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Blackbox Small-Signal Modeling of Grid-Connected Inverters in Asymmetrical Power Grids
Airan Frances, Dionisio Ramirez, Javier Uceda
TL;DR
Blackbox models for three-phase converters have largely assumed symmetric, balanced conditions, leaving asymmetrical-grid behavior insufficiently addressed. The paper develops a sequence-domain small-signal structure for grid-connected inverters and experimentally validates it, showing accurate prediction under unbalanced conditions and a 0.7 pu phase-voltage sag.
Problem
Existing three-phase blackbox models assume symmetric, balanced conditions, although grid asymmetry and positive-negative sequence coupling affect converter dynamics.
Method
The paper uses positive and negative symmetrical components in their respective synchronous frames to build a blackbox small-signal model for grid-connected inverters.
Results
A 0.7 pu voltage sag in one phase showed that the proposed model provides a good representation of the three-phase inverter in highly asymmetrical conditions.
Takeaways & Limitations
The structure can represent static and dynamic behavior of three-phase inverters under asymmetrical and unbalanced conditions.
Abstract
from arXiv · showhide
Power electronic converters are envisaged to be key enablers of modern electric power distribution systems. Grid-connected three-phase inverters are widely used in Smart Grids and microgrids, but also in standard grids. They provide controllability and dynamic decoupling capabilities, which are fundamental in the integration of renewable sources and storage systems and in specialized applications such as FACTS. Nevertheless, power electronics-based systems can exhibit dynamic interactions, which may lead to power quality issues. Although the electrical model of each element of these systems is important for their system-level analysis, they are rarely available. Blackbox modeling strategies are useful for obtaining behavioral models of commercial electronic converters. Most blackbox modeling strategies are focused on dc-dc electronic converters; however, some works have extended these concepts to three-phase converters by means of the dq framework, under the assumption of symmetrical and balanced conditions. This work proposes a new structure in the sequence domain to represent the dynamic behavior of grid-connected commercial converters in asymmetrical conditions. Experimental tests have been performed on a three-phase inverter to identify its blackbox model and its performance has been validated during an asymmetric voltage sag.
I. INTRODUCTION
Blackbox models address scarce electrical-model information for power electronic converters, but prior three-phase approaches generally assume balanced conditions. This paper proposes a blackbox small-signal structure for grid-connected inverters under asymmetrical conditions and validates it experimentally.
- System identification supports applications including control design, stability analysis, system integration, filter design, and protection design.
- Blackbox models use system identification to represent converter dynamics when device information is scarce.
- Most blackbox structures target dc-dc converters, while extensions to three-phase converters use rotating d-q coordinates to obtain dc variables.
- Existing three-phase blackbox extensions assume symmetric, balanced conditions, although analytical studies show that positive- and negative-sequence coupling matters in asymmetrical grids.
- The paper proposes a Fortescue-based sequence-domain structure for asymmetrical grid-connected inverters, supporting dynamic and impedance-based stability analysis of commercial converters.
II. SMALL-SIGNAL STRUCTURE OF THREE-PHASE INVERTERS UNDER ASYMMETRICAL CONDITIONS
The proposed small-signal structure models a grid-feeding inverter connected to an asymmetrical grid by using port voltages and control references as inputs and currents as outputs. It separates positive and negative sequences and represents each in its corresponding synchronous frame.
- The model treats imposed input and output voltages as inputs and input and output currents as outputs, with active and reactive power references as optional control inputs.
- The structure is defined for a grid-feeding converter connected to an asymmetrical power grid.
- Three-phase ac signals are divided into positive and negative sequences before sequence-domain processing; the zero sequence is neglected for the three-wire system.
- The positive sequence uses the ωt synchronous frame, while the negative sequence uses the −ωt frame so both have constant d and q components.
A. Sequence detector
The sequence detector decomposes unbalanced three-phase signals into positive and negative sequences using transformations and quadrature-signal generation. A SOGI-QSG provides a continuous response, but its dynamics constrain the model-identification frequency range.
- Fortescue’s theorem represents a three-wire unbalanced three-phase signal as the sum of balanced positive and negative sequences.
- The detector obtains sequence components through the Lyon transformation, Clarke transformation, and related in-quadrature operations.
- A SOGI-QSG produces a continuous quadrature response and the correct result in about 1 cycle, improving on a quarter-cycle transport delay that can remain discontinuous for more than 2 cycles.
- The sequence detector combines a Clarke transformation with SOGI-QSG implementations of the required signal relationships.
- SOGI-QSG dynamics affect blackbox inputs and limit the maximum identification frequency according to the detector’s post-perturbation settling time.
B. Synchronization
The model uses a synchronization system to obtain the d and q components of positive and negative sequence signals. Under filtered or undistorted conditions, the arc-tangent function provides instantaneous phase-angle estimation, while SOGI-QSG filtering mitigates harmonic-related drawbacks.
- B. Synchronization: The synchronization system obtains the d and q components of the positive and negative sequence signals.
- B. Synchronization: The arc-tangent function is selected for phase-angle estimation because it has an instantaneous response when signals are undistorted or properly filtered.
- B. Synchronization: SOGI-QSG filtering reduces harmonic content before synchronization, mitigating drawbacks of using the arc-tangent function.
C. Small-signal model
The small-signal model represents output responses to perturbed inputs through a collection of transfer functions. Selected operating-point variables are excluded when synchronization or simplifying assumptions make them unnecessary inputs.
- C. Small-signal model: The model contains one transfer function for each input-output pairing, describing every output response when an input is perturbed.Each transfer function is identified with all other inputs held constant.
- C. Small-signal model: The model excludes vq+, Vin, and Qref from its inputs because vq+ is regulated to zero, while Vin and Qref are treated as constant.
- C. Small-signal model: Transfer-function names use a two-character subscript: the first character denotes the input and the second denotes the output.The notation distinguishes relationships among positive- and negative-sequence d and q components.
1) DC operating point:
The model uses a constant DC operating point corresponding to the identification condition, but accounts for input-current oscillations caused by negative-sequence components. Active power is expressed using symmetrical-component d and q variables, including second-harmonic terms.
- 1) DC operating point:: The DC operating point is constant and corresponds to the condition at which the transfer functions are identified.The input current is exceptional because negative sequence can produce steady-state oscillations.
- 1) DC operating point:: The input-current reference is defined through power balance between input power and the active power of the three-phase port.The balance uses input voltage, converter efficiency, and three-phase-port active power.
- 1) DC operating point:: The blackbox electrical equivalent model is presented as the model representation for the converter.
- 1) DC operating point:: The d and q voltage and current components are decomposed into positive and negative symmetrical components.Negative-sequence components in the positive reference frame are related to constant components in the negative reference frame through trigonometric projections.
- 1) DC operating point:: Active power is represented as P = Pa + Pc · cos(2ωt) + Ps · sin(2ωt), combining average power with second-harmonic oscillations.Pa is the average active power, while Pc and Ps define the second-harmonic terms.
III. EXPERIMENTAL TESTS
Experimental tests use a controlled three-phase inverter and a programmable four-quadrant source to identify the proposed transfer functions. Model validation is performed under an asymmetric voltage sag.
- III. EXPERIMENTAL TESTS: A Semikron Semiteach VSC controlled by a dual-core F28M35H52C microcontroller is used for identification and validation experiments.Its control structure uses decoupled current control, sequence detection, and positive-sequence current and voltage signals.
- III. EXPERIMENTAL TESTS: A four-quadrant REGATRON TC.ACS generates perturbations in the positive- and negative-sequence d and q components of grid voltage.These perturbations are used to identify the transfer functions of the proposed model.
- III. EXPERIMENTAL TESTS: The experimental setup supports both identification and validation tests.
A. Model identification
The identification process perturbs one model input at a time, uses measured responses to estimate transfer functions, and validates sequence-domain relationships under asymmetrical excitation.
- Test design: Four tests identify the transfer-function columns by perturbing one input while holding the remaining inputs constant.The process applies superposition so each test isolates one input's small-signal contribution.
- Test design: The first test perturbs active-power reference, while the second applies a balanced step to positive-sequence voltage amplitude.The other voltage-sequence components remain constant in both tests.
- Test design: The third test injects a negative-sequence voltage aligned with the positive sequence to perturb Vd−.The amplitudes of the positive and negative sequences are denoted Ap and An, respectively.
- Test design: The fourth test perturbs Vq− by injecting a negative sequence shifted 90 degrees relative to the positive sequence.This isolates the q component of the negative-sequence voltage.
- Identification results: In the third test, Vd− negligibly affects Id+ and Iq+ but significantly affects Id− and Iq−, identifying Yd−d−(s) and Yd−q−(s).The transfer functions were estimated with Vd− as input and Id− and Iq− as outputs using MATLAB's System Identification Toolbox.
- Identification procedure: The measured signals are mean-centered before MATLAB's tfest estimates transfer-function parameters at the resulting operating point.The operating point is defined by the subtracted initial averages.
B. Model validation
The identified model is validated against an inverter during a highly asymmetrical voltage sag, comparing measured and estimated voltage and current sequence responses.
- Validation setup: A validation test imposes a voltage sag in one phase while the other phases remain unaffected.The perturbed phase changes from approximately 190 V to 150 V, roughly 0.2 pu.
- Validation setup: The validation figures compare measured and estimated three-phase signals and positive- and negative-sequence components for voltages and currents.The plots organize voltages on the left, currents on the right, and sequence components beneath the three-phase signals.
- Measured response: The sag strongly affects d components of both voltage sequences, while its effect on the negative-sequence voltage q component is negligible.This identifies the principal voltage-sequence responses under the asymmetric disturbance.
- Measured response: The current response is concentrated in the negative sequence, disturbing both d and q components, with a smaller effect on positive-sequence d current.The observed response is consistent with the sequence-domain model's input-output structure.
- Validation outcome: The model accurately predicts the inverter's static and dynamic three-phase current behavior under asymmetrical and unbalanced conditions.The comparison uses measured currents from the experimental setup and the model estimation.
- Model scope: The averaged model cannot account for high-frequency current components, but it reproduces the sag-induced second-harmonic input-current oscillation through power balance.The power balance computes three-phase-port active power from the model's other dc inputs and outputs.
- Validation outcome: A 0.7 pu sag in one phase is reported as producing a good representation of the three-phase inverter in highly asymmetrical conditions.The result is shown by comparing experimental signals with model estimates.
IV. CONCLUSIONS
The conclusion presents a sequence-domain blackbox structure for asymmetrical three-phase inverter operation and details its identification procedure and dynamic scope.
- IV. CONCLUSIONS: The framework represents positive- and negative-sequence d-q components in their corresponding synchronous frames, where they remain constant under asymmetrical conditions.This extends the balanced-condition formulation used in prior three-phase blackbox models.
- IV. CONCLUSIONS: The model includes a sequence detector and synchronization system to represent dynamic relationships among system variables.These are identified as the main structural elements of the small-signal model.
- IV. CONCLUSIONS: The SOGI-QSG adds dynamics to sequence detection, limiting the model's capabilities, whereas arc-tangent synchronization is instantaneous.The conclusion explicitly links the capability limit to the sequence detector's dynamics.
- IV. CONCLUSIONS: The work details the tests needed to identify each transfer function and proposes a method to account for the second harmonic in averaged input-current response.The second harmonic is associated with negative-sequence voltage in the averaged response.