Data-Driven Probabilistic Power Flow for Energy-Storage Planning Considering Interconnected Grids

Abstract

As renewable energy penetration increases, the volatility and uncertainty of photovoltaic generation and load demand pose significant challenges to power-system stability. This paper proposes a data-driven probabilistic load-flow method that employs a Gaussian mixture model (GMM) to model uncertainties in photovoltaic generation and load demand. Cumulative quantity analysis is then applied to conduct probabilistic load-flow studies, quantifying the impact of these uncertainties on the power system. Building upon this foundation, a two-layer optimization model is constructed to optimize the siting, capacity, and operational strategies of energy storage systems. Experimental results demonstrate that this method effectively reduces the probability of voltage-limit violations, ensures the reliability of supply–demand balance, and enhances system stability and reliability even under fluctuating PV generation and load-demand conditions.

1. Introduction

With profound transformations in the global energy landscape, the adoption of clean renewable energy—particularly wind and solar photovoltaic (PV) power—has become increasingly widespread [1]. As the world’s largest energy producer and consumer, China is actively promoting renewable energy development to align with its “dual-carbon” goals [2]. The growing integration of renewable energy into power systems has led to significant changes in traditional power architectures [3]. However, unlike conventional thermal power generation, wind and solar outputs exhibit significant intermittency, randomness, volatility, and high uncertainty. Their generation is strongly influenced by meteorological conditions, making accurate forecasting challenging [4]. PV output may plummet sharply during cloudy weather, nighttime, or seasonal variations [5]. This uncertainty poses significant challenges to power system stability and reliability and increases the risk of supply–demand imbalances [6].

Meanwhile, complexity on the demand side is intensifying. With the integration of emerging loads such as electric vehicles, the demand side of power systems also exhibits complex stochastic fluctuations [7]. This dual uncertainty from both generation and demand makes maintaining supply–demand balance during real-time operation highly challenging [8,9,10]. Traditional approaches to power-system planning and operation largely rely on deterministic models and simplified assumptions, often overlooking complex dependence structures between PV output and load demand while failing to effectively quantify inherent uncertainties [11,12]. To more accurately assess these risks, academia has proposed various probabilistic modeling approaches. Classical Monte Carlo simulation (MCS) is widely regarded as a benchmark for probabilistic power flow and risk assessment due to its minimal distributional assumptions and its ability to handle strong nonlinearities [13]. However, achieving stable estimates of tail-related metrics typically requires a large number of samples, resulting in a significant computational burden—especially for large-scale systems and long-horizon planning tasks, where direct coupling with iterative optimization solvers becomes challenging [13]. Point estimate methods (PEMs) reduce the number of power-flow evaluations by matching low-order moments using a small set of representative points, thereby improving computational efficiency [14,15]. Nevertheless, when input uncertainties exhibit heavy tails, multimodality, or strong skewness, low-order moment approximations can lead to amplified errors, and quantile approximations such as the Cornish–Fisher expansion may suffer from non-negligible truncation errors when distributions deviate substantially from normality [16].

To overcome these limitations, data-driven scenario generation and machine learning (ML) techniques have become research hotspots. Kernel density estimation (KDE) can nonparametrically approximate complex distributions for scenario construction; however, its performance heavily depends on bandwidth selection, and it is susceptible to the curse of dimensionality with high-dimensional inputs or complex correlations, leading to unstable generalization [17]. Generative adversarial networks (GANs) can fit data distributions through adversarial learning and have been utilized to generate realistic renewable energy scenarios [18]. Improved variants such as Wasserstein GAN (WGAN) can partially mitigate training instability and improve gradient behavior [19]. Even so, GAN-based methods may still suffer from mode collapse and convergence sensitivity, potentially resulting in insufficient coverage of extreme tail risks or neglect of critical patterns in generated samples [20]. Furthermore, deep reinforcement learning (DRL) and end-to-end learned policies demonstrate strong adaptability in online control and scheduling applications [21,22,23]. However, these approaches often lack physical interpretability, and their performance may degrade under out-of-distribution (OOD) conditions caused by sudden disturbances or distribution shifts, which can translate into difficult-to-quantify operational risks [24]. Consequently, significant fluctuations in PV output or abrupt demand changes may still trigger security issues such as voltage violations [25]. Fundamentally, these problems arise because the system fails to maintain a reliable probabilistic balance between supply and demand under uncertainty, i.e., active and reactive power generation cannot match consumption at a specified probability level, thereby compromising grid security and stability [26,27,28]. A comparison of the methods used in this paper with those in other existing studies in this field is shown in

Table 1. Comparison between this work and relevant literature.

Method	High Accuracy	Fast Calculation	Physical Interpretability	Stability/Convergence
[13]	✓		✓	✓
[14]	×	✓	✓	✓
[15]	×	✓	✓	✓
[16]	×	✓	✓	×
[17]	✓	×	×	×
[18]	✓	×	×	×
[19]	✓	×	×	×
[21]	✓	✓	×	×
[23]	✓	✓	×	×
This Work	✓	✓	✓	✓

.

Against this backdrop, energy storage systems have garnered increasing attention as an effective solution to supply–demand imbalances [29]. These systems provide bidirectional power regulation by storing surplus electricity during periods of high PV generation and releasing energy when PV output is insufficient or load demand increases. Such functionality mitigates real-time power shortages caused by uncertainty and helps maintain system equilibrium [30,31]. However, the capital and operational costs of ESS remain high [32,33]. In practical applications, how to rationally plan the capacity, location, and operational strategies of ESSs to ensure supply–demand balance has become a critical issue of guaranteeing the long-term stability and economic operation of power systems [34,35,36,37,38,39,40].

It is important to emphasize that lithium-ion batteries inevitably experience capacity degradation and increased internal resistance during long-term operations. Their degradation rates are typically closely related to operational stresses such as the state-of-charge (SOC) range, charge/discharge rates, depth of cycle, and temperature. If degradation effects are overlooked during energy storage planning, it often leads to underestimating the depreciation/replacement costs over the entire lifecycle, causing capacity allocation and operational strategies to deviate from true optimization on a long-term scale. In recent years, research has begun embedding empirical or semi-empirical degradation models into energy storage planning and scheduling optimization as degradation costs. For instance, the authors in [41] systematically reviewed mainstream approaches for degradation modeling, stress factor characterization, and lifetime cost accounting in BESS operational optimization.Reference [42] further explicitly coupled battery aging models into optimal energy management and capacity design for isolated microgrids to quantify trade-offs between degradation and economics. Research indicates that while high-precision energy storage degradation models offer more accurate lifespan predictions, their nonlinear characteristics significantly increase computational complexity in optimization problems. Given this, while this study does not incorporate complex energy storage degradation models within its current scope, we fully recognize the importance of battery state-of-health (SOH) evolution. Future work will explore integrating more refined battery decay models while ensuring computational efficiency.

In summary, the main contributions of this paper are as follows:

(1)

This paper proposes a probabilistic load flow calculation method designed for interconnected power systems. Its principal novelty lies in: the development of a joint Gaussian Mixture Model (GMM) to concurrently model the correlated uncertainties of photovoltaic output and load demand—a step beyond typical single-source or radial-network GMM applications; and the integration of this joint model with the cumulant method for probabilistic propagation, which effectively circumvents the non-physical negative value problem inherent in traditional series expansion techniques. This integrated approach enhances both the modeling fidelity for complex uncertainties and the numerical robustness of the calculation.

(2)

We established a two-layer energy-storage planning framework that explicitly incorporates PV and load uncertainties. The model optimizes both the location and capacity of energy storage systems to mitigate voltage violation risks induced by intermittent renewable generation and variable demand.

(3)

The effectiveness of the proposed planning scheme is rigorously validated through extreme scenario analysis and robustness testing, with comprehensive simulations conducted on the IEEE 33-node partitioned grid standard system.

2. Uncertainty Modeling for Load and Photovoltaic Generation Based on Gaussian Mixture Model

In power systems, load demand and photovoltaic output exhibit significant uncertainty and constitute critical factors affecting the probabilistic balance of supply and demand. Particularly with a high penetration of renewable energy, the supply–demand balance of power systems faces considerable challenges. Traditional forecasting methods of load demand and photovoltaic output are often based on a single probability distribution model or simplified statistical assumptions, making it difficult to accurately capture multimodal, asymmetric, and other complex characteristics. Therefore, accurately characterizing the probability distributions of photovoltaic generation and load demand has become a critical issue.

To address this issue, this study employs the GMM to conduct uncertainty modeling of photovoltaic output and load demand. The GMM is a model that fits complex probability distributions through a weighted sum of multiple Gaussian distributions. It is capable of effectively handling complex randomness and nonlinear distributions, making it suitable for capturing the multimodal characteristics of photovoltaic generation and load demand. The pseudocode of the GMM method is shown in Algorithm 1.

Algorithm 1: PV And Load Uncertainty Modeling Based On GMM

Input: Historical data of PV output or load demand(dimension d): $X=\left\{x_1,x_2,\dots ,x_n\right\}$    Maximum number of Gaussian components: Kmax Output: Optimal number of Gaussian components: Kopt     GMM parameters: Weight vector $\pi$, mean vector $\mu$, covariance matrix $\mathsf{\Sigma }$     Probability density function of PV output or load demand: $p(x)$ 1: Procedure Use BIC to select the best number of Gaussian components Kopt 2:   Initialize BIC_best = ∞, Kopt = 1 3:   for k = 1 to Kmax do 4:     $\left\{{\pi }_k,{\mu }_k,{\mathsf{\Sigma }}_k,\log (p(X))\right\}=\mathrm{EMAlgorithm}(X,k)$ 5:     Calculate $\mathrm{BIC}=-2\log p(X)+p\log N$ 6:     if BIC < BIC_best then 7:       BIC_best = BIC, Kopt = k 8:     end if 9:   end for 10: end Procedure 11: Procedure Using the BIC_best to retrain GMM 12:   $\left\{\pi ,\mu ,\mathsf{\Sigma }\right\}=\mathrm{EMAlgorithm}(X,k_{\mathrm{opt}})$ 13:   return Kopt, $\pi$, $\mu$, $\mathsf{\Sigma }$ 14: end Procedure 15: function EMAlgorithm (X, K) 16:   if iteration < max_iteration then 17:     for n = 1 to N do 18:       for k = 1 to K do 19:         Calculating Gaussian PDF $p_r(k,x_n)$ 20:       end for 21:       Calculate $p_r(k,x_n)=p_r(k,x_n)/{\displaystyle {\sum }_{k=1}^K{p}_r(k,x_n)}$ 22:     end for 23:     Recalculation $\log (p(X))$ 24:     if $\left|\mathsf{\Delta }\log (p(X))\right|<\epsilon$ then 25:       break 26:     end if 27:     Update ${\pi }_k,{\mu }_k,{\mathsf{\Sigma }}_k$ 28:   end if 29:   return ${\pi }_k,{\mu }_k,{\Sigma }_k,\log (p(X))$ 30: end function

2.1. Mathematical Formulation of GMM

The probability density function of a random variable x under the GMM is expressed as a weighted sum of K Gaussian distributions:

$$p(x)={\displaystyle \sum _{k=1}^K{\pi }_k}\mathcal{N}\left(x|{\mu }_k,{\Sigma }_k\right)$$

(1)

where p(x) denotes the probability density function of load demand or photovoltaic output, x represents the corresponding load demand or photovoltaic output, K is the optimal number of Gaussian components, which can be determined using the Bayesian Information Criterion (BIC) method. The value of K corresponding to the smallest BIC value is considered the optimal number of Gaussian components. The BIC formula is given by $\mathrm{BIC}=-2\log p(X)+p\log N$, where $\log p(X)$ is the log-likelihood of the dataset X, and $p=k(d+d(d+1)/2+1)-1$ is the total number of free parameters in the GMM, $d$ is the data dimensionality, $N$ is the total number of samples in the dataset $X$. ${\pi }_k$ is the weight of the Gaussian component, satisfying ${\displaystyle \sum _{k=1}^K{\pi }_k}=1$. $\mathcal{N}\left(x|{\mu }_k,{\Sigma }_k\right)$ denotes a Gaussian distribution with mean ${\mu }_k$ and covariance ${\mathsf{\Sigma }}_k$. The probability density function is given by:

$$\mathcal{N}(x|{\mu }_k,{\mathsf{\Sigma }}_k)={\displaystyle \frac{e^{\left(-\frac{1}{2}{(x-{\mu }_k)}^{\mathrm{TR}}{\mathsf{\Sigma }}_k^{-1}(x-{\mu }_k)\right)}}{{(2{\pi }_k)}^{d/2} | {\mathsf{\Sigma }}_k {|}^{1/2}}}$$

(2)

where ${\mu }_k$ is the mean of component k, and ${\mathsf{\Sigma }}_k$ is the corresponding covariance matrix.

2.2. Parameter Estimation of GMM Using Expectation–Maximization (EM) Algorithm

In practical applications, the parameters of GMM are unknown and must be estimated from historical data. A commonly used estimation method is the Expectation–Maximization (EM) algorithm, which iteratively updates the parameters to gradually approximate the maximum likelihood estimate. Through iterative updates of parameters, convergence is achieved when the changes in the model parameters between consecutive iterations become negligible. The goal of the EM algorithm is to maximize the maximum likelihood function:

$$\log p(X)={\displaystyle \sum _{n=1}^N\log }\left({\displaystyle \sum _{k=1}^K{\pi }_k}\mathcal{N}\left(x_n | {\mu }_k,{\mathsf{\Sigma }}_k\right)\right)$$

(3)

The basic steps of the EM algorithm are as follows:

(1)

Initialization: The K-means clustering algorithm is first applied to the data for preliminary clustering, in order to obtain the initial mean, covariance matrix, and weight for each component k.

(2)

E-step (Expectation): Based on the current mean and covariance, the posterior probability of component k for data point is computed as:

$$P_r\left(k,x_n\right)={\displaystyle \frac{{\pi }_k^{\mathrm{Last}}\mathcal{N}\left(x_n | {\mu }_k^{\mathrm{Last}},{\mathsf{\Sigma }}_k^{\mathrm{Last}}\right)}{{\displaystyle \sum _{k=1}^K{\pi }_k^{\mathrm{Last}}}\mathcal{N}\left(x_n | {\mu }_k^{\mathrm{Last}},{\mathsf{\Sigma }}_k^{\mathrm{Last}}\right)}}$$

(4)

where $P_r\left(k,x_n\right)$ denotes the posterior probability that sample x_n belongs to component k, ${\mu }_k^{\mathrm{Last}}$ is the mean of component k from the previous iteration, and ${\mathsf{\Sigma }}_k^{\mathrm{Last}}$ is the covariance of component k from the previous iteration, and ${\pi }_k^{\mathrm{Last}}$ is the weight of component k from the previous iteration.

(3)

M-step (Maximization): Using the posterior probabilities obtained in the E-step, the parameters, ${\mu }_k$ ,${\pi }_k$ and ${\Sigma }_k$ for component k are recalculated, thereby updating the maximum likelihood function.

(4)

Iteration: The E-step and M-step are repeated until the maximum likelihood function converges, that is, until the parameter values show no significant change. Once convergence is achieved, the current ${\mu }_k$ ,${\pi }_k$ and ${\Sigma }_k$ for each component k are the optimal output.

3. Probabilistic Load Flow Calculation Based on the Cumulant Method

The power flow equations of an electrical power system constitute a set of nonlinear equations characterized by complex mathematical relationships between input and output variables. When large amounts of random renewable energy sources and loads are integrated into the system, traditional deterministic power flow calculations can only provide the system state under specific input conditions, failing to capture the system state and probability distribution affected by uncertainty. Probabilistic Load Flow (PLF) calculation, however, is a method for handling load flow under uncertainty. It converts system outputs such as node voltages and bus-to-bus currents into probability distributions.

3.1. Principle of the Cumulant Method

After processing the uncertainties in photovoltaic output and load demand using GMM, their probability density functions can be obtained. This paper employs the cumulative quantity method for probabilistic power flow calculations and utilizes the C-type Gram–Charlier series expansion to compute the probability density functions of node voltages and branch currents.

Based on the mean, variance, and weights of photovoltaic output and load demand obtained from the GMM, calculate the vth-order cumulative photovoltaic output and v-order cumulative load demand. ${\kappa }^{(1)}$ is the first-order cumulant, representing the mean; ${\kappa }^{(2)}$ is the second-order cumulant, representing the variance; ${\kappa }^{(3)}$ is the third-order cumulant, representing the skewness; ${\kappa }^{(4)}$ is the fourth-order cumulant, representing the kurtosis. These cumulants characterize the distributional features of the random variables and provide the basis for subsequent PLF calculations.

3.2. Cumulant Calculation

In PLF analysis, the power balance equations of each interconnected power system region, including nodal voltage magnitudes and branch power flows, can be expressed as:

$$\left\{\begin{array}{l}{P}_i=U_i{\displaystyle \sum _{j=1}^J{U}_j}(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij})\hfill \\ Q_i=U_i{\displaystyle \sum _{j=1}^J{U}_j}(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij})\hfill \end{array}\right.$$

(5)

$$P_{ij}=U_i{U}_j(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij})-G_{ij}{U}_i^2$$

(6)

where $J$ denotes the total number of partitioned sub-grids, P_i and Q_i denote the active and reactive powers of node i in the interconnected system, U_i is the voltage magnitude at node i, and ${\theta }_{ij}$ is the voltage phase angle difference between nodes i and j. G_ij and B_ij represent the real and imaginary parts of the admittance between nodes i and j, respectively. P_ij represents the active power on the line between node i and node j.

To incorporate probabilistic load flow into power flow analysis, Equation (5) is linearized, and disturbance terms are introduced to describe variations in voltages and branch power flows. The nonlinear power flow equations in Equation (6) are rewritten in matrix form and expanded into a Taylor series at the reference operating point using the Newton–Raphson method, while higher-order terms are neglected. Consequently, a linearized power flow model is obtained:

$$\left\{\begin{array}{c}L=L_0+J_0\mathsf{\Delta }U\\ K=K_0+G_0\mathsf{\Delta }U\end{array}\right.$$

(7)

where L denotes the vector of active and reactive power injections of the interconnected system, K is the vector of active and reactive power transmitted through branches, U is the vector of voltage magnitudes and phase angles of the interconnected nodes, and L₀ and K₀ are the corresponding deterministic values. ΔU represents the perturbation vector of the voltage state, while J₀ and G₀ are the Jacobian matrices used in the final calculation of the Newton–Raphson method.

By rearranging and simplifying Equation (7), we can obtain the nth-order cumulative quantities of voltage disturbances at each node and branch-flow disturbances in the partitioned power grid:

$$\left\{\begin{array}{l}{\kappa }_{\mathsf{\Delta }U}^{(v)}=S_0^{(v)}{\kappa }_{\mathsf{\Delta }L}^{(v)}\hfill \\ {\kappa }_{\mathsf{\Delta }K}^{(v)}=T_0^{(v)}{\kappa }_{\mathsf{\Delta }L}^{(v)}\hfill \end{array}\right.$$

(8)

where ${\kappa }_{\mathsf{\Delta }L}^{(v)}={\kappa }_{\mathrm{PV}}^{(v)}-{\kappa }_{\mathrm{Load}}^{(v)}$ represents the cumulative power injection, calculated from the cumulative photovoltaic output and load demand, $S_0=J_0^{-1},T_0=G_0{J}_0^{-1}$. $S_0^{(v)}$ denotes the vth-order sensitivity matrix of the cumulative node voltage disturbance ${\kappa }_{\mathsf{\Delta }U}^{(v)}$ to the cumulative power injection ${\kappa }_{\mathsf{\Delta }L}^{(v)}$, which is $S_0$ raised to the power of v; $T_0^{(v)}$ is the vth-order sensitivity matrix of the cumulative branch power ${\kappa }_{\mathsf{\Delta }K}^{(v)}$ to the cumulative power injection ${\kappa }_{\mathsf{\Delta }L}^{(v)}$, and is the vth power of $T_0$.

Since the reference operating point $U_0$ of the node voltage in the partitioned power grid is constant, the cumulative quantities of the node voltage state variable U and the node voltage disturbance $\mathsf{\Delta }U$ at each order are related as follows:

$${\kappa }_U^{(1)}=U_0+{\kappa }_{\mathsf{\Delta }U}^{(1)}, {\kappa }_U^{(v)}={\kappa }_{\mathsf{\Delta }U}^{(v)}{\kappa }_U^{(v)}={\kappa }_{\mathsf{\Delta }U}^{(v)}(v\ge 2)$$

(9)

The node voltage state variables and disturbances share identical central moments, differing only in their means by $U_0$. Therefore, solving for the cumulative quantity of $\mathsf{\Delta }U$ yields the probability distribution characteristics of the node voltage state variable $U$. Similarly, for branch power flow, the reference operating point $K_0$ of the branch power flow is also constant. The cumulative quantities of the branch power flow state variable $K$ and the branch power flow disturbance $\mathsf{\Delta }K$ also satisfy the relationship given by Equation (9). Through the above transformation, the cumulative quantity ${\kappa }_U^{(v)}$ of the node voltage in the partitioned grid and the cumulative quantity ${\kappa }_K^{(v)}$ of the branch power flow can be obtained.

3.3. Cumulant Propagation and Probabilistic Load Flow Calculation

By calculating the cumulants of nodal voltages and branch power flows in the interconnected power system, the output probability distributions under uncertainty can be derived. To obtain the probability density functions of nodal voltages and branch power flows, the C-type Gram–Charlier series expansion is employed. This method avoids the shortcomings of the traditional A-type Gram–Charlier expansion, where the probability density function may become negative in some regions, and ensures that the probability density function remains non-negative and normalized. The mathematical formulation of the C-type Gram–Charlier expansion is expressed as:

$$h(y)={\displaystyle \frac{\exp \left[{\displaystyle \sum _m^{+\infty }\frac{1}{m}}{\delta }_m{H}_m(y)\right]}{{\displaystyle \int \exp }\left[{\displaystyle \sum _m^{+\infty }\frac{1}{m}}{\delta }_m{H}_m(y)\right]dy}}$$

(10)

${\delta }_m$ denotes the expansion coefficients, $\delta ={\left[{\delta }_1,{\delta }_2,\dots {\delta }_{m_{\max }}\right]}^T$.where $m$ is the order of expansion and $m_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum expansion order. $H_m(y)$ represents the Hermite polynomial, which is defined as

$$H_m(y)={(-1)}^m{e}^{{\displaystyle \frac{y^2}{2}}}{\displaystyle \frac{d^m}{d{y}^m}}{e}^{-{\displaystyle \frac{y^2}{2}}}$$

(11)

Equation (19) does not directly use the cumulants obtained; instead, the cumulants are transformed into expansion coefficients by solving the linear system $ℛ\delta =\lambda$, where $ℛ$ is a symmetric matrix and $\lambda$ is an $m$-dimensional column vector. Since ${\lambda }_1=0$, the remaining terms can be obtained using Equation (11):

$${\lambda }_m=(1-m){\displaystyle \frac{T_{m-2}}{{\sigma }^{m-2}}}(m=2,3,\dots ,m_{\max })$$

(12)

Here, $\sigma$ denotes the standard deviation. Because the constructed probability density function follows the standard normal distribution, $\sigma =1$. $T_m$ can be computed from the $m$-th order cumulant ${\kappa }_m$ as follows $T_0=1, T_1=T_2=0$ and for $m\ge 3$,

$$T_m={\kappa }_m+{\displaystyle \sum _{q=3}^{m-3}\frac{(m-1)!}{q!(m-q-1)!}}{\kappa }_{m-q}{T}_q(m\ge 3)$$

(13)

Through this process, the probability density functions of nodal voltages and branch power flows in the interconnected power system can be derived based on limited cumulant information. These probability distributions can further be applied to the assessment of voltage-limit violation risks in the power grid.

4. Bi-Level Optimization Model for Energy-Storage Planning

Energy storage systems can maintain probabilistic supply–demand balance through flexible charging and discharging. However, the high investment cost of storage makes large-scale deployment challenging. Therefore, both technical feasibility and economic factors must be considered when determining the installation location and capacity of storage systems. To ensure probabilistic supply–demand balance in interconnected power systems, while simultaneously considering economic efficiency and system security, this study develops a bi-level optimization model for storage planning. The model consists of two sub-models: the upper level for site selection and the lower level for capacity allocation and operational strategy.

4.1. Upper-Level Optimization Model

For the upper-level optimization, the comprehensive effectiveness coefficient is used to measure the improvement of grid performance by installing energy storage. The objective function is as follows:

$$\min \mathsf{\Phi }=(w_1\cdot f_1+w_2\cdot f_2+w_3\cdot f_3)$$

(14)

where $f_1$, $f_2$, $f_3$ are the effectiveness coefficients of the net power fluctuation $P_1$, voltage exceedance probability $P_2$, and network loss $P_3$ of the partitioned power grid. The coefficients w₁, w₂, and w₃ are weighting factors. The weight coefficients w₁, w₂, w₃ in the objective function reflect the decision-maker’s preference levels for different optimization objectives. To balance system safety and economic efficiency, we assume equal importance among these performance factors, setting w₁ = w₂ = w₃, related studies [43,44,45] have also adopted this approach. These weight coefficients can be flexibly adjusted based on actual engineering requirements.

The power fluctuation $P_1$ is the average peak-to-valley difference of the net load power during the production simulation period, which serves as an indicator for measuring the peak-shaving and valley-filling capability of the selected energy storage configuration. The calculation formula is as follows:

$$P_1={\displaystyle \frac{{\displaystyle \sum _{t=1}^T(P_{\max }(\mathsf{\Delta }t)-P_{\min }(\mathsf{\Delta }t)})}{T}}$$

(15)

where $\mathsf{\Delta }t$ is the sampling time interval, $P_{\max }(\mathsf{\Delta }t)$ and $P_{\min }(\mathsf{\Delta }t)$ are the maximum and minimum values of the net power within the sampling time interval.

The voltage limit exceeding probability $P_2$ refers to the probability that the node voltage exceeds the preset safe-operating range. It is calculated based on the probability density function of the regional network voltage, and the formula is as follows:

$$P_2={\displaystyle \sum _{j=1}^J\left[{\displaystyle {\int }_{-\infty }^{V_{\min }}{h}_j}(v)dv+{\displaystyle {\int }_{V_{\max }}^{\infty }{h}_j}(v)dv\right]}$$

(16)

where $h_j(v)$ is the voltage probability density function of subnetwork $j$, and $V_{\max }$ and $V_{\min }$ are the upper and lower voltage limits. In this study, to ensure the safe operation of the power grid, the voltage limits are set to $V_{\min }=0.95 \mathrm{p}.\mathrm{u}.$ and $V_{\max }=1.05 \mathrm{p}.\mathrm{u}.$

The network loss $P_3$ is used as an index to evaluate the effectiveness of an energy storage system in reducing network losses, and it is calculated as follows:

$$P_3={\displaystyle \sum _{j=1}^J{P}_{\mathrm{loss}}}$$

(17)

The efficacy function method can transform a multi-objective problem into a single-objective one, effectively solving the multi-objective optimization problem. The efficacy coefficient $f_d$ is calculated as:

$$f_d=\left\{\begin{array}{cc}{\displaystyle \frac{p_d-p_{\mathrm{s},d}}{p_{\mathrm{h},d}-p_{\mathrm{s},d}}}\times 40+60& p_d>p_{\mathrm{h},d}\\ 100& p_d\le p_{\mathrm{h},d}\end{array}\right.$$

(18)

In the above equation, $f_d$ is the efficacy coefficient value of the $d$ variable, $p_d$ denotes the actual value of the $d$ evaluation index, $p_{\mathrm{h},d}$ presents the satisfactory value of the $d$ evaluation index, and $p_{\mathrm{s},d}$ indicates the unsatisfactory value of the $d$ evaluation index.

4.2. Lower-Level Optimization Model

Once the optimal installation region of storage systems is determined by the upper-level model, the lower-level optimization further determines the storage capacity and operational strategies. The objective function is formulated as:

$$\min C_{\mathrm{total}}=\mathrm{CRF}{\displaystyle \sum _{i=1}^{N_{\mathrm{ESS}}}(C_E{E}_i+C_P{P}_i)}+W{\displaystyle \sum _{j=1}^J(V_{j,t}^{+}+V_{j,t}^{-})}+365{\displaystyle \sum _{i=1}^{N_{\mathrm{ESS}}}(k_{op}+C_{\mathrm{deg}}){\displaystyle {\int }_0^t\cdot (P_{i,t}^{\mathrm{ch}}+P_{i,t}^{dis})\cdot dt}}$$

(19)

where the first term represents the investment cost of energy storage. C_E and C_P denote the unit investment costs of energy storage capacity and power, respectively; E_i and P_i are the rated capacity and rated power of energy storage unit i; and CRF is the capital recovery factor, which annualizes the investment cost and is calculated as $\mathrm{CRF}={\displaystyle \frac{r{(1+r)}^{{\mathrm{N}}_{\mathrm{L}}}}{{(1+r)}^{{\mathrm{N}}_{\mathrm{L}}}-1}}$, r is the discount rate and N_L is the lifetime of the storage system. The second term represents the voltage-limit violation penalty, where W is the penalty coefficient, and V⁺_j,t and V⁻_j,t represent the upper and lower voltage-limit violations of node j at time t. The third term represents the annual operational cost, where k_op is the operation and maintenance cost coefficient for charging and discharging, $C_{\mathrm{deg}}$ is the degeneration cost coefficient. This objective function jointly considers investment, operation, and voltage violation risks to ensure both economic efficiency and reliability of the power system. The lower-level model is further subject to a set of physical and operational constraints for storage systems, defined as follows:

$$\left\{\begin{array}{cc}{E}_i=\rho \cdot P_i& \left(20\mathrm{a}\right)\\ 0\le P_{i,t}^{\mathrm{ch}}\le P_i& \left(20\mathrm{b}\right)\\ 0\le P_{i,t}^{\mathrm{dis}}\le P_i& (20\mathrm{c})\\ P_{i,t}^{\mathrm{ch}}\cdot P_{i,t}^{\mathrm{dis}}=0& (20\mathrm{d})\\ {\displaystyle \sum _{j=1}^J{P}_{j,t}^{\mathrm{in}}}={\displaystyle \sum _{j=1}^J{P}_{j,t}^{\mathrm{load}}}-{\displaystyle \sum _{j=1}^J{P}_{j,t}^{\mathrm{PV}}}+{\displaystyle \sum _{i=1}^{N_{\mathrm{ESS}}}({\eta }_{\mathrm{ch}}{P}_{i,t}^{\mathrm{ch}}-P_{i,t}^{\mathrm{dis}}/{\eta }_{\mathrm{dis}})}& (20\mathrm{e})\\ {\mathrm{SOC}}_{\min }\le {\mathrm{SOC}}_{i,t}\le {\mathrm{SOC}}_{\max }& (20\mathrm{f})\\ {\mathrm{SOC}}_i^0-{\mathsf{\Delta }}_{\mathrm{SOC}}\le {\mathrm{SOC}}_{i,24}\le {\mathrm{SOC}}_i^0+{\mathsf{\Delta }}_{\mathrm{SOC}}& (20\mathrm{g})\\ {\mathrm{SOC}}_{i,t+1}={\mathrm{SOC}}_{i,t}+({\eta }_{\mathrm{ch}}{P}_{i,t}^{\mathrm{ch}}-P_{i,t}^{\mathrm{dis}}/{\eta }_{\mathrm{dis}})/E_i& (20\mathrm{h})\end{array}\right.$$

(20)

where ρ is the ratio between storage capacity and power. Equation (20b) represents the charging power limit, and Equation (20c) represents the discharging power limit. $P_{i,t}^{\mathrm{ch}}$ and $P_{i,t}^{\mathrm{dis}}$ denote the charging and discharging power of storage unit i at time t. Equation (20d) defines the charging–discharging exclusivity constraint, which ensures that storage cannot charge and discharge simultaneously. Equation (20e) denotes the power balance constraint, where $P_{j,t}^{in}$ represents the net injected active power of node j at time t, and $P_{j,t}^{load}$ and $P_{j,t}^{PV}$ denote the load demand and photovoltaic generation at node j at time t. ${\eta }_{\mathrm{ch}}=0.95$ and ${\eta }_{\mathrm{dis}}=0.95$ represent charging and discharging efficiencies [46]. Equation (20f) defines the state-of-charge (SOC) limits, where SOC_i denotes the SOC of storage unit i at time t, and ${\mathrm{SOC}}_{\min }=0.1$ and ${\mathrm{SOC}}_{\max }=0.9$ are the minimum and maximum SOC levels, respectively. Equation (20g) defines the end-of-cycle SOC constraint, at the beginning of each scheduling period, the initial SOC of the energy storage system is set to ${\mathrm{SOC}}^0=0.5$. Equation (20h) describes the SOC update constraint of storage.

5. Results and Discussion

The flowchart of the data-driven probabilistic power flow method for energy storage planning considering interconnected grids is illustrated in Figure 1.

To verify the effectiveness of the proposed data-driven storage planning method for interconnected power systems based on Probabilistic Load Flow (PLF), experiments were conducted on a typical interconnected power system model. The experimental data consist of historical photovoltaic generation and load demand data, as well as the network topology and parameters of the interconnected power system. The photovoltaic and load data are obtained from actual measurements, covering fluctuation characteristics under different seasonal and weather conditions. Hourly load and photovoltaic output data are used for modeling, thereby reflecting typical uncertainties in power systems.

This experiment conducted simulation tests on the IEEE 33-node standard reference system [43]. For pure photovoltaic systems, PV modules are connected to specific nodes; whereas in PV-storage integrated configurations, energy storage units are co-located with PV modules at the same nodes, forming hybrid generation-storage resources. To validate the applicability of the methodology in interconnected grid scenarios, we logically divided the 33-node standard system into three sub-regions and established interconnection lines, its schematic diagram is shown in Figure 2, the energy storage system shown in the figure is derived from simulation results in subsequent content. The PV output data utilized in this study originated from actual historical operational records of a typical regional grid in Northwest China, covering a time span of 8760 h with a time resolution of Δt = 60 min. The computational experiments were performed on a personal computer equipped with an Intel Core i5 processor and 16 GB of RAM, using Python 3.10 as the programming environment. The optimization model was solved with the Gurobi solver. Photovoltaic sources are connected at nodes 10 in Region I, node 3 and10 in Region II, and node 8 in Region III. Historical profiles of load demand and photovoltaic generation are presented in Figure 3 and Figure 4, respectively.

The Bayesian Information Criterion (BIC) is employed to determine the optimal number of components K in the GMM. For photovoltaic or load data, the BIC values differ across nodes and time points. Figure 5 illustrates the variation curve of photovoltaic output at Node 4 in Region I over 2000 h. As K increases, the BIC value first decreases and then rises. The minimum BIC value occurs at K = 4, thus selecting K = 4 as the optimal number of components.

The Monte Carlo simulation method is commonly used as a benchmark for Probabilistic Load Flow (PLF) calculations. In this paper, the accuracy and computation time of the proposed cumulant-based method are compared with those of the Monte Carlo method. Using Monte Carlo simulation, 5000 scenarios are generated for each time period, and conventional power flow calculations are performed for every scenario to obtain the corresponding voltage magnitudes. The resulting probability density curves are then fitted and compared with those obtained using the cumulant-based probabilistic power flow method proposed in this study.

To analyze the differences between the two methods in fitting the probability density functions of system state variables, simulations are conducted for different buses and time periods. The comparison of the mean and variance of the two methods is shown in

Table 2. Comparison of Mean and Covariance between This Work and Monte Carlo Method.

Case	Mean			Covariance
	This Work	Monte Carlo	Relative Error (%)	This Work	Monte Carlo	Relative Error
Node 6, 12:00	0.978372	0.978645	0.03%	0.006018	0.006169	2.44%
Node 6, 16:00	0.977908	0.977769	0.01%	0.006216	0.006363	2.31%
Node 19, 12:00	0.998174	0.998196	0.01%	0.000362	0.000375	3.64%
Node 19, 16:00	0.998169	0.998167	0.01%	0.000377	0.000367	2.62%

, Figure 6 illustrates the probability density function (PDF) curves calculated by the two methods under different conditions. The results indicate that the relative errors between the proposed method and the Monte Carlo method are within 5%, demonstrating that the proposed method is both effective and accurate.

Furthermore,

Table 3. Comparison of Computational Speed between This Work and Monte Carlo Method.

Method	Number of Calculations (Times)	Computation Times (s)
Monte Carlo	5000	2331.81
This Work	37	13.43

presents the comparison of computational speed between the two methods. As can be seen, the computation time of the proposed method is much shorter than that of the Monte Carlo simulation, indicating a substantial reduction in computational burden for power flow analysis.

The final siting results of the optimization model indicate that storage units are installed at node 5 in Region I, node 1 in Region II, and node 3 in Region III. The optimization model prioritizes nodes with the highest voltage sensitivity for energy storage deployment. These are locations where fluctuations from PV generation and load demand cause the most pronounced voltage deviations. Consequently, strategically placing storage at these critical nodes allows the system to mitigate overvoltage issues most effectively with minimal capacity. The optimal energy capacities and rated power capacities of the storage units are summarized in

Table 4. Regional Energy Storage Planning Results for the Interconnected Power Grid.

Storage Location	Rated Energy Capacity (kWh)	Rated Power Capacity (kW)
Node 5 in Region I	11,188	5594
Node 2 in Region II	6760	3380
Node 3 in Region III	11,739	5869

. The storage units maintain supply–demand balance through charging and discharging operations. The annual charging/discharging profiles and SOC variations of the storage units in Regions I, II, and III are presented in Figure 7 and Figure 8, Figure 9 and Figure 10, and Figure 11 and Figure 12, respectively.

Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 display the optimized annual charge–discharge power curves for energy storage systems in Region I, II, and III. Direct comparison with the PV generation curve and load demand curve reveals the operational logic underpinning the energy storage strategy. As shown in Figure 3 and Figure 4, PV generation peaks during midday, while load demand typically reaches its highest point in the evening. This time mismatch results in daytime power surpluses and potential power shortages at night. Accordingly, the energy storage units are scheduled to charge during peak PV generation periods and discharge during evening load peaks. This coordinated operation effectively balances power generation and consumption demands.

Figure 13 compares the voltage-limit violation probabilities across different regions before and after the installation of storage, while Figure 14 illustrates the corresponding average voltages. It can be observed that without storage, the voltage violation probability reaches 32% in Region II and 47% in Region III, which seriously affects power quality and supply reliability. In addition, the average voltages in Region II and Region III are both below 0.96 p.u. After the integration of storage systems, the voltage violation probabilities are reduced to 5% and 6% for Region II and Region III, respectively, demonstrating that the occurrence of voltage violations is largely eliminated. Furthermore, the average voltages of these regions are improved to values above 0.96 p.u., confirming that the storage systems effectively mitigate voltage violations by injecting power at appropriate times and locations, thereby enhancing system stability.

To further validate the effectiveness of storage under extreme conditions, the connection between nodes 9 and 10 in Region II was intentionally disconnected. Figure 15 shows the comparison of voltage violation probabilities with and without storage under the extreme scenario, while Figure 16 presents the corresponding average voltages. Without storage, the voltage violation probabilities in Region II and Region III approach 40%, and the average voltages in both regions fall below 0.96 p.u., with the average voltage in Region III approaching the lower limit. After storage deployment, the average voltages in all interconnected regions increase significantly, returning to higher levels, while the voltage violation probabilities in Regions II and III are reduced to approximately 5%. These results confirm that the proposed storage configuration effectively improves voltage performance even under extreme scenarios.

To further validate the robustness of the planning scheme, simulation analysis was conducted under another extreme operating condition. This scenario features a sudden severe weather event causing a sharp drop in photovoltaic power generation while system load remains at peak levels. As shown in Figure 17, node voltage distributions under such extreme scenarios were analyzed. A Monte Carlo simulation generated 2000 random operating scenarios, covering conditions ranging from normal to severe fluctuations. Among these, scenarios featuring significant PV output reduction coupled with surging load demand were specifically selected for detailed evaluation.

Results indicate, no voltage violations occurred under conventional operating conditions, but certain extreme scenarios caused voltage overshoot. As shown in

Table 5. Voltage Levels in Each Area Before and After Energy Storage Installation.

Storage Location		Probability of Voltage Exceeding Limits (%)	Node Average Voltage (kV)
Node 2 in Region II	Extreme	4.7	0.9616
	Normal	0	0.9899
Node 3 in Region III	Extreme	11.65	0.9658
	Normal	0	0.9910

, the voltage overshoot probabilities for the two selected extreme cases were 4.7% and 11.65%, respectively. Although these operating conditions increased the risk of overshoot and reduced the average node voltage compared to normal operation, the system performance still outperformed configurations without energy storage. This result confirms the enhanced robustness and disturbance resistance of the proposed planning scheme under extreme events.

Additionally, we conducted a comparative analysis using a deterministic planning approach that disregards uncertainty, with results shown in Figure 18 and Figure 19. The results indicate that the deterministic approach reduces investment costs and operational maintenance costs by 54.2% and 48.3%, respectively. However, this method simultaneously increases the probability of voltage violations by 15.5% in Region 2 and 25% in Region 3. It is evident that while the deterministic approach lowers economic costs, the accompanying increase in voltage violation risks significantly compromises the voltage security of the partitioned grid.

Historical PV generation and load curves are used to construct a GMM-based probabilistic distribution, explicitly capturing the uncertainty characteristics of actual source-load data. These data-driven distributions then serve as inputs for probabilistic power flow calculations within a two-layer planning model, which optimizes energy storage siting and capacity. Consequently, the planning outcomes and annual performance projections are intrinsically linked to the statistical patterns of the real PV and load data employed. The effectiveness of this approach is confirmed by the results: after storage deployment, the probability of voltage violations in Region II and Region III decreased by 27% and 41%, respectively, while the average voltages in Regions I, II, and III increased by 0.01 p.u., 0.06 p.u., and 0.07 p.u., respectively. These improvements directly contribute to enhanced system reliability.

In summary, the proposed storage planning strategy achieves remarkable improvements by significantly reducing the probability of voltage-limit violations and maintaining the probabilistic supply–demand balance of the interconnected power system.

6. Conclusions

This paper presents a data-driven probabilistic power flow method for energy storage planning, designed to enhance the reliability of interconnected grids with high penetration of renewable energy. The framework first employs a Gaussian Mixture Model to accurately capture the complex uncertainties in photovoltaic generation and load. A probabilistic power flow analysis, based on an improved Cumulant Method, is then conducted to quantify voltage stability risks. Informed by this risk assessment, a bi-level optimization model is established to determine the optimal location, capacity, and operational strategy for energy storage, effectively balancing system security and economy. Experimental results validate the method’s effectiveness, demonstrating a significant reduction in the probability of voltage-limit violations. The configured energy storage system exhibits strong robustness, maintaining grid stability even under extreme fault scenarios. This study provides a viable solution for the safe and reliable integration of renewable energy.

While the proposed method demonstrates strong performance in energy storage planning, it is subject to several limitations. First, the computational complexity associated with the Gram–Charlier series expansion increases nonlinearly with the scale of the grid nodes, which may constrain its application in larger systems. Second, the current model does not yet incorporate a physics-based state-of-health representation for the storage units, limiting its ability to capture detailed aging dynamics. Future work could focus on improving computational efficiency through sparse-modeling techniques and integrating refined battery health models to enable more accurate and physically representative planning.