Bayesian Inertia Estimation via Parallel MCMC Hammer in Power Systems

Abstract

The stability of modern power systems has become critically dependent on precise inertia estimation of synchronous generators, particularly as renewable energy integration fundamentally transforms grid dynamics. Increasing penetration of converter-interfaced renewable resources reduces system inertia, heightening the grid’s susceptibility to transient disturbances and creating significant technical challenges in maintaining operational reliability. This paper addresses these challenges through a novel Bayesian inference framework that synergistically integrates PMU data with an advanced MCMC sampling technique, specifically employing the Affine-Invariant Ensemble Sampler. The proposed methodology establishes a probabilistic estimation paradigm that systematically combines prior engineering knowledge with real-time measurements, while the Affine-Invariant Ensemble Sampler mechanism overcomes high-dimensional computational barriers through its unique ensemble-based exploration strategy featuring stretch moves and parallel walker coordination. The framework’s ability to provide full posterior distributions of inertia parameters, rather than single-point estimates, helps for stability assessment in renewable-dominated grids. Simulation results on the IEEE 39-bus and 68-bus benchmark systems validate the effectiveness and scalability of the proposed method, with inertia estimation errors consistently maintained below $1\%$ across all generators. Moreover, the parallelized implementation of the algorithm significantly outperforms the conventional M-H method in computational efficiency. Specifically, the proposed approach reduces execution time by approximately $52\%$ in the 39-bus system and by $57\%$ in the 68-bus system, demonstrating its suitability for real-time and large-scale power system applications.

1. Introduction

As the primary source of rotational inertia in conventional power grids, synchronous generators play a crucial role in maintaining frequency stability during system disturbances [1,2]. The inertia stored in the rotating masses provides a vital buffer that mitigates the initial impact of imbalances between power generation and consumption, thus slowing the rate of frequency decline immediately after disturbances. This inherent inertial response is fundamental to preventing large frequency excursions that could otherwise lead to system instability or even widespread blackouts [3].

The rapid integration of renewable energy sources, such as wind and solar photovoltaic systems, into modern power grids has significantly altered grid dynamics. These sources are typically connected via power electronic converters, which replace traditional synchronous generators and decouple mechanical inertia from the grid [4]. As a result, the total available synchronous inertia has declined, making power systems more vulnerable to sharp and fast frequency deviations during disturbances [5]. Accurate estimation of system inertia has thus become a pressing requirement for dynamic stability analysis, real-time control decisions, and emergency response planning.

The inertia time constant is a fundamental parameter that characterizes the dynamic behavior of synchronous generators. It reflects the amount of kinetic energy stored in rotating masses relative to the machine’s rated power and directly affects key frequency stability metrics, including the rate of change in frequency and the frequency nadir during transient events [6]. Precise inertia information is vital for designing effective primary frequency control strategies and ensuring protection systems can handle large disturbances.

Traditionally, system operators rely on inertia values provided by generator manufacturers at the time of installation. These values, however, may become outdated due to mechanical wear, equipment aging, or changes in operating conditions. To maintain system reliability in increasingly dynamic and renewable-rich environments, it is essential to estimate generator inertia in real time or update it periodically [7].

Existing inertia estimation methods typically focus on online approaches due to their superior time efficiency compared to offline techniques [8]. The method proposed in [9] estimates inertia in Japan’s 60 Hz power system by analyzing phase differences and power exchanges between interconnected areas. However, this approach requires complex computations that may hinder real-time implementation. The study in [10] proposes estimating equivalent inertia by analyzing electromechanical oscillations obtained through Fast Fourier Transform of generator electrical power and rotor speed measurements. Tuttelberg et al. developed an autoregressive moving average exogenous model for effective system inertia estimation [11]. However, this method requires electrical frequency from selected buses as inputs, without providing clear criteria or justification for the bus selection process in the paper. Bizzarri et al. introduced a covariance matrix-based method using ambient measurements to estimate inertia and damping coefficients for both synchronous generators and converter-based resources [12]. While innovative, this linearized derivation risks inaccuracy in highly nonlinear power system conditions. Recent advancements in machine learning have introduced powerful data-driven paradigms for inertia estimation. For instance, Linaro et al. employed convolutional neural networks to model the nonlinear relationships between time-series data and generator inertia constants [13], while a two-stage hybrid deep learning framework was proposed in [14] to enable real-time inertia estimation with improved speed and accuracy. Beyond conventional neural networks, more recent studies have explored advanced architectures. Tuo and Li [15] proposed a novel graph convolutional neural network-based approach that extracts informative features from ambient measurements collected via PMUs, and embeds spatial structural information by modeling generator interconnections as graph topologies. While these deep-learning-based methods have shown promising empirical performance, they generally require large volumes of labeled data for training, are highly sensitive to the training conditions, and often lack physical interpretability. Moreover, model generalization to unseen system configurations or disturbance types remains a concern.

Bayesian inference has emerged as a compelling framework for power system parameter estimation, offering not only point estimates but also full posterior distributions of dynamic parameters [16,17,18]. Despite these advantages, most existing Bayesian methods for inertia estimation rely on the M-H algorithm to approximate posterior distributions. However, M-H suffers from several fundamental drawbacks that severely limit its effectiveness in high-dimensional and nonlinear settings, such as the simultaneous estimation of inertia parameters for all generators in large-scale power grids. First, M-H depends on manually designed proposal distributions, which are notoriously difficult to tune in high-dimensional spaces. Poorly chosen proposals result in low acceptance rates, redundant samples, and inefficient exploration of the parameter space. Second, the sampling process in M-H is inherently sequential, which not only hinders parallelization but also leads to slow mixing and long autocorrelation lengths. These issues make it computationally prohibitive to generate sufficient effective samples in practical time frames, especially when the posterior is highly non-Gaussian or exhibits strong correlations between parameters.

To overcome the aforementioned limitations of the M-H algorithm, this paper introduces a parallelized implementation of the MCMC Hammer algorithm, an Affine-Invariant Ensemble Sampler, for Bayesian posterior approximation in inertia estimation. Unlike M-H, which relies on a single-chain and manually tuned proposal distribution, the MCMC Hammer algorithm leverages an ensemble of walkers that collaboratively explore the parameter space through stretch moves. This ensemble-based design fundamentally changes the sampling dynamics: each walker proposes its new position based on the relative positions of other walkers, rather than a fixed proposal kernel. Such stretch move operations automatically incorporate information about the shape, orientation, and scale of the posterior distribution without explicit tuning, making the sampler highly robust to parameter anisotropy and correlation. Moreover, by incorporating a parallel stretch move strategy, the proposed method enables concurrent updates of multiple walkers, thereby overcoming the sequential bottleneck of traditional MCMC schemes. This parallelism significantly accelerates convergence and improves effective sample generation, making the approach well-suited for online inertia estimation. The main contributions of this paper include:

• Establish a Bayesian inference framework that enables online estimation of inertia time constants for all synchronous generators while simultaneously providing the full posterior probability density functions of all estimated parameters;
• Leverage an ensemble of walkers for collaborative exploration of the parameter space, effectively addressing the limitations of traditional Bayesian methods in high-dimensional settings and improving estimation accuracy in large-scale power systems;
• Introduce a parallel sampling strategy to accelerate computation, overcoming the low efficiency of conventional MCMC methods and enhancing the feasibility of online inertia estimation.

The remainder of this paper is organized as follows: Section 2 formulates the inertia estimation framework, Section 3 details the proposed MCMC Hammer methodology, Section 4 validates the approach through IEEE test cases, and Section 5 concludes the paper.

2. Mathematical Representation and Estimation Framework

This section provides a detailed mathematical description of the synchronous generator dynamics and outlines a probabilistic approach for inertia estimation.

2.1. Dynamic Modeling of Synchronous Generators

The dynamic behavior of synchronous generators is mathematically represented using the well-established second-order swing equation model, as originally formulated by Sauer and Pai [19]. This model captures the essential electromechanical dynamics of synchronous machines and is widely adopted in power system stability studies. The differential equations are expressed as follows:

$${\displaystyle \frac{d\delta }{dt}}=\omega -{\omega }_0,$$

(1)

$${\displaystyle \frac{2H}{{\omega }_0}}{\displaystyle \frac{d\omega }{dt}}=T_M-P_e-D(\omega -{\omega }_0),$$

(2)

$$P_e={\displaystyle \frac{EV}{X_d^{\prime }}}\sin (\delta -\theta ),$$

(3)

$$Q_e=-{\displaystyle \frac{V^2}{X_d^{\prime }}}+{\displaystyle \frac{EV}{X_d^{\prime }}}\cos (\delta -\theta ),$$

(4)

where $\delta$ represents the generator’s rotor angle. $\omega$ and ${\omega }_0$ denote its instantaneous and nominal rotor speeds, respectively. The parameter H corresponds to the generator’s inertia time constant, which is a key quantity to be estimated. The term D captures the damping effect associated with the generator. Mechanical torque, represented by $T_M$, is equivalent to mechanical power in per-unit form, and $P_e$ signifies the active power output of the generator. Reactive power output is denoted as $Q_e$. Variables E, V, and $\theta$ refer to the magnitude of the internal voltage, the terminal voltage magnitude, and the voltage phase angle, respectively, while $X_d^{\prime }$ indicates the transient reactance of the generator.

The increasing deployment of PMUs in modern power systems has recently drawn significant research attention to observability analysis of nonlinear dynamic models in power grids [20]. To enhance the observability of dynamic states during transient conditions, the approach proposed by [21] is adopted, utilizing generator output powers as primary measurement variables. This framework forms the basis for developing a transient simulation model specifically designed for synchronous generator analysis.

2.2. Bayesian Inference

After identifying the relevant PMU measurements, a Bayesian inference framework is established for estimating the inertia constants of synchronous generators [22]. The relationship between the measurements and the inertia parameters is formulated as [23]:

$$\mathit{z}=\mathit{f}(\mathit{H})+\mathit{\nu },$$

(5)

where $\mathit{z}=[z_1;z_2]$ represents the measurements, consisting of active power $P_e$ and reactive power $Q_e$. The vector $\mathit{H}=[H_1,H_2,\dots ,H_N]$ denotes the set of inertia time constants for all synchronous generators in the system, with N representing the number of generators. The function $\mathit{f}(·)$ maps $\mathit{H}$ to the measurements $\mathit{z}$ based on the differential-algebraic Equations (1)–(4), and $\mathit{\nu }$ represents the measurement error vector from the PMUs. The elements of $\mathit{\nu }$ are assumed to be independent and identically distributed random variables. Physically, this assumption implies that each measurement channel, such as active and reactive power outputs, is subject to its own local sensing and transmission noise, which originates from separate and uncorrelated sources such as sensor electronics, analog-to-digital conversion, and communication latency. Moreover, assuming identical distributions reflects the engineering practice that PMUs deployed across the system often operate under similar hardware configurations and calibration standards, resulting in comparable statistical noise characteristics. This modeling simplification facilitates tractable formulation of the joint noise distribution, which is expressed as ${\varphi }_{\mathit{\nu }}={\prod }_{i=1}^2{\psi }_i({\nu }_i)$.

The Bayesian inference framework enables systematic estimation of generator inertia parameters $\mathit{H}$ by incorporating prior knowledge and observed system measurements $\mathit{z}$. According to Bayes’ theorem, the posterior pdf is given by [24]:

$${\varphi }_{\mathrm{post}}(\mathit{H})\propto {\varphi }_{\mathcal{L}}(\mathit{z}|\mathit{H}){\varphi }_{\mathrm{prior}}(\mathit{H}),$$

(6)

where ${\varphi }_{\mathrm{prior}}(\mathit{H})$ denotes the prior distribution of $\mathit{H}$ and ${\varphi }_{\mathcal{L}}(\mathit{z}|\mathit{H})$ represents the likelihood function. To simplify the prior distribution, we assume that the inertia time constants of all synchronous generators are mutually independent. Under this assumption, the joint prior pdf of the parameter vector $\mathit{H}$ can be expressed as the product of marginal distributions:

$${\varphi }_{\mathrm{prior}}(\mathit{H})=\prod _{i=1}^N{\varphi }_i(H_i).$$

(7)

This choice is motivated by both computational and practical considerations. In real-world power systems, generator inertias may exhibit some correlation due to shared design, operational, or geographical factors. However, the prior in Bayesian inference does not need to capture such dependencies precisely. By assuming independence, we avoid the need to model complex joint distributions, making the inference process more tractable. In this study, each ${\varphi }_i(H_i)$ is chosen as a univariate normal distribution centered around a nominal value, with variance selected to reflect reasonable physical uncertainty. This simplified prior facilitates efficient sampling without significantly compromising posterior accuracy, especially when sufficient measurement data are available.

The likelihood function evaluates how well a candidate parameter set $\mathit{H}$ explains the observed data by comparing it with simulated outputs $\mathit{f}(\mathit{H})$ over a finite time horizon. It is constructed as:

$${\varphi }_{\mathcal{L}}(\mathit{z}|\mathit{H})=\prod _{i=1}^2{\psi }_i(z_i-f_i(\mathit{H})),$$

(8)

where each ${\psi }_i(·)$ measures the agreement between the measured and simulated trajectories for the i-th observable. To incorporate temporal dynamics, the individual likelihood terms ${\psi }_i(·)$ are expanded in logarithmic form as:

$$\log {\psi }_i(z_i-f_i(\mathit{H}))=\sum _{t=0}^{t_{\mathrm{end}}}\log {\psi }_i^t(z_i^t-f_i^t(\mathit{H})),$$

(9)

where $t_{\mathrm{end}}$ is the total simulation length and superscript t denotes the time step. Combining the above expressions, the posterior distribution in the log-domain becomes:

$$\log {\varphi }_{\mathrm{post}}(\mathit{H})\propto \sum _{i=1}^2\sum _{t=0}^{t_{\mathrm{end}}}\log {\psi }_i^t(z_i^t-f_i^t(\mathit{H}))+\sum _{i=1}^N\log {\varphi }_i(H_i).$$

(10)

Due to the inherent nonlinearity of power systems, the likelihood function ${\varphi }_{\mathcal{L}}(\mathit{z}|\mathit{H})$ often exhibits a complex structure. While the prior pdf ${\varphi }_{\mathrm{prior}}(\mathit{H})$ is typically assumed to be Gaussian, the posterior pdf ${\varphi }_{\mathrm{post}}(\mathit{H})$ may display multimodality or asymmetry, making analytical solutions challenging [25]. To overcome this, the MCMC Hammer algorithm is employed, which facilitates efficient Bayesian inference for nonlinear problems by generating a large number of samples to approximate the posterior distribution. The detailed implementation of this approach is discussed in the next section.

3. Proposed Methodology

This section presents the integration of the M-H algorithm into the Bayesian inference framework, followed by the introduction of the MCMC Hammer algorithm for enhanced inertia estimation in power systems.

3.1. M-H Algorithm

The M-H algorithm, a foundational component of the MCMC framework, constructs a Markov chain that converges to a target distribution. In Bayesian inference, this target distribution corresponds to the posterior pdf ${\varphi }_{\mathrm{post}}(\mathit{H})$, enabling the approximation of the posterior distribution.

The algorithm begins by initializing the Markov chain with an initial sample. At each iteration, a candidate sample ${\mathit{H}}_{★}$ is proposed based on the current sample ${\mathit{H}}_m$ using a proposal distribution q. During the mth iteration, the acceptance probability in logarithmic form is computed as:

$$\alpha =\log \left(\min \left\{1,{\displaystyle \frac{{\varphi }_{\mathrm{post}}({\mathit{H}}_{★})}{{\varphi }_{\mathrm{post}}({\mathit{H}}_m)}}·{\displaystyle \frac{q({\mathit{H}}_{★},{\mathit{H}}_m)}{q({\mathit{H}}_m,{\mathit{H}}_{★})}}\right\}\right),$$

(11)

where $\alpha$ determines whether the proposed sample ${\mathit{H}}_{★}$ is accepted. Specifically, ${\mathit{H}}_{m+1}$ is set to ${\mathit{H}}_{★}$ with probability $e^{\alpha }$ and retains ${\mathit{H}}_m$ with probability $1-e^{\alpha }$. After sufficient iterations, the sequence of samples converges to the target distribution ${\varphi }_{\mathrm{post}}(\mathit{H})$. The Maximum A Posteriori MAP estimation is then used to determine the inertia time constants:

$${\widehat{\mathit{H}}}_{\mathrm{MAP}}=\underset{\mathit{H}}{\arg \min }\{-\log {\varphi }_{\mathrm{post}}(\mathit{H})\},$$

(12)

where ${\widehat{\mathit{H}}}_{\mathrm{MAP}}$ represents the estimated inertia parameters. The detailed implementation of the M-H algorithm is outlined in Algorithm 1.

Algorithm 1 Bayesian inference using the M-H algorithm

1: Draw the initial sample ${\mathit{H}}_1$ from the prior pdf ${\varphi }_{\mathrm{prior}}(\mathit{H})$;   2: Compute $\log ({\varphi }_{\mathrm{post}}({\mathit{H}}_1))$;   3: for $m=1$ to M do   4:       Propose a candidate sample ${\mathit{H}}_{★}$ from q;   5:       Compute $\log ({\varphi }_{\mathrm{post}}({\mathit{H}}_{★}))$;   6:       Compute $\alpha$ using (11);   7:       Draw $u\sim U([0,1))$;   8:       if $\log (u)<\alpha$ then   9:              Accept: Set ${\mathit{H}}_{m+1}={\mathit{H}}_{★}$; 10:       else 11:              Reject: Set ${\mathit{H}}_{m+1}={\mathit{H}}_m$; 12:       end if 13: end for 14: Approximate ${\varphi }_{\mathrm{post}}(\mathit{H})$ and compute ${\widehat{\mathit{H}}}_{\mathrm{MAP}}$ using (12).

Despite its utility, the M-H algorithm faces limitations, particularly in designing an effective proposal distribution. Poorly chosen proposal distributions can result in low acceptance rates and inefficient exploration of the parameter space, especially for high-dimensional or multimodal distributions. Additionally, the algorithm requires careful tuning of hyperparameters, such as the step size in the proposal distribution, which can significantly impact performance. To address these challenges, we introduce the MCMC Hammer algorithm, which leverages an ensemble of walkers to improve sampling efficiency and robustness in its ability to tackle challenges with minimal complexity.

3.2. MCMC Hammer Algorithm

The MCMC Hammer algorithm, also known as the Affine-Invariant Ensemble Sampler [26], employs an ensemble of walkers to explore the parameter space collectively. This approach eliminates the need for a finely tuned proposal distribution and enhances exploration efficiency, even for complex posterior distributions.

The algorithm relies on a key mechanism: the stretch move update strategy. The stretch move adjusts the position of each walker based on the relative positions of other walkers, ensuring affine invariance and robustness across diverse parameter spaces. The algorithm initializes an ensemble of walkers, $\mathit{S}=\{{\mathit{H}}^1,{\mathit{H}}^2,\dots ,{\mathit{H}}^K\}$. In each iteration, the position of each walker ${\mathit{H}}^k$ is updated using the positions of other walkers in the complementary ensemble ${\mathit{S}}^{[-k]}=\{{\mathit{H}}^i\mid i\ne k\}$. The update for walker ${\mathit{H}}_m^k$ in the mth iteration is given by:

$${\mathit{H}}_{★}^k={\mathit{H}}_m^k+\zeta ·({\mathit{H}}_m^j-{\mathit{H}}_m^k),$$

(13)

where ${\mathit{H}}_{★}^k$ is the proposed walker, and $\zeta$ is a random scaling factor drawn from a distribution $g(\zeta )$. Following Algorithm 2, $g(\zeta )$ is defined as:

$$g(\zeta )\propto \left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{\zeta }}}\hfill & \mathrm{if}\zeta \in \left[{\displaystyle \frac{1}{a}},a\right],\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(14)

where a is an adjustable parameter. The symmetry condition $g({\zeta }^{-1})={\zeta }^{-1}g(\zeta )$ ensures detailed balance. A conceptual illustration of the stretch move is shown in Figure 1, which depicts two consecutive updates of a three-walker ensemble. The blue markers represent the initial positions of the walkers, the orange markers indicate their positions after the first update, and the green markers correspond to their positions after the second update. In this example, walker $H^1$ is updated using walker $H^2$, walker $H^2$ is updated using walker $H^3$, and walker $H^3$ is updated using walker $H^1$.

The acceptance probability for the proposed walker in logarithmic form is adjusted as:

$$\alpha =\log \left(\min \left\{1,{\zeta }^{N-1}{\displaystyle \frac{{\varphi }_{\mathrm{post}}({\mathit{H}}_{★}^k)}{{\varphi }_{\mathrm{post}}({\mathit{H}}_m^k)}}\right\}\right).$$

(15)

The ensemble-based approach of MCMC Hammer promotes parallelism and independence among walkers, enabling efficient exploration of the parameter space and reducing the risk of convergence to local modes. This collaborative mechanism enhances mixing and convergence, particularly in complex and high-dimensional spaces. The workflow of the MCMC Hammer algorithm is illustrated in Figure 2. In this diagram, the black arrows represent the stretch move operation, where a new candidate position is proposed for each walker based on the relative position of another walker from the complementary subset. The green arrows indicate the acceptance step, where the proposed walker is either accepted or rejected according to the acceptance rate. The detailed implementation of Bayesian inference using the MCMC Hammer algorithm is provided in Algorithm 2.

Algorithm 2 Bayesian inference using the MCMC Hammer algorithm

1: Initialize the ensemble ${\mathit{S}}_{\mathbf{0}}=\{{\mathit{H}}_0^1,{\mathit{H}}_0^2,\dots ,{\mathit{H}}_0^K\}$ from ${\varphi }_{\mathrm{prior}}(\mathit{H})$;   2: for $m=1$ to M do   3:       for $k=1$ to K do   4:             Draw $\zeta$ from (14) and ${\mathit{H}}_m^j$ ($j\ne k$) from ${\mathit{S}}_m^{[-k]}$;   5:             Compute ${\mathit{H}}_{★}^k$ using (13);   6:             Compute $\log ({\varphi }_{\mathrm{post}}({\mathit{H}}_{★}^k))$ and $\log ({\varphi }_{\mathrm{post}}({\mathit{H}}_m^k))$ by (10);   7:             Compute $\alpha$ using (15);   8:             Draw $u\sim U([0,1))$;   9:             if $\log (u)<\alpha$ then 10:                     Accept: Set ${\mathit{H}}_{m+1}^k={\mathit{H}}_{★}^k$; 11:             else 12:                     Reject: Set ${\mathit{H}}_{m+1}^k={\mathit{H}}_m^k$; 13:             end if 14:       end for 15:       Update the ensemble ${\mathit{S}}_{\mathit{m}+\mathbf{1}}=\{{\mathit{H}}_{m+1}^1,{\mathit{H}}_{m+1}^2,\dots ,{\mathit{H}}_{m+1}^K\}$; 16: end for 17: Approximate ${\varphi }_{\mathrm{post}}(\mathit{H})$ and compute ${\widehat{\mathit{H}}}_{\mathrm{MAP}}$ using (12).

To enhance computational efficiency for large-scale power systems, we propose a parallelized implementation of the MCMC Hammer algorithm. This extension leverages the parallel stretch move mechanism, which preserves detailed balance while enabling efficient distributed computing.

The parallel implementation divides the ensemble of walkers $\mathit{S}=\{{\mathit{H}}^1,{\mathit{H}}^2,\dots ,{\mathit{H}}^K\}$ into two complementary subsets [27]:

$${\mathit{S}}^{(0)}=\{{\mathit{H}}^k\mid k=1,\dots ,K/2\}, {\mathit{S}}^{(1)}=\{{\mathit{H}}^k\mid k=K/2+1,\dots ,K\}.$$

(16)

The algorithm alternates between updating ${\mathit{S}}^{(0)}$ and ${\mathit{S}}^{(1)}$ in a two-step process: all walkers in ${\mathit{S}}^{(i)}$ are updated simultaneously using only walkers from the complementary subset ${\mathit{S}}^{(\sim i)}$; updated positions from ${\mathit{S}}^{(i)}$ are used to subsequently update ${\mathit{S}}^{(\sim i)}$. The flowchart of the parallel MCMC Hammer algorithm is shown in Figure 3, where the black arrows represent the stretch move operation and the green arrows indicate the acceptance step.

The parallel workflow is formalized in Algorithm 3, which preserves detailed balance while enabling parallel computation of walker updates.

Algorithm 3 Bayesian inference using parallel MCMC Hammer

1: Partition ${\mathit{S}}_0$ into ${\mathit{S}}_0^{(0)}$ and ${\mathit{S}}_0^{(1)}$   2: for $m=1$ to M do   3:       for $i\in \{0,1\}$ do   4:             for all walker k in ${\mathit{S}}^{(i)}$ do   5:                   Draw a complementary walker ${\mathit{H}}^j$ from ${\mathit{S}}^{(\sim i)}$;   6:                   Draw $\zeta \sim g(\zeta )$    (Equation (14));   7:                   Propose ${\mathit{H}}_{★}^k={\mathit{H}}^j+\zeta [{\mathit{H}}_m^k-{\mathit{H}}^j]$;   8:                   Compute $\log ({\varphi }_{\mathrm{post}}({\mathit{H}}_{★}^k))$ and $\log ({\varphi }_{\mathrm{post}}({\mathit{H}}_m^k))$ by (10);   9:                   Compute $\alpha$ using (15); 10:                   Draw $u\sim U([0,1))$ 11:                   if $\log (u)<\alpha$ then 12:                          Accept: Set ${\mathit{H}}_{m+1}^k={\mathit{H}}_{★}^k$; 13:                   else 14:                          Reject: Set ${\mathit{H}}_{m+1}^k={\mathit{H}}_m^k$; 15:                   end if 16:             end for 17:       end for 18: end for

4. Results and Discussions

This section presents the experimental validation of the proposed algorithm’s capability for estimating generator inertia, using MATLAB 2022b and a maximum of 12 parallel MATLAB workers. Performance comparisons are conducted against the conventional M-H algorithm on both the IEEE 39-bus and IEEE 68-bus benchmark systems. The ground truth values of the generator inertia time constants are taken directly from the dynamic models specified in the standard IEEE 39-bus and 68-bus test systems. The simulation employs a time step of $0.01$ s and simulates a three-phase line-to-ground fault at 1 s after simulation initiation. The fault is cleared after a brief duration, and the total simulation time is 6 s. This setup provides sufficient post-disturbance dynamic response for inertia estimation. For both the M-H and MCMC Hammer algorithms, the initial $30\%$ of samples are discarded as burn-in to ensure convergence toward the target posterior distribution. All random processes in the experiment, including initialization, walker selection, and proposal generation, are performed without setting a random seed. This allows assessment of algorithm robustness under purely stochastic conditions. All experiments are executed on a workstation equipped with an Intel Core i7-13790 processor (Intel Corporation, Santa Clara, CA, USA), 32 GB RAM.

The comparative analysis focuses on two key performance metrics: estimation accuracy relative to known system parameters and computational efficiency in terms of execution time. The parameter a in (14) is set to 2. The PMU measurement errors, $\mathit{\nu }$, are assumed to follow mutually independent Gaussian distributions with means of 0 and a standard deviation of $0.001$. We assume that the prior distributions of the inertia time constants follow Gaussian distributions, with means deviating by approximately $5\%$ from the actual inertia time constants, and the standard deviation set to $10\%$ of the corresponding means. In the mth iteration of the M-H algorithm, the proposal distribution q is set as a Gaussian distribution with means of ${\mathit{H}}_m$ and a standard deviation of $5\%\ast {\mathit{H}}_m$.

4.1. Estimation in IEEE 39-Bus System

In this part of the study, we apply the proposed algorithms, including the MCMC Hammer algorithm, the parallel MCMC Hammer algorithm, and the M-H algorithm, to the IEEE 39-bus system. The fault occurs on Bus 15 and Bus 16. Both algorithms use a total of 50,000 samples. Specifically, for the M-H algorithm, $M=\mathrm{50,000}$, and for the MCMC Hammer algorithm, $M\times K=1000\times 50=\mathrm{50,000}$.

The estimation results of the three algorithms are presented in

Table 1. Estimation results and errors of the M-H and MCMC Hammer algorithm in IEEE 39-bus system.

	Prior	True	M-H		MCMC Hammer		Parallel Hammer
			MAP	Error (%)	MAP	Error (%)	MAP	Error (%)
$H_1(s)$	525	500	498.27	0.35%	501.31	0.26%	504.31	0.86%
$H_2(s)$	29	30.3	30.70	1.31%	30.21	0.31%	30.27	0.11%
$H_3(s)$	34	35.8	35.69	0.31%	35.86	0.16%	35.82	0.05%
$H_4(s)$	27.5	28.6	29.03	1.50%	28.56	0.15%	28.61	0.02%
$H_5(s)$	24.5	26	25.86	0.52%	26.01	0.04%	25.99	0.04%
$H_6(s)$	36	34.8	34.64	0.47%	34.88	0.22%	34.85	0.14%
$H_7(s)$	28	26.4	26.88	1.81%	26.42	0.09%	26.34	0.23%
$H_8(s)$	25	24.3	24.21	0.36%	24.23	0.30%	24.27	0.11%
$H_9(s)$	36	34.5	34.44	0.18%	34.50	0.00%	34.41	0.27%
$H_{10}(s)$	40	42	41.62	0.91%	42.00	0.01%	41.98	0.04%

, which shows the MAP points and estimation errors of the inertia time constants for each generator obtained by the different methods. In terms of computational efficiency, the parallel MCMC Hammer algorithm achieves the shortest computation time of 142.61 s, demonstrating significant advantages in processing speed. In contrast, both the standard MCMC Hammer and M-H algorithms require considerably longer computation times of 301.97 s and 297.02 s, respectively, showing a nearly identical computational cost. Furthermore, among the 35,000 collected samples, only 3478 unique samples were generated by the M-H algorithm, indicating a high level of sample redundancy and poor exploration efficiency. In comparison, the standard MCMC Hammer and its parallel version produced 18,408 and 18,354 unique samples, respectively, highlighting their superior mixing behavior and sampling diversity.

The experimental findings clearly demonstrate the effectiveness of all three algorithms in estimating the inertia time constants of synchronous generators within the IEEE 39-bus test system. Specifically, the M-H algorithm yields an average estimation error of $0.77\%$ across the ten generators, with the maximum and minimum errors reaching $1.81\%$ and $0.18\%$, respectively. In comparison, the MCMC Hammer algorithm achieves significantly improved accuracy, with an average error of only $0.15\%$. Its worst-case error is limited to $0.22\%$, while the best-case error is as low as $0.004\%$, indicating a high level of precision. The parallel implementation of the MCMC Hammer algorithm also maintains strong performance, producing an average error of $0.19\%$, with maximum and minimum errors of $0.86\%$ and $0.02\%$, respectively.

Beyond point accuracy, the reliability of the estimation results is also critically important. As shown in

Table 2. Stds and $95\%$ CIs of the M-H and MCMC Hammer algorithm in IEEE 39-bus system.

	M-H		MCMC Hammer		Parallel Hammer
	std	95% CI	std	95% CI	std	95% CI
$H_1(s)$	40.59	[444.98, 598.39]	6.20	[488.75, 512.94]	6.49	[490.47, 516.40]
$H_2(s)$	1.27	[27.40, 32.35]	0.14	[29.97, 30.52]	0.12	[30.04, 30.53]
$H_3(s)$	1.10	[33.51, 37.76]	0.13	[35.49, 35.99]	0.11	[35.57, 36.01]
$H_4(s)$	1.42	[26.26, 31.79]	0.17	[28.29, 28.97]	0.17	[28.27, 28.94]
$H_5(s)$	1.45	[22.63, 28.28]	0.19	[25.60, 26.33]	0.21	[25.51, 26.34]
$H_6(s)$	1.76	[31.54, 38.43]	0.22	[34.32, 35.19]	0.20	[34.39, 35.20]
$H_7(s)$	1.79	[23.57, 30.55]	0.26	[26.00, 27.04]	0.23	[26.02, 26.92]
$H_8(s)$	2.28	[20.62, 29.75]	0.65	[23.04, 25.55]	0.64	[22.95, 25.49]
$H_9(s)$	2.18	[30.78, 39.13]	0.33	[34.05, 35.36]	0.27	[34.05, 35.12]
$H_{10}(s)$	1.02	[39.79, 43.89]	0.09	[41.83, 42.18]	0.10	[41.81, 42.21]

, the M-H algorithm consistently exhibits greater standard deviations and wider credible intervals, suggesting higher variability and reduced confidence in the inferred inertia parameters. In contrast, both versions of the MCMC Hammer algorithm produce more concentrated posterior distributions, reflecting greater robustness and reliability in their estimates.

The superior performance of the MCMC Hammer algorithms can be attributed to their advanced sampling mechanisms. The ensemble sampling approach enables more thorough exploration of the parameter space, leading to more accurate estimation of the posterior distribution. The remarkable efficiency of the parallel implementation stems from its ability to distribute computational load across multiple processors while maintaining sampling effectiveness. This makes it particularly suitable for real-time applications where both accuracy and speed are crucial. Although the absolute differences in estimation accuracy may appear minor, the combined advantages of precision and computational efficiency position the parallel MCMC Hammer as the most practical choice for modern power system operation.

The Bayesian inference framework can also provide their complete posterior distributions of the inertia time constants, as demonstrated in Figure 4. The figure clearly visualizes the probability density functions of the posterior distributions for each generator’s inertia time constant. The varying widths of the distributions reflect different confidence levels in the estimation between generators, providing useful information on the reliability of the values obtained. The results reveal an important finding regarding the distribution characteristics: While the prior distributions of inertia time constants were initially assumed to be Gaussian, the posterior distributions obtained through the MCMC Hammer algorithm demonstrate significantly non-Gaussian characteristics.

This observation fundamentally validates the necessity of employing advanced random sampling techniques to approximate rather than relying on simplified distributional assumptions. The divergence between the assumed Gaussian priors and the complex, often skewed or multimodal posterior distributions underscores several critical aspects of the estimation problem. First, it highlights the inherent nonlinearity in the relationship between power system dynamics and inertia parameters, which cannot be adequately captured through conventional Gaussian approximations. Second, it demonstrates how measurement data and system constraints can fundamentally reshape the parameter distributions in ways that simple analytical forms cannot represent. Third, the results prove that the MCMC Hammer algorithm successfully captures these complex distributional features through its sophisticated sampling mechanism, which would be impossible under traditional Gaussian assumptions.

4.2. Estimation in IEEE 68-Bus System

In this part, we further compare the performance of the algorithms on the IEEE 68-bus system. In this case, the fault is applied between Bus 32 and Bus 33. To address the increased dimensionality, the total number of samples used by both algorithms is set to 100,000. Specifically, for the M-H algorithm, we set M = 100,000, while for the MCMC Hammer algorithm, we set $M\times K=1000\times 100$ = 100,000.

The experimental results presented in

Table 3. Estimation results and errors of the M-H and MCMC Hammer algorithm in IEEE 68-bus system.

	Prior	True	M-H		MCMC Hammer		Parallel Hammer
			MAP	Error (%)	MAP	Error (%)	MAP	Error (%)
$H_1(s)$	40	42	40.19	4.32%	42.06	0.14%	42.04	0.09%
$H_2(s)$	28.5	30.2	30.25	0.18%	30.22	0.08%	30.13	0.23%
$H_3(s)$	34	35.8	35.74	0.17%	35.82	0.05%	35.84	0.11%
$H_4(s)$	30	28.6	29.96	4.74%	28.64	0.13%	28.71	0.39%
$H_5(s)$	24.5	26	26.18	0.70%	26.08	0.32%	25.83	0.66%
$H_6(s)$	36.5	34.8	35.73	2.67%	34.97	0.50%	34.68	0.34%
$H_7(s)$	25	26.4	25.70	2.64%	26.33	0.27%	26.49	0.32%
$H_8(s)$	25.5	24.3	24.69	1.60%	24.24	0.24%	24.35	0.20%
$H_9(s)$	36	34.5	34.56	0.19%	34.60	0.29%	34.54	0.12%
$H_{10}(s)$	29.5	31	31.06	0.18%	31.02	0.05%	30.98	0.08%
$H_{11}(s)$	30	28.2	28.18	0.07%	28.20	0.01%	28.20	0.00%
$H_{12}(s)$	88	92.3	92.50	0.21%	92.30	0.00%	92.29	0.01%
$H_{13}(s)$	260	248	243.91	1.65%	247.26	0.30%	248.12	0.05%
$H_{14}(s)$	280	300	286.75	4.42%	301.12	0.37%	299.96	0.01%
$H_{15}(s)$	320	300	320.45	6.82%	300.90	0.30%	302.09	0.70%
$H_{16}(s)$	200	225	204.15	9.26%	223.01	0.88%	222.93	0.92%

and

Table 4. Stds and $95\%$ CIs of the M-H and MCMC Hammer algorithm in IEEE 68-bus system.

	M-H		MCMC Hammer		Parallel Hammer
	std	95% CI	std	95% CI	std	95% CI
$H_1(s)$	2.49	[36.45, 45.87]	0.43	[40.24, 42.54]	0.36	[40.23, 42.65]
$H_2(s)$	1.43	[26.79, 32.39]	0.15	[29.53, 30.36]	0.16	[29.82, 30.43]
$H_3(s)$	1.12	[33.70, 38.14]	0.13	[35.61, 36.16]	0.12	[35.67, 36.14]
$H_4(s)$	2.07	[25.35, 32.91]	0.56	[28.25, 30.46]	0.75	[28.49, 30.88]
$H_5(s)$	1.90	[20.83, 28.41]	0.58	[25.01, 26.28]	0.55	[24.28, 26.19]
$H_6(s)$	1.94	[31.97, 39.59]	0.54	[34.51, 36.19]	0.31	[34.59, 35.79]
$H_7(s)$	1.84	[22.05, 29.11]	0.57	[25.55, 26.71]	0.37	[25.28, 26.65]
$H_8(s)$	1.01	[22.56, 26.47]	0.14	[24.07, 24.74]	0.20	[24.09, 24.87]
$H_9(s)$	1.22	[32.30, 37.02]	0.21	[34.19, 34.80]	0.22	[34.30, 35.13]
$H_{10}(s)$	0.26	[30.49, 31.51]	0.03	[30.96, 31.07]	0.04	[30.95, 31.10]
$H_{11}(s)$	0.10	[28.00, 28.41]	0.01	[28.17, 28.22]	0.01	[28.17, 28.21]
$H_{12}(s)$	0.50	[91.31, 93.30]	0.06	[92.23, 92.44]	0.05	[92.21, 92.43]
$H_{13}(s)$	9.28	[230.82, 266.89]	1.19	[245.41, 249.77]	1.13	[245.36, 249.83]
$H_{14}(s)$	20.44	[252.18, 330.96]	6.01	[280.13, 303.98]	3.82	[293.12, 308.56]
$H_{15}(s)$	22.25	[270.07, 350.64]	7.93	[293.45, 337.00]	6.33	[289.28, 313.20]
$H_{16}(s)$	15.13	[184.07, 241.20]	6.56	[210.45, 229.77]	4.89	[210.67, 228.67]

provide a comprehensive comparison of the three algorithms’ performance in estimating generator inertia time constants for the IEEE 68-bus system. Although all three methods operate with the same total number of samples, namely 100,000, the M-H algorithm exhibits substantial estimation errors across multiple generators. This performance disparity highlights its limitations when applied to high-dimensional parameter spaces, where convergence efficiency and sampling accuracy become critical. In contrast, both the standard MCMC Hammer algorithm and its parallel implementation consistently demonstrate superior estimation performance across all generators, maintaining errors below $1\%$ throughout the IEEE 68-bus test system.

Specifically, the M-H algorithm yields an average inertia estimation error of $2.49\%$ for the 16 generators, with individual errors ranging from a minimum of $0.07\%$ to a maximum of $9.26\%$. The standard MCMC Hammer algorithm significantly improves upon this performance, reducing the average error to just $0.25\%$. Its most accurate estimate deviates by only $0.002\%$ from the true value, while even the largest deviation does not exceed $0.88\%$. The parallel MCMC Hammer algorithm delivers comparably strong results, achieving an average error of $0.27\%$, with errors ranging from $0.004\%$ to $0.92\%$.

In addition to estimation accuracy, uncertainty quantification results further underscore the advantages of the proposed algorithms. The M-H algorithm exhibits noticeably larger standard deviations and wider credible intervals, indicating high uncertainty in its estimates. In contrast, both the standard and parallel MCMC Hammer algorithms produce significantly narrower credible intervals. These tighter intervals suggest that the sample distributions are more concentrated around the true values, meaning the inferred inertia parameters have higher statistical confidence. This reflects the robustness and reliability of the MCMC Hammer-based methods, particularly when precise and trustworthy parameter estimation is required for system monitoring and control.

From a computational perspective, the proposed parallel MCMC Hammer algorithm offers a clear advantage. It completes the entire estimation process in $364.49$ s, considerably faster than the standard MCMC Hammer, which takes $856.31$ s, and the M-H algorithm, which requires $843.17$ s. The parallel implementation, therefore, not only enhances estimation accuracy but also improves practical applicability for large-scale and real-time power system analysis. In addition, out of the 70,000 collected samples, the M-H algorithm yielded only 9121 unique samples, reflecting significant redundancy and poor mixing performance. In contrast, the standard MCMC Hammer and its parallel variant produced 30,952 and 30,939 unique samples, respectively. This demonstrates the proposed method’s superior ability to efficiently explore high-dimensional posterior distributions and generate diverse, informative samples.

The posterior distributions of the generator inertia constants in the IEEE 68-bus system, obtained using the parallel MCMC Hammer algorithm, are shown in Figure 5. An important observation from the posterior analysis is the presence of multimodal distributions for certain inertia parameters. To quantitatively validate and characterize this multimodality, the Hartigan Dip Test is applied to the posterior samples of each generator’s inertia. This statistical test evaluates the departure of a given empirical distribution from unimodality by computing the dip statistic, denoted as $\gamma$, which represents the maximum vertical distance between the empirical distribution function and the best-fitting unimodal distribution. Typically, a $\gamma$ value greater than $0.05$ indicates significant evidence against unimodality. The computed dip statistics for all generators are summarized in

Table 5. $\gamma$ of generators’ posterior pdfs in IEEE 68-bus system.

No.	$H_1(s)$	$H_2(s)$	$H_3(s)$	$H_4(s)$	$H_5(s)$	$H_6(s)$	$H_7(s)$	$H_8(s)$
$\gamma$	0.1107	0.0226	0.0269	0.0318	0.0244	0.0168	0.0371	0.0222
No.	$H_9(s)$	$H_{10}(s)$	$H_{11}(s)$	$H_{12}(s)$	$H_{13}(s)$	$H_{14}(s)$	$H_{15}(s)$	$H_{16}(s)$
$\gamma$	0.0322	0.0178	0.0283	0.0339	0.0231	0.0333	0.0681	0.0289

. Notably, generators 1 and 15 exhibit dip values well above the $0.05$ threshold, confirming the presence of multimodal features in their posterior distributions. This observation reinforces the inherently non-Gaussian nature of the inferred parameter distributions, which poses substantial challenges for the M-H algorithm.

The traditional M-H algorithm demonstrates fundamental limitations when applied to such complex estimation problems. The degraded performance of the M-H algorithm in this higher-dimensional system originates from its single-chain sampling mechanism, which struggles to adequately explore the complex posterior distributions of multiple coupled inertia parameters. The method’s sensitivity to manually tuned proposal distributions becomes increasingly problematic as system dimensionality grows, often resulting in inefficient exploration of the parameter space and convergence to suboptimal solutions.

The MCMC Hammer algorithm addresses these challenges through its innovative ensemble-based sampling approach. The method’s stretch move mechanism generates informed, geometry-adaptive proposals by leveraging the relative positions of walkers in the ensemble. This strategy provides several advantages for power system parameter estimation. First, the affine-invariant property ensures consistent performance regardless of parameter scaling or correlation structure. Second, the ensemble automatically adapts to the local geometry of the posterior distribution, enabling efficient exploration of anisotropic parameter spaces. Third, the parallel implementation maintains these benefits while significantly improving computational efficiency through distributed walker updates.

These results conclusively demonstrate the superiority of the parallel MCMC Hammer algorithm for inertia estimation in large-scale power systems. By combining high estimation accuracy with significant computational efficiency gains, the proposed method addresses two of the most critical challenges in modern power system analysis. The algorithm’s ensemble-based sampling strategy enables effective exploration of complex, high-dimensional parameter spaces, ensuring accurate identification of generator inertia time constants even in systems with heterogeneous dynamic behaviors. Furthermore, the parallelized implementation substantially reduces computational time without sacrificing estimation quality, making the method particularly well-suited for real-time or near-real-time applications. This advantage is especially important for modern power grids with increasing penetration of renewable energy sources, where rapid and precise inertia estimation is vital for maintaining frequency stability and ensuring system reliability. The consistent performance observed across generators with varying inertia values further validates the robustness and versatility of the proposed approach. These qualities position the parallel MCMC Hammer algorithm as a highly promising and preferred solution for parameter estimation tasks in large-scale, complex power networks, offering a strong foundation for future developments in dynamic system monitoring and control.

5. Conclusions

Based on the results and analysis presented above, it is clear that the parallel MCMC Hammer algorithm represents a significant advancement in the field of estimation of power system parameters. By efficiently combining ensemble-based Bayesian inference with parallel computation, the proposed method successfully overcomes the limitations of traditional approaches such as the M-H algorithm, particularly in high-dimensional and computationally demanding settings. Extensive simulation studies on the IEEE 39-bus and 68-bus systems confirm the superior performance of the proposed MCMC Hammer algorithm and its parallel implementation in estimating generator inertia. Across all generators, both variants achieve estimation errors consistently below $1\%$, with average errors as low as $0.15\%$ in the 39-bus system and $0.25\%$ in the more complex 68-bus system. In contrast, the traditional M-H algorithm exhibits significantly larger deviations, with average estimation errors exceeding $2\%$ and maximum errors reaching up to 9.26% in the 68-bus case. Moreover, the parallel MCMC Hammer reduces computational time by over $50\%$ compared to both the standard Hammer and M-H methods. This acceleration is achieved without compromising accuracy or robustness. Furthermore, the proposed method produces posterior distributions with narrower confidence intervals, indicating greater certainty and stability in the estimated values. These results demonstrate that the parallel MCMC Hammer not only enables near real-time inertia estimation but also maintains high precision and reliability across generators with diverse dynamic behaviors. The method is therefore well-suited for modern power systems with high penetration of renewable generation, where accurate, fast, and scalable inertia estimation is critical for maintaining grid stability.

Future work will focus on extending the proposed methodology to accommodate more complex system dynamics. A key direction is to address potential challenges arising from real-world PMU data, which may exhibit delays, losses, or temporal sparsity due to communication constraints or limited sampling resolution. To improve robustness under such conditions, we plan to explore data compensation and imputation strategies that can maintain estimation accuracy even with incomplete or irregular measurements. Although the current method relies on measurements from the terminal buses of all generators, it may not perform well when data from certain generator buses are missing. To overcome this, future work will investigate dimension-reduction and decoupling techniques, enabling localized, one-to-one estimation of a generator’s inertia based solely on its own measurements. This would enhance the method’s scalability and applicability in partially observable systems.

At present, the Bayesian inference framework employs the widely used second-order classical generator model, which captures dominant electromechanical behavior. Moving forward, we aim to extend the Bayesian inference formulation to support higher-order generator models, such as fourth-order or ninth-order synchronous machine representations. This will allow more accurate inertia estimation under a wider range of operational scenarios and model complexities. Lastly, as modern power systems increasingly integrate converter-interfaced renewable resources, the dynamics of synthetic or emulated inertia—introduced via fast control loops rather than physical rotating masses—must also be considered. Future work will adapt the proposed methodology for hybrid systems, combining synchronous machine inertia and converter-based virtual inertia, thereby enhancing its relevance to next-generation low-inertia grids.