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Reinsertion-Aware Parameter Estimation of the WECC Composite Load Model via Closed-Loop Equivalent-Network Co-Estimation

Jong Ha Woo, Victor Daldegan Paduani, Zishuo Yang, Yu Ma, Qi Xiao, Ning Lu

arXiv:2610.34497v1eess.SYcs.SY

TL;DR

Measurement-based CMLD estimation often uses voltage replay, which omits load–grid interaction and leaves voltage prescribed rather than fitted. This paper alternately estimates CMLD parameters and a five-parameter equivalent network to restore that interaction. In T&D co-simulation, it reduces P, Q, and V reinsertion errors by 73%, 64%, and 62%, respectively, and maintains improved performance on unseen disturbances.

  • Problem

    Voltage replay fits CMLD active and reactive power under prescribed voltage, but does not jointly capture the load–grid interaction and voltage response relevant after reinsertion.

    Source: §I · p. 1

  • Method

    The framework alternately co-estimates CMLD parameters and a five-parameter equivalent network so P, Q, and V jointly constrain the model.

    Source: §I · p. 2

  • Results

    In T&D co-simulation, P, Q, and V reinsertion errors fell by 73%, 64%, and 62%, respectively, versus the conventional baseline for each metric.

    Source: D. Reinsertion Performance in T&D… · p. 8, §IV · p. 10

  • Takeaways & Limitations

    The identified model maintained improved reinsertion accuracy under disturbances not used for parameter estimation.

    Source: §I · p. 2, §IV · p. 10

Abstract

from arXiv · show

Measurement-based parameter estimation of the WECC composite load model (CMLD) commonly relies on voltage replay, where measured voltage is imposed on an isolated CMLD and its parameters are optimized to reproduce measured active and reactive power. However, this open-loop formulation neglects the two-way dynamic interaction between the load and grid, potentially causing large errors when the estimated model is reinserted into dynamic grid simulations. To address this limitation, we propose a closed-loop parameter estimation framework that alternately estimates the CMLD parameters and a five-parameter equivalent network representing the external grid. This framework restores load-grid interaction during estimation and enables voltage, active power, and reactive power to jointly constrain the parameter estimation. Compared with four baseline methods, the proposed approach achieves the highest reinsertion accuracy in both the CMLD benchmark and transmission and distribution (T&D) co-simulation environments. In the T&D case, it reduces active-power, reactive-power, and voltage errors by 73%, 64%, and 62%, respectively. The identified model also maintains improved reinsertion accuracy under disturbances not used for estimation, demonstrating transferability beyond the identification event.

I. INTRODUCTION

Voltage replay can fit measured power while missing the load–grid feedback that determines behavior after reinsertion. The paper introduces closed-loop co-estimation to address this gap and evaluates it in a controlled benchmark and a T&D co-simulation.

I. INTRODUCTION

  • Practical estimation favors a single disturbance and no full dynamic network model, since severe events are infrequent and full-network simulation can be costly or unavailable.The paper identifies these as considerations for practical application.Source: §I · p. 1
  • Voltage replay prescribes measured voltage and fits P and Q, omitting load–grid feedback and leaving voltage unable to act as an independent fitting objective.A low replay error therefore does not ensure accurate system response after reinsertion.Source: §I · p. 1
  • The proposed method alternately estimates CMLD and five-parameter equivalent-network parameters, allowing P, Q, and V to jointly constrain the model without repeated full-network simulations.The equivalent network restores load–grid interaction during estimation.Source: §I · p. 2
  • In the T&D case, the proposed method reduced P, Q, and V reinsertion errors by 73%, 64%, and 62%, respectively, for the identification event, while maintaining improved average performance across unseen fault events.It also showed improved transferability beyond the disturbance used for parameter estimation.Source: §I · p. 2

A. CMLD Parameter Estimation

The CMLD represents aggregated load behind a transmission-level measurement point through a transformer, equivalent feeder, and multiple load components. Its heterogeneous dynamic responses motivate estimating a parameter subset that captures influential behavior.

A. CMLD Parameter Estimation

  • The CMLD aggregates static, power-electronic, three-phase-motor, and single-phase-motor loads behind a substation transformer and equivalent feeder.The model is used to represent load behind a transmission-level measurement point.Source: A. CMLD Parameter Estimation · p. 2, Figure 1 · p. 2
  • The study estimates a 50-dimensional subset of CMLD parameters because estimating all parameters from one disturbance is generally impractical.The subset includes influential component dynamics, load-composition fractions, and feeder parameters.Source: A. CMLD Parameter Estimation · pp. 2–3
  • During FIDVR, stalled single-phase motors draw reactive power that diminishes as thermal protection trips them, while power-electronic responses may persist through post-fault recovery.Similar immediate aggregate deviations can arise from components with different temporal behavior, so post-fault evolution matters for reproducing voltage recovery.Source: A. CMLD Parameter Estimation · p. 3, Figure 2 · p. 3

B. Conventional Voltage-Replay Estimation

Conventional voltage replay fixes the measured voltage while fitting CMLD power, so candidate parameters cannot affect the grid-side voltage during estimation. The proposed equivalent-network approach restores feedback and makes voltage part of the coupled response being fitted.

B. Conventional Voltage-Replay Estimation

  • Voltage replay imposes measured boundary voltage on an isolated CMLD and fits active and reactive power, leaving voltage independent of candidate parameters.Consequently, a small P–Q replay error need not reproduce the coupled response after reinsertion.Source: B. Conventional Voltage-Replay… · p. 3

C. Proposed Closed-Loop Equivalent-Network Co-Estimation

  • The proposed method replaces prescribed voltage with a low-order equivalent grid, so changes in CMLD power affect boundary voltage and the resulting voltage feeds back into load response.This converts open-loop replay into a closed-loop load–grid simulation.Source: C. Proposed Closed-Loop… · p. 3, Figure 3 · p. 4

1) From Voltage Replay to Closed-Loop Estimation:

  • Closed-loop estimation evaluates the CMLD under reinsertion-like interaction and includes voltage deviation alongside P and Q errors.The resulting parameters target the coupled P–Q–V response rather than power response under a fixed voltage trajectory.Source: 1) From Voltage Replay to Closed-Loop… · pp. 3–4

2) Equivalent-Network Model:

A low-order equivalent network restores load–grid feedback without reconstructing the external grid in detail. Alternating updates use voltage-inclusive errors to refine the CMLD and network together.

2) Equivalent-Network Model:

  • The external grid is represented by one equivalent source and one equivalent line, with five adjustable parameters chosen to reproduce dominant voltage response while limiting compensation for CMLD errors.The source uses GENROU with a simplified SEXS excitation system; the line is represented by Zr + jZx.Source: 2) Equivalent-Network Model · p. 4

3) Voltage-Inclusive Objective and Co-Estimation:

  • The closed-loop objective adds simulated-versus-measured voltage mismatch to active- and reactive-power trajectory errors, with wV = 100 balancing those terms.The voltage error is informative because simulated voltage depends on the candidate CMLD interacting with the equivalent network, unlike prescribed replay voltage.Source: 3) Voltage-Inclusive Objective and… · p. 4
  • Alternating updates minimize J2 for CMLD parameters at fixed network parameters, then calibrate the network using the voltage component of J2.The two steps repeat until convergence, refining both sides of the closed-loop interaction.Source: 3) Voltage-Inclusive Objective and… · p. 4

4) CMLD Parameter Update:

The CMLD update is performed within the feasible parameter space and is not tied to a particular optimizer. Evaluation distinguishes replay from reinsertion errors and compares the proposed method with four replay-based baselines.

4) CMLD Parameter Update:

  • The CMLD parameters are optimized over the feasible space, with candidates violating physical constraints projected back into that space before evaluation.The nonlinear, nonconvex update can in principle use any capable search algorithm.Source: 4) CMLD Parameter Update · p. 4

D. Evaluation Metrics and Baseline Methods

  • Replay error measures active- and reactive-power mismatch under prescribed voltage, whereas reinsertion error measures active power, reactive power, and voltage after the CMLD is returned to the original grid.The equivalent network is used during estimation, not reinsertion validation; RMSE excludes samples immediately around fault application and clearing.Source: D. Evaluation Metrics and Baseline… · pp. 4–5
  • Span-normalized parameter-estimation error is evaluated only in the CMLD Benchmark, where ground-truth CMLD parameters are available.NMAE normalizes each estimated-versus-ground-truth parameter difference by its search-bound span.Source: D. Evaluation Metrics and Baseline… · p. 5
  • PSO, GA, CMA-ES, and TRF are compared as voltage-replay baselines, sharing the proposed method’s parameter space, constraints, and data.Using CMA-ES for both the conventional baseline and proposed method controls for the search algorithm when comparing formulations.Source: D. Evaluation Metrics and Baseline… · p. 5, Table II · p. 5

III. SIMULATION RESULTS

The study evaluates the method in two validation environments with different model fidelity, including a controlled benchmark and a more realistic T&D co-simulation.

III. SIMULATION RESULTS

  • Two validation environments test the method at different levels of model fidelity: one supports direct parameter-recovery assessment, while the other uses a detailed distribution feeder without an exact CMLD parameter vector.Environment A has known ground-truth CMLD parameters; Environment B assesses reproduction of heterogeneous distribution-system terminal behavior after reinsertion.Source: §III · p. 5

A. Validation Environments

Environment A provides a controlled benchmark with known CMLD parameters, while Environment B tests aggregate-load reproduction against a detailed, heterogeneous distribution network.

1) Environment A (CMLD Benchmark):

  • Environment A uses an IEEE 118-bus system with a CMLD at bus 1 and known parameters to generate 240-Hz disturbance measurements.Because the estimation model matches the ground-truth load structure, parameter recovery and reinsertion response can both be assessed.Source: 1) Environment A (CMLD Benchmark) · p. 6

2) Environment B (T&D Co-Simulation):

  • Environment B couples an IEEE 118-bus transmission system to a detailed IEEE 123-bus feeder, whose heterogeneous load nodes are CMLDs with independently varied non-static fractions.The distribution network, rather than a predefined CMLD, is ground truth; no unique ground-truth CMLD parameter vector is assumed.Source: 2) Environment B (T&D Co-Simulation) · p. 6

B. Identification and Validation Protocol

The proposed method improves reinsertion accuracy in the CMLD benchmark by including load-grid interaction during estimation. The protocol also tests whether that accuracy transfers from the identification fault to unseen faults.

B. Identification and Validation Protocol

  • Both validation environments share the IEEE 118-bus system and four fault buses; estimation uses only the five-cycle bus-1 fault at t = 1 s.Boundary-bus voltage and active and reactive powers are recorded at 240 Hz over a 15-s window.Source: B. Identification and Validation… · p. 6, Figure 5 · p. 6

C. Replay-to-Reinsertion Gap in the CMLD Benchmark

D. Reinsertion Performance in T&D Co-Simulation

In the T&D co-simulation, the proposed method reduces reinsertion errors relative to the best conventional baselines. Its average advantage persists across four fault locations, although individual baselines can outperform it for particular signals at unseen contingencies.

D. Reinsertion Performance in T&D Co-Simulation

  • Averaged over four fault locations, reinsertion errors are reduced by approximately 34%, 19%, and 22% for P, Q, and V, respectively.The proposed method has the lowest average error across all three quantities, though a baseline can win for a particular signal at an unseen contingency.Source: D. Reinsertion Performance in T&D… · p. 8, Figure 10 · p. 9
  • Under model mismatch, conventional methods fit closely in voltage replay but separate substantially after reinsertion into the T&D co-simulation.The proposed method reproduces post-fault P, Q, and V recovery more consistently with load-grid interaction represented during estimation.Source: D. Reinsertion Performance in T&D… · p. 8, Figure 9 · p. 8, Figure 8 · p. 8
  • Environment B evaluates whether an aggregate CMLD reproduces the detailed feeder’s terminal dynamics; it assumes no unique ground-truth CMLD parameter vector.Figure 10 summarizes average reinsertion errors over four contingencies for both validation environments.Source: D. Reinsertion Performance in T&D… · p. 8, Figure 10 · p. 9
  • At the identification contingency, P, Q, and V reinsertion errors are reduced by 73%, 64%, and 62%, respectively, versus the best conventional baseline for each metric.The proposed method achieves the lowest reinsertion errors in all three measured quantities.Source: D. Reinsertion Performance in T&D… · pp. 7–8, Table IV · p. 8

E. Parameter Accuracy Analysis

Parameter recovery is difficult even when the CMLD structure is known, and parameter error does not closely track reinsertion accuracy. In the T&D environment, estimation quality is instead judged by reinsertion response because no exact CMLD parameter vector exists.

E. Parameter Accuracy Analysis

  • Environment B has no exact ground-truth CMLD parameter vector because a single CMLD only approximates the heterogeneous distribution feeder.Its estimation quality is assessed through reinsertion response accuracy, not parameter error.Source: E. Parameter Accuracy Analysis · pp. 9–10
  • The proposed method achieves the best overall reinsertion performance with 21.3% overall parameter NMAE, while CMA-ES has the lowest parameter NMAE at 17.4% but larger reinsertion errors.Across tested methods, overall parameter NMAE falls within approximately 17–21%.Source: E. Parameter Accuracy Analysis · p. 9, Table V · p. 9
  • No method is uniformly superior across CMLD components, and component-wise parameter rankings vary substantially among estimators.The component-wise NMAE is reported for Environment A, the environment with a ground-truth parameter vector.Source: E. Parameter Accuracy Analysis · p. 9, Table V · p. 9
  • Parameter recovery and dynamic-response reproduction are distinct objectives, making reinsertion accuracy the more meaningful criterion when no unique CMLD parameter vector exists.Different parameter sets can match similar replay trajectories while differing from ground truth and one another.Source: E. Parameter Accuracy Analysis · pp. 9–10

F. Computational Cost Analysis

The proposed framework improves reinsertion accuracy in benchmark and T&D validation but requires longer offline estimation; future work includes field-PMU evaluation.

F. Computational Cost Analysis

  • 11.9 h versus 2.0–6.1 h total estimation time; each closed-loop evaluation takes approximately 4.6 s versus 0.2 s for replay, with the added cost primarily from restoring load-grid interaction.Source: F. Computational Cost Analysis · p. 10, Table VII · p. 10
  • Because identification is offline and optimization uses a low-order equivalent network rather than the full transmission system, the computational requirement remains practical for the intended application.Source: F. Computational Cost Analysis · p. 10

IV. CONCLUSION

  • 73%, 64%, and 62%: the proposed method reduced T&D active-power, reactive-power, and voltage errors, respectively, for the identification event, while improving average performance across unseen contingencies.Source: §IV · p. 10
  • Future work will examine equivalent-network structure and parameterization, then evaluate field PMU measurements under uncertainty, unmodeled load dynamics, and real-world disturbances.Source: §IV · p. 10
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