed generalized model reproduces detailed ESS behavior across fast and slow dynamics, including energy limits and stochastic load conditions. Simplified models can match early transients but deviate later.

  • Contingency: 60% lower overshoot and approximately 15 s settling time characterize the SMES response after the fault, versus about 40 s without ESS.Without ESS, the post-fault frequency variation is around 1%.
  • Contingency: 57 MJ is the maximum SMES stored-energy increase during and after the fault, with about 20 MJ remaining above the initial condition at steady state.The energy base is 100 MJ.
  • Contingency: Very small differences from the detailed model remain when the SMES reaches its maximum storable energy and a 40 MJ energy variation limit is imposed.
  • Contingency: The proposed SMES model tracks the detailed model in both fast transient and slow postcontingency dynamics, whereas the simplified model is accurate only during the first few seconds.The simplified model’s later deviation is attributed to its reduced dynamic order.
  • Stochastic Load Variations: Across hundreds of stochastic-load simulations, the proposed model outperforms the simplified model for every initial state of charge, with best average accuracy at 50%.Accuracy is higher when the simulated state of charge stays closer to the linearization’s initial condition.
  • Contingency: For a 15 MW loss of load, the proposed CAES model accurately reproduces the detailed CAES response despite the detailed model’s complexity.The loss occurs at bus 5 at t = 10 s, and the load is reconnected at t = 80 s.

1) Contingency:

Under stochastic load variations and load-loss contingencies, the generalized ESS model closely reproduces detailed-model behavior, while the commonly used simplified CAES model can differ substantially. The generalized model preserves accuracy with fewer variables and shared control parameters across generalized and detailed models.

  • Stochastic Load Variations: The simplified CAES response differs considerably from the detailed model under stochastic load perturbations, whereas the generalized model appears very accurate.The comparison uses identical control parameters for all models before separate simplified-model tuning is considered.
  • Stochastic Load Variations: A 29-variable detailed CAES model is reduced to only 5 variables without an apparent loss of generalized-model accuracy.The reported accuracy is not affected by the complexity or order reduction of the original detailed model.
  • Stochastic Load Variations: Simplified-model accuracy cannot be guaranteed even after lengthy, careful control-parameter tuning.Because tuning is required, control strategies cannot be based on the simplified model.
  • Stochastic Load Variations: The generalized and detailed ESS models can use exactly the same control parameters, unlike the simplified CAES comparison requiring gain adjustment.The simplified CAES gains were reduced to one-third in the alternate comparison.
  • Loss of Load: For a 40 MW BES regulating center-of-inertia frequency after a 40 MW load loss, the generalized model tracks the detailed model across charging and discharging.This remains accurate at 85% initial state of charge despite battery nonlinearities and switching between operating modes.
  • Contingency Findings: Overall, the proposed model offers reduced, constant dynamic order while faithfully reproducing complex ESS behavior, though simpler models may be imprecise when windup limiters bind.Detailed ESS models are still needed to define the generalized model's parameters.
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