Hybrid Load Perturbation in AC/DC Grids

• Hybrid load perturbation is a method that quantifies stochastic fluctuations in both electrical load and PV generation in interconnected AC/DC power systems.
• It employs cumulant-based probabilistic load flow techniques to analytically propagate uncertainties and obtain detailed PDF/CDF profiles of grid parameters.
• This approach achieves high computational efficiency and sub-percent error accuracy, making it practical for managing renewable-infused power networks.

Hybrid load perturbation refers to the treatment and analysis of stochastic fluctuations in electrical loads and renewable energy injection within interconnected AC/DC power systems, particularly those incorporating both alternating current (AC) and voltage sourced converter high voltage direct current (VSC-HVDC) technology linked with photovoltaic (PV) generation. The probabilistic modeling of such hybrid systems, which contain both traditional and converter-based nodes, addresses the uncertainty and correlations inherent in modern power networks. Cumulant-based probabilistic load flow (PLF-CM) methods efficiently propagate input uncertainties—due to both load and PV generation—through complex hybrid grids, yielding detailed statistical characterizations such as the probability density function (PDF) and cumulative distribution function (CDF) of nodal voltages and line flows (Sun et al., 2022).

1. Hybrid AC/DC System Architecture and Control Modes

Hybrid networks consist of conventional AC transmission grids electrically coupled, via VSC stations, to DC networks that may be configured point-to-point or as multiterminal DC islands. The steady-state equations for DC-side nodal behavior incorporate both injected currents and active power balances: $$P_{dk} = U_{dk} \sum_{j=1}^{n_{dc}} G_{dk,j} (U_{dk} - U_{dj})$$ where $P_{dk}$ is the net power injection at DC node $k$ and $G_{dk,j}$ is the DC conductance.

Grid interconnection leverages VSCs configured in distinct control modes:

• Master–slave control: One converter maintains the DC voltage, others regulate power and AC-side quantities.
• Droop control: All converters share voltage regulation following a droop characteristic,

$$(U_{dk} - U_{dk}^{\text{ref}}) + k_{\text{droop}}(P_{dk} - P_{dk}^{\text{ref}}) = 0$$

This structure allows flexible assignment of PV, PQ, and voltage-holding roles, aligning with component technical capabilities and system operational objectives.

2. Deterministic Power Flow Calculation for Coupled AC/DC Systems

The unified iterative method is employed to solve the steady-state power flows, integrating AC and DC domains via block-augmented Newton-Raphson procedures. The core set of mismatch equations includes AC nodal power mismatches (incorporating converter injections) and DC-side power mismatch, compactly written as: $$\begin{bmatrix} \Delta \mathbf{P} \ \Delta \mathbf{Q} \ \Delta \mathbf{P}_{dc} \end{bmatrix} = J_{ac/dc} \begin{bmatrix} \Delta \boldsymbol\theta \ \Delta \mathbf{U} \ \Delta \mathbf{U}_{dc} \end{bmatrix}$$ Here, the Jacobian $J_{ac/dc}$ includes off-diagonal blocks accounting for converter-mediated coupling. The system is classified at each node according to Table 1 in the paper, designating PQ, PV, or voltage-controlled nodes as per converter settings.

3. Probabilistic Modeling of Loads and PV Generation

Stochastic inputs are characterized using empirically justified probability distributions:

• Load active/reactive powers $(P_L, Q_L)$: Treated as normal distributions,

$$P_L \sim N(\mu_P, \sigma_P^2), \quad Q_L \sim N(\mu_Q, \sigma_Q^2)$$

• PV active output $P_M$: Modeled as a Beta distribution on $[0, P_M^{\max}]$, with shape parameters derived from empirical mean and variance.

Correlation among inputs—such as spatially or temporally linked load and PV variations—is handled via Cholesky factorization of the empirical correlation matrix $C_Z$. Whitening transforms the correlated sources to independent variables: $$\mathbf{Y} = G^{-1} \mathbf{Z}; \quad \mathrm{Cov}(\mathbf{Y}) = I$$ with $G$ the Cholesky factor of $C_Z$.

4. Cumulant Propagation and High-Order Moment Calculation

The cumulant-based framework underpins the efficient propagation of probabilistic perturbations through the network:

• Cumulant calculation: For input vectors, raw moments $m_k = E[X^k]$ yield cumulants recursively,

$$\kappa_v = m_v - \sum_{j=1}^{v-1}\binom{v-1}{j-1} \kappa_j m_{v-j}, \quad v \ge 2$$

• Distribution-specific cumulant rules: PV (Beta) and load (Normal) have closed-form cumulants. When unavailable, Monte Carlo sample-based moment estimates are used.
• Additivity: For independent sources, cumulants add directly: $$\kappa_k(W_1 + W_2) = \kappa_k(W_1) + \kappa_k(W_2)$$.
• Linear transformation: Cumulants propagate as $$\kappa_1(Y) = a \kappa_1(X) + b,\; \kappa_{k>1}(Y) = a^k \kappa_k(X)$$ for $Y = aX + b$.

This approach replaces computationally intensive multi-dimensional convolutions with algebraic cumulant operations, streamlining uncertainty quantification.

5. Analytical PDF/CDF Construction via Gram–Charlier Series

The cumulative effect of random load and generation perturbations on network state variables is expressed analytically through Gram–Charlier (GC) expansions: $$f(x) = \phi(x)\left[1 + \frac{g_3}{6}H_3(x) + \frac{g_4}{24}H_4(x) + \cdots \right]$$ where $x = (X - \mu) / \sigma$ is standardized, $g_v = \kappa_v / \sigma^v$ are normalized cumulants, and $H_n(x)$ are Hermite polynomials. The closed-form CDF is analogous,

$$F(x) = \Phi(x) + \phi(x) \left[ \frac{g_3}{6} H_2(x) + \frac{g_4}{24} H_3(x) + \cdots \right]$$

This analytical representation enables direct extraction of mean, variance, skewness, and tail probabilities for physical state variables (nodal voltages, line flows) under hybrid load perturbation.

6. Computational Workflow and Algorithmic Steps

The cumulant-based PLF procedure unfolds as follows:

1. System input: Load/PV distributions, control modes, correlations.
2. Cumulant computation: Closed-form for analytical distributions; else, Monte Carlo estimation.
3. Correlation handling: Cholesky-based whitening of correlated sources.
4. Deterministic power flow: Nominal hybrid AC/DC solution yields sensitivities.
5. Sensitivity propagation: Input cumulants mapped to state cumulants via linear transformations.
6. Distribution construction: State PDFs/CDFs expanded via GC series.
7. Metric extraction: Calculation of mean, variance, and reliability indices.

This hybrid approach avoids large-sample Monte Carlo simulation of the full system, achieving sub-percent errors in PDF/CDF estimation with significantly reduced computational cost (Sun et al., 2022).

7. Significance and Efficiency of the Cumulant Approach

The cumulant-based PLF-CM enables rapid, accurate analysis of stochastic load and PV effects in AC/DC hybrid grids:

• Computational tractability: Reduces numerical burden to primarily linear algebra and series expansions, requiring a single deterministic solve for sensitivity matrices.
• High-order statistics: Skewness and kurtosis from non-Gaussian PV generation and other renewables are preserved.
• Correlation fidelity: Exact treatment of input dependencies via whitening transformation ensures correct risk representation.
• Practical accuracy: Validation on IEEE 34-bus and 57-bus systems confirms sub-percent error in output PDFs/CDFs relative to Monte Carlo benchmarks, with orders-of-magnitude speedup.

A plausible implication is that cumulant-based PLF-CM constitutes a robust solution for modern, uncertainty-prone hybrid grid analysis, directly supporting probabilistic reliability assessment and operational decision-making in large-scale, renewable-integrated networks (Sun et al., 2022).