Optimal Planning Method of Wind Farms Considering Spatiotemporal Frequency Characteristics Under Uncertain Disturbances

Abstract

With the large-scale integration of wind farms, the spatial distribution characteristics of system dynamic frequency response become increasingly prominent. Existing wind farm planning methods considering frequency constraints mostly rely on predefined disturbance scenarios or system-level average frequency indicators, failing to account for the spatial randomness of disturbance locations. This leads to potential safety risks of local bus frequency limit violations when the systems planned accordingly are subjected to uncertain disturbances in actual operation. To address this issue, this paper proposes an optimal wind farm planning method based on frequency response upper bound constraints. The proposed method utilizes the algebraic connectivity index to characterize the theoretical upper bound of system frequency response under uncertain disturbances, which is then incorporated into the planning model as a constraint, thereby avoiding the risk of frequency limit violations in weak areas of the network. First, based on the closed-loop transfer function model and infinity norm theory, the theoretical upper bound of frequency response under uncertain disturbances is derived, and an upper bound evaluation index explicitly correlated with algebraic connectivity is constructed to identify the worst-case frequency response under uncertain disturbances. Second, the upper bound evaluation index is transformed into constraints for the wind farm siting and sizing model and an optimal planning method based on the frequency upper bound constraint is proposed. Simulation results on the IEEE 39-bus system show that, when a 1.2 p.u. active-power disturbance is applied at each bus individually, the proposed planning method reduces the average maximum frequency deviation by 36.78% relative to the unoptimized siting-and-sizing scheme (arithmetic mean across 39 bus-wise scenarios); for eight preset topology change scenarios, it yields a further 7.03% average reduction in maximum frequency deviation relative to the unoptimized scheme under each scenario topology.

1. Introduction

Driven by the dual-carbon target, the high penetration of wind power replacing traditional synchronous generators has led to significant low-inertia characteristics in power systems [1,2,3]. With the decline of the overall system inertia level, the power grid is prone to severe frequency fluctuations when subjected to power disturbances. Consequently, the frequency stability problem has become increasingly prominent, posing a severe challenge to the safe operation of the system [4,5]. Recently, some studies [6,7] have highlighted the emerging frequency stability challenges in modern low-inertia grids and proposed various robust control strategies to enhance real-time operational resilience. However, integrating these complex dynamic controls directly into the long-term planning stage often introduces significant computational challenges.

For system frequency security assessment, existing analytical methods mainly include time domain simulation, data-driven algorithms, and analytical models [8,9,10,11]. Among them, time domain simulation has high computational accuracy but is time-consuming, and data-driven methods heavily rely on sample quality, while analytical models are widely applied due to their clear mechanism. Regarding analytical models, the traditional COI [12] or system average frequency models can only characterize the global average features. However, with the localized integration of renewable energy, the uneven distribution of system inertia [13] makes the spatiotemporal distribution characteristics of frequency response increasingly prominent [14]. Therefore, existing studies have developed modal analysis, bus transfer functions, and other methods [15,16,17], and proposed various evaluation parameters covering the frequency drop slope coefficient, the frequency stability index (FSI), and the short-circuit ratio (SCR). These methods and indices effectively reveal that different disturbance locations might lead to significantly different frequency responses across buses. Nevertheless, these methods are mostly limited to analyzing specific predefined disturbance scenarios. In actual power grid operations, due to the high randomness of disturbance locations, evaluating and planning for only a limited number of deterministic scenarios might ignore the most severe conditions of the system. Conversely, performing a full-bus traversal scan introduces a heavy computational burden. Therefore, existing evaluation indices face challenges in accurately characterizing the worst-case frequency response boundary under uncertain disturbances, making it difficult to effectively prevent unpredictable safety risks caused by local frequency limit violations. Although algebraic connectivity has been recognized as a key index of network structural strength, its prior applications primarily focus on transient rotor angle synchronization rather than bounding the dynamic maximum frequency deviation [18].

The above frequency response boundary is not only critical for operational evaluation but also a safety constraint that high-penetration renewable energy system planning (e.g., wind farm siting and capacity sizing) must satisfy. In existing work that incorporates frequency constraints into indicators and planning: Reference [19] derives the lowest frequency point and RoCoF constraints from a simplified frequency response model for unit commitment and related problems but does not embed all-bus frequency data into security constraints. References [20,21] incorporate bus maximum frequency deviation constraints into optimization models to limit RoCoF and reduce reserve costs but do not systematically analyze the effect of disturbance location on bus frequency response. References [22,23] consider wind frequency support and frequency constraints in generation expansion planning or stochastic planning but tend to use simplified system-level indicators or a single extreme scenario when imposing constraints. These approaches are all based on single-scenario or deterministic-scenario assumptions or simplified indices. Planning schemes derived therefrom may encounter unforeseen local disturbances in actual operation, which may limit their adaptability to spatially random disturbance events [24,25,26]. While existing robust planning approaches [27,28] attempt to address such uncertainties, they typically rely on computationally heavy scenario sampling or complex min–max optimizations, suffering from the curse of dimensionality as the system scale increases.

Motivated by these challenges, extending algebraic connectivity to the infinity norm boundary, this paper proposes a topological index to analytically evaluate worst-case frequency responses under uncertainties. Unlike traditional robust methods relying on computationally expensive scenario sampling, this index dimensionally reduces complex spatial–temporal scenarios. Incorporating it as a security constraint, an efficient wind farm siting and sizing planning method is then proposed. The main contributions of this paper are summarized as follows:

• A theoretical upper bound evaluation index of frequency response oriented towards uncertain disturbances is constructed. Based on the closed-loop transfer function and norm theory, the theoretical upper bound of frequency response under uncertain disturbances is derived, and algebraic connectivity is extracted as the core characterization parameter, revealing the inherent influence mechanism of network topology on the worst-case frequency response.
• An optimal wind farm siting and sizing planning method based on the theoretical frequency response upper bound constraint is proposed. Instead of relying on predefined deterministic scenarios, the proposed method embeds the index into the planning model as a core security constraint to prevent local frequency limit violations caused by uncertain disturbances.

The research framework of this paper is arranged as follows: Section 2 establishes the frequency response model of the power system with renewable energy and analyzes its spatial frequency characteristics. Section 3 derives the upper bound index of frequency response and proposes a spatial frequency response evaluation method based on this index. Section 4 presents the system optimal planning method based on the index. Section 5 details the simulation results of simple and complex systems, respectively. Section 6 discusses the theory and application of the proposed method, highlighting its advantages and limitations. Finally, Section 7 concludes the paper.

2. System Modeling

According to the assumption in references [29,30], the fluctuations in the generator terminal voltage amplitude, as well as nonlinear and time-varying factors, are ignored, and only the dynamic relationship between frequency and active power is considered. In general, the dynamic characteristics of the system integrating renewable energy sources, synchronous generators and other devices can be expressed as Equation (1):

$$\Delta P_{\mathrm{Gen}}(s)=-G_{\mathsf{\omega },\mathrm{p}}(s)\Delta {\omega }_{\mathrm{Gen}}(s)$$

(1)

where $\Delta P_{\mathrm{Gen}}(s)$ and $\Delta {\omega }_{\mathrm{Gen}}(s)$ are the active-power vector and frequency vector of the device, respectively, and $G_{\mathsf{\omega },\mathrm{p}}(s)$ is the diagonal matrix of the frequency–active-power transfer functions of all devices, with a negative sign indicating that the active power output is positive when disturbed. In order to facilitate the analysis of boundary derivation, this paper adopts a representative simplified second-order model, as referenced in the literature [31], with the specific form provided in the Appendix A.

The phase angle–active-power matrix of each bus in the dynamic model of the network side is shown in Equation (2). To ensure consistency with the generation side, passive buses must be eliminated, retaining only the buses with power generation equipment. First, the standard network-side model matrix can be expressed as follows:

$$\left[\begin{array}{c}\Delta P_{\mathrm{Gen}}(s)\\ \Delta P_{\mathrm{Bus}}(s)\end{array}\right]=\left[\begin{array}{cc}{L}_{11}& L_{12}\\ L_{21}& L_{22}\end{array}\right]\left[\begin{array}{c}\Delta {\theta }_{\mathrm{Gen}}(s)\\ \Delta {\theta }_{\mathrm{Bus}}(s)\end{array}\right]$$

(2)

where the subscript Bus represents the network buses, $\Delta \theta$ is denoted as the phase angle response vector of each bus, and $L$ is the network Laplacian matrix, also known as the phase angle–active-power Jacobian matrix.

Since the network buses will not respond to the frequency regulation process, i.e., $\Delta P_{\mathrm{Bus}}(s)=0$, Equation (2) can be rewritten by combining Equation (1) and after eliminating passive buses and performing model reduction:

$$\Delta P_{\mathrm{Gen}}(s)=L_{\mathrm{ro}}\Delta {\theta }_{\mathrm{Gen}}(s)$$

(3)

where $L_{\mathrm{ro}}=L_{11}-L_{12}{L}_{22}^{-1}{L}_{21}$ is the reduced-order network Laplacian matrix.

Since $\Delta \theta ={\omega }_0{s}^{-1}\Delta \omega$, where ${\omega }_0$ is the nominal frequency and $s$ is the Laplace operator, Equation (3) can be expressed as follows, considering the equivalence between phase angle and frequency:

$$\Delta P(s)={\omega }_0{L}_{\mathrm{ro}}{s}^{-1}\Delta \omega (s)$$

(4)

It should be noted that the derivation of Equation (4) is based on the electromechanical transient approximation, which neglects the fast electromagnetic dynamics of network lines. Additionally, it assumes a linearized active-power flow model where passive nodes have been eliminated via Kron reduction.

The closed-loop frequency response of the renewable energy system, as shown in Equation (5) and Figure 1, can be obtained by combining the generation unit side and network side models under an active-power step disturbance. The system then distributes the disturbance power to each device:

$$\Delta \omega (s)={(G_{\mathsf{\omega },\mathrm{p}}+{\omega }_0{L}_{\mathrm{ro}}{s}^{-1})}^{-1}\Delta P_{\mathrm{d}}(s)$$

(5)

where $\Delta P_{\mathrm{d}}(s)={\left[\begin{array}{ccc}\Delta P_{\mathrm{d}1}& \dots & \Delta P_{\mathrm{dn}}\end{array}\right]}^{\mathrm{T}}$ is the disturbance power vector.

It should be noted that in the sign convention of this dynamic model, a positive disturbance ($\Delta P_{\mathrm{d}}(s)>0$) represents an active-power deficit (e.g., load increase). A positive power deviation ($\Delta P_{\mathrm{Gen}}(s)>0$) indicates an increase in generation output to compensate for the deficit, while a positive frequency deviation ($\Delta \omega (s)>0$) represents a frequency rise. Therefore, a positive disturbance naturally leads to a negative frequency deviation.

Define the closed-loop transfer function matrix:

$$H(s)={(G_{\omega ,\mathrm{p}}(s)+{\omega }_0{L}_{\mathrm{ro}}{s}^{-1})}^{-1}$$

(6)

Therefore Equation (5) reads $\Delta \omega (s)=H(s)\Delta P_{\mathrm{d}}(s)$. Assume $\Delta P_{dk}(s)$ is the disturbance at the k-th bus corresponding to the k-th component. If the disturbance occurs only at bus k, then $\Delta P_{\mathrm{d}}(s)=e_k\Delta P_{\mathrm{d}k}(s)$, where $e_k={[0,\dots ,1,\dots ,0]}^T$ is the k-th standard basis vector. Consequently, the frequency deviation of bus i can be expanded as $\Delta {\omega }_i(s)=H_{ik}(s) \Delta P_{\mathrm{d}k}(s)$, where $H_{ik}(s)$ represents the element located in the i-th row and k-th column of the matrix $H(s)$. Under the same disturbance magnitude, a different observation bus i corresponds to different transfer function elements $H_{ik}\left(s\right)$, which fundamentally leads to the spatial differences in frequency responses.

When disturbances occur at different buses (k = 1, 2,…, n), it is mathematically equivalent to selecting different columns of $H(s)$ to be multiplied by the corresponding disturbance magnitude $\Delta P_{dk}(s)$. Therefore, the spatial distribution of the frequency deviation $\Delta \omega$ varies significantly with the disturbance location. As defined in Equation (6), $H(s)$ is determined by the system transfer function matrix and the Laplacian matrix. This indicates that the unit dynamic characteristics and the network topology jointly determine the closed-loop transfer function matrix $H(s)$, thereby establishing an explicit mapping relationship between the spatial disturbance locations and the system frequency responses.

3. Evaluation Method for Frequency Response Boundaries of Systems Under Uncertain Disturbances

3.1. Derivation of Upper Bound Response Index

In control theory, the $H_{\infty }$ norm is an effective mathematical tool for measuring the worst-case response (i.e., the maximum system gain) of a system under uncertain disturbances [32,33]. For any linear system, the $H_{\infty }$ norm of its transfer function matrix $G(j\omega )$ is defined as the maximum singular value across the entire frequency range:

$${‖G‖}_{\infty }=\underset{\omega }{\sup }{\sigma }_{\max }(G(j\omega ))$$

(7)

where ${\sigma }_{\max }(G(j\omega ))$ is the maximum singular value at frequency $\omega$. $\underset{\omega }{\sup }$ denotes the supremum over all frequencies.

In practical power systems, to characterize the worst-case response of the system when a disturbance occurs at a specific bus i, the directional $H_{\infty }$ column norm ${‖g_i‖}_{\infty }$ is further defined as:

$${‖g_i‖}_{\infty }=\underset{\omega }{\sup }\Vert G(j\omega )e_i\Vert$$

(8)

where $e_i$ is the standard basis vector (representing the unit disturbance of bus i), and ${‖g_i‖}_{\infty }$ reflects the maximum response of the system when bus i is used as the disturbance source.

To apply the aforementioned infinity norm theory to the frequency security assessment of power systems, the closed-loop time domain model of the system needs to be transformed into the frequency domain. The frequency domain transfer function matrix of the system can be derived as:

$$G(j\omega )={\left(G_{\omega ,\mathrm{p}}(j\omega )+{\displaystyle \frac{{\omega }_0{L}_{\mathrm{ro}}}{j\omega }}\right)}^{-1}$$

(9)

$G_{\omega ,\mathrm{p}}(j\omega )$ is the frequency–power transfer function matrix of the power generation side, and $j\omega$ is the complex frequency variable.

From a mathematical perspective, observing Equations (7) and (8) reveals that the magnitude of the infinity norm, which characterizes the worst-case frequency response, is entirely determined by the frequency domain transfer function matrix. Furthermore, combining this with Equation (9), it can be found that the structure of $G(j\omega )$ depends not only on the dynamic parameters of the generators $G_{\omega ,\mathrm{p}}(j\omega )$, but its inverse matrix also explicitly contains the reduced-order Laplacian matrix term $L_{\mathrm{ro}}$ that represents the physical network topology. This implies that there is a direct mathematical mapping relationship between the worst-case frequency response of the system and the grid topology. Since the structural characteristics of the system can be intuitively quantified by connectivity in graph theory, an analytical relationship between the infinity norm and network connectivity can be further established.

To this end, the power grid is considered as a graph, where nodes correspond to buses and edges correspond to transmission lines. In graph theory, the topological structure of the system can be represented by the Laplacian matrix containing the bus degree vector $\delta ={({\delta }_1,\dots ,{\delta }_n)}^T$ composed of the degrees of the buses [34]. ${\delta }_i$ is the number of lines connected to bus i (or the sum of the edge weights associated with the bus in a weighted graph), reflecting the scale of the connection between the bus and the rest of the network.

The synchronous power matrix has an inherent relationship with the reduced-order Laplacian matrix: $K_{\mathrm{sync}}=-L_{\mathrm{r}0}$. The graph theory Laplacian matrix is as follows:

$$L_{\mathrm{con}}=\mathrm{diag}(\delta )-K_{\mathrm{sync}}^{\mathrm{sym}}$$

(10)

where $L_{\mathrm{con}}$ is the graph Laplacian matrix, and $K_{\mathrm{sync}}^{\mathrm{sym}}$ is the symmetrized synchronous power matrix, $\mathrm{diag}(\delta )$ is the degree matrix.

To quantify this correlation, this section introduces algebraic connectivity to characterize network connectivity, which is the smallest non-zero eigenvalue of the connectivity Laplacian matrix. According to linear system theory, the $H_{\infty }$ norm of the system’s transfer function characterizes its maximum worst-case gain, which is strictly bounded by the dominant eigenvalues of the embedded graph Laplacian matrix $L_{\mathrm{con}}$. Specifically, this maximum low-frequency gain is inversely dominated by its smallest non-zero eigenvalue. The core contribution of this section is establishing an explicit quantitative relationship between the frequency response upper bound and algebraic connectivity, embedding network topology into frequency safety assessment:

$${\alpha }_{\mathrm{AC}}={\lambda }_{\min ,\mathrm{non}-\mathrm{zero}}(L_{\mathrm{con}})$$

(11)

where ${\alpha }_{\mathrm{AC}}$ is the algebraic connectivity. This value measures the strength of network connectivity and reflects the tightness of the topological structure [35]. The network structure is more compact and has a stronger disturbance rejection ability when ${\alpha }_{\mathrm{AC}}$ is larger. Otherwise, the disturbance propagation range is greater and the compactness is weaker.

Based on the algebraic connectivity and directional gain, the theoretical upper bound index of frequency response is established:

$$B_i={\displaystyle \frac{{\kappa }_i}{{\alpha }_{\mathrm{AC}}}}$$

(12)

where ${\kappa }_i=\left|\Delta P_{\mathrm{d},i}\right|\cdot {‖g_i‖}_{\infty }$, $B_i$ is the upper bound of the frequency response when bus i is the disturbance source, ${\kappa }_i$ is the directional gain (reflecting the combined effect of disturbance magnitude and local device dynamics, independent of network topology). In order to ensure the effectiveness and safety of the upper bound index in practical engineering applications, the equivalent eigenvalue and the engineering safety margin are introduced. The equivalent eigenvalue is defined as:

$${\lambda }_{\mathrm{equ}}(L_{\mathrm{conn}})={\lambda }_{\mathrm{ref}}\sqrt{{\displaystyle \frac{{\lambda }_{\max }(L_{\mathrm{con}})}{{\lambda }_{\mathrm{ref}}}}}{\zeta }_{\mathrm{saf}}$$

(13)

where ${\lambda }_{\mathrm{ref}}$ is the reference eigenvalue (used to normalize the eigenvalue distribution), ${\lambda }_{\max }(L_{\mathrm{con}})$ is the maximum eigenvalue of the connectivity Laplacian matrix, and ${\zeta }_{\mathrm{saf}}$ is the engineering safety margin coefficient. It balances bound tightness and safety, compensating for unmodeled nonlinearities. Based on extensive simulations, a larger ${\zeta }_{\mathrm{saf}}$ (1.35–1.45) is recommended for large-scale systems with severe nonlinearities or uneven inertia distributions to ensure strict safety, while a smaller ${\zeta }_{\mathrm{saf}}$ (1.05–1.15) provides a tighter bound for uniform, highly linear systems [36,37].

Based on the theoretical upper bound index and equivalent eigenvalues, combined with the $H_{\infty }$ norm of the transfer function matrix, the upper bound inequality of the frequency response is established:

$$\max \left|\Delta {\omega }_i\right|\le \sqrt{{\displaystyle \frac{{\lambda }_{\mathrm{equ}}(L_{\mathrm{con}})}{{\alpha }_{\mathrm{AC}}}}}\cdot \Vert \Delta P_{\mathrm{d},\mathrm{sp}}\Vert \cdot \Vert G_{\omega ,\mathrm{p}}(s){\Vert }_{\infty }$$

(14)

where $\max \left|\Delta {\omega }_i\right|$ is the maximum frequency deviation of bus i, $‖\Delta P_{\mathrm{d},\mathrm{sp}}‖$ is the norm of the spatial disturbance vector (corresponding to the scalar $\left|\Delta P_{\mathrm{d},i}\right|$ in Equation (11), representing the spatial disturbance amplitude considering all possible disturbance locations), and ${‖G_{\omega ,\mathrm{p}}(s)‖}_{\infty }$ is the $H_{\infty }$ norm of the transfer function matrix. Substituting Equation (12) into Equation (13):

$$\max \left|\Delta {\omega }_i\right|\le \sqrt{{\lambda }_{\mathrm{ref}}{\zeta }_{\mathrm{saf}}{\displaystyle \frac{\sqrt{\frac{{\lambda }_{\max }(L_{\mathrm{con}})}{{\lambda }_{\mathrm{ref}}}}}{{\alpha }_{\mathrm{AC}}}}}\cdot \Vert \Delta P_{\mathrm{d},\mathrm{sp}}\Vert \cdot \Vert G_{\omega ,\mathrm{p}}(s){\Vert }_{\infty }$$

(15)

For detailed dimensions and units of symbols such as the mixed norm in the above formulas, please refer to the symbol

Table A4. Symbol table.

Symbol	Mathematical Type	Unit
$\max \left|\Delta {\omega }_i\right|$	Absolute value	p.u.
${\alpha }_{\mathrm{AC}}$	Eigenvalue	dimensionless
${\lambda }_{\max }(L_{\mathrm{conn}})$	Eigenvalue	dimensionless
${\lambda }_{\mathrm{ref}}$	Constant	dimensionless
${\lambda }_{\mathrm{equ}}(L_{\mathrm{conn}})$	Composite Scalar	dimensionless
${\zeta }_{\mathrm{saf}}$	Constant	dimensionless
$‖\Delta P_{\mathrm{d},\mathrm{sp}}‖$	2-norm (Vector norm)	p.u.
${‖G_{\omega ,\mathrm{p}}(s)‖}_{\infty }$	Infinity norm	dimensionless
$H_{\mathrm{eff}}$	Constant	s
$D_{\mathrm{eff}}$	Constant	dimensionless
$K_{\mathrm{eff}}$	Constant	dimensionless

in Appendix A.

It can be seen that the algebraic connectivity is jointly determined by the network topology and line parameters. However, the generator unit parameters influence the closed-loop gain and ${\kappa }_i$ through the transfer function $G_{\omega ,\mathrm{p}}(s)$, thereby affecting the value and robustness of the upper bound. Therefore, appropriately analyzing the sensitivity of the upper bound index to system parameters can verify the response characteristics of the theoretical formula on the unit side and provide a reference for planning and parameter tuning.

3.2. Spatial Frequency Response Boundary Evaluation Method

Based on the previous derivations, the quantitative relationship between algebraic connectivity and frequency response is established, characterizing the spatial features of the frequency response from the perspective of network structure. There is the following upper bound relationship between algebraic connectivity and worst frequency response:

$$\max \left|\Delta \omega \right|\le {\displaystyle \frac{c}{a_{{\mathrm{AC}}^b}}}$$

(16)

where c is the coefficient related to the system parameters and the size of the disturbance, b is the power exponent (close to 1). This inverse relationship mathematically proves that a smaller a indicates a weaker overall network connectivity, which in turn leads to a larger worst-case frequency response under uncertain disturbances.

Guided by this theoretical boundary, an evaluation method for spatial frequency response characteristics under uncertain disturbances is proposed:

First, weak links are identified by calculating the algebraic connectivity and the bus connectivity distribution, which effectively pinpoints the inherently weak areas in the network topology.

Second, the worst-case disturbance location is identified by combining the boundary index with individual bus directionality, which accurately pinpoints the specific bus that maximizes the dynamic amplification effect.

Finally, the theoretical upper bound of the worst-case frequency deviation for the current operational state is rapidly estimated using the fitted parameters in Equation (16).

The effectiveness of the aforementioned theoretical upper bound fitting formula and the proposed evaluation method will be thoroughly verified in the subsequent simulation section, in conjunction with specific system parameters.

4. Optimal Planning of Wind Farms Based on Frequency Response Evaluation Index

The quantitative relationship derived above shows that the upper bound of the frequency response is inversely proportional to the algebraic connectivity. Therefore, the planning problem of minimizing the upper bound of the frequency can be equivalently transformed into an optimization problem of maximizing algebraic connectivity, thereby avoiding complex simulations of all disturbance locations during the optimization process and significantly improving computational efficiency. This section establishes the following objective function and constraints:

4.1. Optimization Model Construction

By Equation (14), the frequency response upper bound $B_{i,\mathrm{upper}}$ is inversely related to algebraic connectivity. Writing the bound as a function of the planning variable x (siting and capacity), $B_{i,\mathrm{upper}}(x)=\Gamma \cdot \Psi (x)$ where $\Gamma$ is constant term and independent of x, where $\Psi (x)$ is:

$$\Psi (x)=\sqrt{{\displaystyle \frac{{\lambda }_{\max }(L_{\mathrm{conn}}(x))/{\lambda }_{\mathrm{ref}}}{{\alpha }_{\mathrm{AC}}(x)}}}$$

(17)

Therefore, minimizing the upper bound is equivalent to $\underset{x\in \Phi }{\min }\Psi (x)$. Numerical tracking confirms that ${\lambda }_{\max }(L_{\mathrm{conn}}(x)$ is relatively insensitive to x (change less than 7%), whereas ${\alpha }_{\mathrm{AC}}$ is highly sensitive to the spatial distribution of wind farms (increased by more than 10 times).

Table 1. Numerical tracking before and after optimization under IEEE 39-bus system.

Configuration	${\mathit{\alpha }}_{\mathbf{AC}}$	${\mathit{\lambda }}_{\mathbf{max}}({\mathit{L}}_{\mathbf{conn}}(\mathit{x}))$
Before optimization	0.2327	37.4512
After optimization	2.7729	39.8135

shows that ${\alpha }_{\mathrm{AC}}$ rises by more than tenfold whereas ${\lambda }_{\max }(L_{\mathrm{conn}}(x)$ changes by only 6.3%. From Equation (17), the factor $\sqrt{1.063}\approx 1.031$ from ${\lambda }_{\max }$ is very small relative to the order of magnitude increase in ${\alpha }_{\mathrm{AC}}$. Thus, $\min \Psi (x)$ is aligned with $\max {\alpha }_{\mathrm{AC}}(x)$, the equivalent objective is adopted:

$$F(x)=\underset{x\in \Phi }{\max }{\alpha }_{\mathrm{AC}}(x)$$

(18)

where x is the optimization variable, including siting $S(x)$ (a subset of N buses from candidate set B) and capacity $s(x)$. $\Phi$ is the feasible set (the constraint sets are given below). The upper bound index of Equation (15) is thus directly used as the planning objective, and Equation (18) is its implementable form.

For the optimal configuration of wind power integration, this paper mainly considers the constraints of eigenvalue, frequency change rate and capacity limit. This section discusses the optimal configuration constraints for DFIG wind farms from three aspects: system stability, bus frequency safety and equipment capacity limit.

4.1.1. System Constraints

System constraints are the basis for ensuring the frequency stability after the integration of renewable energy equipment. By setting reasonable thresholds, the system can maintain stable operation under various disturbance conditions. The minimum eigenvalue and the rate of change in frequency are constrained to ensure the effectiveness of the theoretical boundary.

Minimum eigenvalue constraint:

$${\alpha }_{\mathrm{AC}}(x)\ge {\alpha }_{\mathrm{thr}}$$

(19)

where ${\alpha }_{\mathrm{thr}}$ is the minimum eigenvalue threshold, set according to the system baseline operating state (typically 1.2–1.5 times the baseline value), ensuring the system has sufficient network synchronization capability and frequency stability margin.

The rate of change in frequency constraint:

$${\delta }_{RoCoF}\le \left|{\delta }_{i,RoCoF}^{\lim }\right|$$

(20)

${\delta }_{i,RoCoF}^{\lim }$ represents the upper and lower limits of the rate of change in frequency, where the RoCoF is defined as the instantaneous derivative of the system frequency with respect to time ($df(t)/dt$) in Hz/s. To ensure system security, the limit value used in the simulations is set to $\pm 0.5$ Hz/s.

4.1.2. Capacity Allocation and Location Constraints

To ensure reasonable wind farm capacity allocation and site selection, constraints are imposed on wind power penetration rate, number of connected buses, and spatial area layout.

Penetration rate constraint:

$${\rho }_{\min }\le {\rho }_{\mathrm{WF}}(x)\le {\rho }_{\max }$$

(21)

${\rho }_{\mathrm{WF}}(x)$ is the penetration rate of single point wind farm, ${\rho }_{\min }$ and ${\rho }_{\max }$ represent the lower and upper limits of the capacity for an individual wind farm, respectively.

Number of location constraints:

$$\left|S(x)\right|=N$$

(22)

$S(x)$ is the set of selected buses, and N is the number of wind farms.

Area coverage constraint:

$$S(x)\cap A\ne \varnothing ,\hspace{1em}S(x)\cap B\ne \varnothing$$

(23)

where A represents the set of key regional interconnection buses, and B represents the set of major load center buses. From a frequency security perspective, this constraint prevents the extreme concentration of wind farms, ensuring distributed active-power support to enhance spatial synchronization and mitigate severe local frequency limit violation.

4.1.3. Renewable Equipment Constraints

Equipment capacity constraints ensure that each wind farm capacity is within a reasonable range and meets the total penetration rate requirement:

Single point capacity constraint:

$$s_{\min }\le s_i(x)\le s_{\max },\forall i\in S(x)$$

(24)

$s_i(x)$ is the capacity share of a single wind farm, $s_{\min }$ is the lower limit of the minimum capacity, and $s_{\max }$ is the maximum capacity of a single point.

Total penetration constraint:

$${\displaystyle \sum _{i\in S(x)}{s}_i}(x)={\rho }_{\mathrm{WF}}(x)$$

(25)

In summary, the objective function aims to maximize the algebraic connectivity to ensure optimal spatial coupling among resources. Meanwhile, the diverse constraints jointly guarantee that the resulting configuration strictly satisfies the transient frequency safety thresholds, equipment capacity bounds, and regional layout requirements. Therefore, the wind farm planning method model considering the frequency spatial distribution characteristics is as follows:

$$\begin{array}{cc}{\min }_{x\in \phi }\hspace{1em}\hfill & \underset{x\in \Phi }{\max }{\alpha }_{\mathrm{AC}}(x)\hfill \\ \mathrm{s}.\mathrm{t}.\hspace{1em}\hfill & {\alpha }_{\mathrm{AC}}(x)\ge {\alpha }_{\mathrm{thr}}\hfill \\ \hspace{0.17em}\hfill & -{\delta }_{rocof}^{\lim }\le {\delta }_{rocof}\le {\delta }_{rocof}^{\lim }\hfill \\ \hspace{0.17em}\hfill & |S(x)|=N\hfill \\ \hspace{0.17em}\hfill & S(x)\cap A\ne \varnothing ,\hspace{1em}S(x)\cap B\ne \varnothing \hfill \\ \hspace{0.17em}\hfill & {\rho }_{\min }\le {\rho }_{\mathrm{WF}}(x)\le {\rho }_{\max }\hfill \\ \hspace{0.17em}\hfill & s_{\min }\le s_i(x)\le s_{\max },\hspace{1em}\forall i\in S(x)\hfill \\ \hspace{0.17em}\hfill & {\displaystyle \sum _{i\in S(x)}{s}_i}(x)={\rho }_{\mathrm{WF}}(x)\hfill \end{array}$$

(26)

4.2. Solving Method for Optimization Planning Based on Whale Optimization Algorithm

Since the planning method model involves discrete location variables and highly non-convex topological constraints, it constitutes a complex Mixed-Integer Nonlinear Programming (MINLP) problem. Therefore, exact convex methods are mathematically intractable, and the Whale Optimization Algorithm (WOA) [38,39] is employed due to its superior global exploration capability in avoiding local optima compared to traditional metaheuristics like GA or PSO. The solution procedure is illustrated in Figure 2 and solution procedure consists of three main phases:

(1)

Initialization phase

Set optimization parameters (candidate bus set, number of sites N, penetration rate range, etc.). Specifically, the population size is set to 50, and the maximum number of iterations is 150. The convergence factor $\gamma$ decreases linearly from 2 to 0 during iterations. For constraint handling, a static penalty function is employed: penalty factors of 10, 5, and 20 are applied to violations of the algebraic connectivity lower bound, total capacity limit, and RoCoF limit, respectively. The random number generator is initialized with seed 42 before population generation. Each whale is encoded as $x\in {ℝ}^{2\left|B\right|+1}$ ($\left|B\right|=12$ candidate buses; $N=4$ as defined above). Decoding yields $S(x)\subset B$ and $s(x)$ in three steps: (I) $S(x)=\{i\in B:x_i>0\}$, adjusted to enforce $|S(x)|=N$; (II) the corresponding capacity entries are normalized on $S(x)$ under per-bus limits; (III) Final component $x_{2\left|B\right|+1}$ sets ${\rho }_{\mathrm{WF}}(x)={\rho }_{\min }+({\rho }_{\max }-{\rho }_{\min })(\tanh (x_{2\left|B\right|+1})+1)/2$ with ${\rho }_{\min }=0.50$ and ${\rho }_{\max }=0.55$. Generate the initial population and perform feasibility checks and constraint handling.

(2)

Iterative optimization phase

For each candidate solution decoded as in step (1) at every iteration, compute ${\alpha }_{\mathrm{AC}}$ by eigenvalue decomposition of the Laplacian matrix and use it as the fitness value; update the population through WOA mechanisms and apply the static penalty function for connectivity, capacity, and RoCoF limits. No time domain simulation is performed during iterations. The simulation environment is as follows: MATLAB R2021, AMD Ryzen 7 5800H CPU, 16 GB RAM. On the IEEE 39-bus system, the mean wall clock time of this iterative optimization phase is approximately 15 s, averaged over five independent runs.

(3)

Termination and output phase

After reaching the maximum number of iterations or meeting the convergence criteria, output the optimal wind power configuration plan (site S and capacity s) and the system algebraic connectivity ${\alpha }_{\mathrm{AC}}$. Subsequently, a single time domain simulation is conducted on the final optimal configuration to verify the frequency response and to generate the comparative curves in the case studies.

5. Simulation Results

5.1. Two-Machine System

The two-machine system shown in Figure 3 is used to verify the validity and accuracy of the theory. The data settings for the two-machine system are shown in

Table 2. Two-machine system data.

Generator	Absolute Value of Eigenvalue	Eigenvector Components	Allocation Ratio	${‖{\mathit{G}}_{\mathit{i}}\left(\mathit{s}\right)‖}_{\mathit{\infty }}$
G1	4.429	[−0.9982, −0.0221]	0.4488	0.21424
G2	3.564	[−0.0685, − 0.9976]	0.5512	0.21818

.

5.1.1. Upper Bound Index Validity Verification

Figure 4 verifies the above theoretical relationship. The left figure shows the quantitative relationship of Equation (16): 20 data points cover the parameter range of ${\alpha }_{\mathrm{AC}}\in [0.88,6.62]$, and the fitting results are $R^2=0.9873$. The fitted constants with 95% confidence intervals are $b=0.98$ (0.92–1.04) and $c=1.08$ (1.02–1.14), which verifies the inverse relationship. The right figure shows the network topology of test system, where the bus color represents the bus connectivity, which is used to identify weak links. The combination of the two figures shows the spatial frequency response characteristics.

Figure 5 shows the time domain frequency deviation of three locations and the theoretical upper bound under different disturbance locations. The bound strictly encloses the actual response with a small gap, verifying the effect of spatial disturbance and the validity of the algebraic-connectivity-based index. The traditional index—which is based on the equivalent Center of Inertia (COI) empirical formula (detailed in the Appendix A)—underestimates the actual maximum deviation by approximately 65%, whereas the relative error of the proposed index with respect to the actual value (0.0348 Hz) is about 3.2%. This demonstrates that the proposed index is closer to the actual response and offers better accuracy, as the COI model aggregates the grid into a single equivalent machine, failing to capture the spatial amplification of disturbances at weakly connected buses.

Table 3. Comparison of data at three disturbance locations.

Location	Actual (Hz)	Theoretical (Hz)
G1	0.0356	0.0367
G2	0.0368	0.0386
Load	0.0320	0.0325

summarizes the key values of the disturbance location of the three different buses, including the actual response value and the theoretical upper bound value. The actual response data of the three buses show that the theoretical upper bound can effectively cover the actual response, and the prediction accuracy is within a reasonable range. Due to the small scale of the current two-machine system and the relatively simple network topology, the differences in frequency responses under disturbances at the three buses are relatively small.

5.1.2. System Parameter Sensitivity Analysis

The previous section indicated that the upper bound simultaneously depends on the algebraic connectivity of the network and the dynamics on the generator side. To illustrate the impact of unit parameters on the index under a fixed network, this section scans the system parameters under the premise of a given topology and algebraic connectivity, examining the sensitivity of the index to changes in the parameters.

Figure 6 intuitively shows the influence of parameters on the upper bound index and the geometric distribution of the feasible region. The three subgraphs show the variation characteristics of the margin under different parameter planes (D-K, H-K, H-D). The frequency margin is defined as the difference between the system’s allowable frequency deviation threshold and the actual worst-case frequency response and the detailed mathematical expression for calculating this margin is provided in the Appendix A.

Table 4. System parameter impact analysis statistics table.

System Plane	Margin Change Amplitude	Standard Deviation	Percentage of Feasible Region
D-K(H = 10.0)	9.080	1.796	99.3%
H-K(D = 2.5)	0.982	0.223	23.1%
H-D(K = 20.0)	7.680	1.404	86.6%

summarizes the key statistics of the system parameter influence analysis. As can be seen from the table, the damping coefficient D has the greatest influence (the range of margin change is 9.080, and the standard deviation is 1.796), the inertia coefficient H is the second (the range of change is 7.680, and the standard deviation is 1.404), and the adjustment coefficient K is the smallest (the range of change is 0.982, and the standard deviation is 0.223). The sensitivity ranking D > H > K verifies the dominant role of the damping coefficient in system stability and reflects the important influence of the inertia constant and the limited influence of the adjustment coefficient in the range of engineering parameters.

Figure 7 verifies the negative correlation between the robustness margin and the infinity norm and proves the physical consistency of the index. The three subgraphs show the relationship between the infinity norm and the margin in different parameter planes (D-K, H-K, H-D), all showing the characteristics of high infinity norm corresponding to low margin and low infinity norm corresponding to high margin. The order of the infinity norm variation range (D-K plane: 0.25–3.05; H-D plane: 0.13–2.13; H-K plane: 0.08–0.37) is consistent with the parameter sensitivity order (D > H > K) in Table 4.

5.2. IEEE 39-Bus System

The proposed optimization scheme is tested on the IEEE 39-bus system integrated with four wind farms using the Matlab/Simulink R2021a platform. The structure of the system is shown in Figure 8, and the parameters of the system generator and DFIG are respectively presented in

Table A1. Nominal power and inertia constant of SGs in IEEE 39-bus system.

Equipment Number	Nominal Power (MW)	Inertia Constant (s)
G 1	1000	5
G 2	520.81	4.5
G 3	650	4.7
G 4	632	4.6
G 5	508	4.5
G 6	650	4.6
G 7	560	4.5
G 8	540	4.5
G 9	830	4.8
G 10	250	4.2
Total	6140.81	45.9

and

Table A2. Parameters of DFIG.

Parameter	Unit	Value
Rated Apparent Power	MVA	5.0
Rated Active Power	MW	4.87
Rated Voltage	V	575
Number of Pole Pairs	/	2
Inertia Constant	s	5.04
Damping Factor	/	0.01
Stator Resistance	p.u.	0.01
Stator Reactance	p.u.	0.1
Magnetizing Inductor	p.u.	2.9
Rotor Resistance	p.u.	0.005
Rotor Reactance	p.u.	0.156

in the Appendix A.

5.2.1. Validity Verification

The four wind farms are initially connected to the preset buses. The determination of these locations is based on the following considerations:

(1)

Load weight: select the vicinity of the system load center;

(2)

System topology: consider the network connectivity and power flow distribution characteristics;

(3)

Typical scenarios: represent the common wind farm integration mode in actual projects. This configuration ensures that the analysis results can reflect the actual system characteristics and provide a basis for subsequent optimization analysis.

Figure 9 shows the comparison between the theoretical upper bounds and the actual maximum frequency deviations of the 10 units. The actual frequency deviation ranges from 0.12 to 0.19 Hz, while the theoretical upper bound ranges from 0.13 to 0.19 Hz. Statistical analysis reveals a Root Mean Square Error (RMSE) of 0.0125 Hz and a Mean Absolute Error (MAE) of 0.0120 Hz between them. Furthermore, the theoretical margin is consistently maintained at about 1.06 across all units. These statistical metrics quantitatively verify that the proposed index can tightly and safely cover the actual worst-case frequency response.

Furthermore, disturbances of the same magnitude are applied to all 39 buses to compare and analyze the system frequency response characteristics before and after WOA optimization.

Figure 10 shows the comparison results of the maximum frequency deviation before and after optimization. Before optimization, the frequency deviation range of each bus is 0.0267–0.2079 Hz, and the frequency deviation of some buses is close to the frequency safety threshold (0.20 Hz), which has the risk of frequency instability. After WOA optimization, the frequency deviation of all buses is reduced to 0.0173–0.1174 Hz, with an arithmetic mean improvement of 36.78% relative to the unoptimized siting-and-sizing scheme; all values are below the safety threshold, verifying effective frequency security under individual bus disturbance scenarios.

Figure 11 shows the comparison of frequency response of five representative buses. Before optimization (dashed line), the frequency response curves of all buses show clear frequency drop, and the response amplitudes of different buses vary significantly different. After optimization (solid line), the frequency response curve is obviously shifted upward and smoother, and the frequency drop is significantly reduced. The frequency deviation of Bus 6 is the largest (0.1888 Hz) before optimization, and it decreases to 0.0921 Hz after optimization. Although the improvement margin of Bus 20 is the smallest (21.23%), the frequency deviation after optimization (0.0215 Hz) is still significantly lower than the safety threshold. Bus 16, as the wind farm bus and system hub, has a significantly improved frequency response after optimization, which verifies the positive impact of the optimized configuration of wind farms on the frequency stability of the system. (other buses data are uniformly displayed in

Table A3. Optimization data results after all buses disturbance.

Position	Max Deviation	Position	Max Deviation	Position	Max Deviation
1	0.0771	16	0.0185	31	0.0809
2	0.0825	17	0.0182	32	0.0929
3	0.0184	18	0.0181	33	0.0790
4	0.0181	19	0.0772	34	0.0814
5	0.0181	20	0.0791	35	0.08
6	0.0921	21	0.0186	36	0.0954
7	0.0183	22	0.0737	37	0.084
8	0.0184	23	0.0843	38	0.1174
9	0.0738	24	0.0173	39	0.1040
10	0.0878	25	0.0724
11	0.0178	26	0.0182
12	0.0180	27	0.0179
13	0.0177	28	0.0180
14	0.0181	29	0.1086
15	0.0184	30	0.0837

in the Appendix A).

Figure 12 shows the spatial distribution of the system response intensity in the form of a heat map, and the left and right figures correspond to the response distribution before and after optimization, respectively. The spatial distribution pattern reveals obvious hotspots before optimization, reflecting differences in the frequency response across various spatial locations. After optimization, the overall color becomes noticeably lighter, indicating a significant reduction in the frequency deviations of all buses. Moreover, the hotspot areas shrink considerably, resulting in a more uniform spatial distribution.

In summary, the proposed wind farms siting and sizing planning method based on the upper bound index can effectively ensure the frequency safety of the 39-bus system under any bus disturbance. Compared with the planning method only for specific disturbance locations, the proposed method accounts for full-bus disturbance scenarios, yielding a more reliable and comprehensive planning scheme. Furthermore, the wide applicability of the index is verified through the full-bus disturbance analysis, providing a novel perspective and methodology for frequency stability planning in power systems with high shares of renewable energy.

5.2.2. Comparison of Planning Results Under Different Evaluation Indices

To verify the applicability of the proposed framework under different evaluation indices, this section compares wind farm location and capacity determination on the 39-bus system when the proposed upper-bound index, FSI [40] and SCR [41,42] are used as optimization objectives (formulas in the Appendix A). FSI is a frequency stability index that reflects the maximum frequency deviation and RoCoF in a given scenario. SCR is a network-strength-related index that characterizes the electrical strength or connectivity of the system.

Table 5. Summary of Optimization Results.

Index	ObjValue	AvgFreDev	${\mathit{\alpha }}_{\mathit{A}\mathit{C}}$
Upper bound	0.1156	0.1082	3.7556
FSI	0.1743	0.1821	3.3080
SCR	9.0264	0.1893	3.1097

summarizes the optimization results under the three indices (WOA). The proposed index achieves the highest ${\alpha }_{\mathrm{AC}}$, the lowest average frequency deviation, and the lowest objective function value. FSI and SCR yield lower ${\alpha }_{\mathrm{AC}}$ and higher frequency deviations under the same setting, indicating that the proposed index performs best in the frequency upper-bound dimension.

Table 6. Site selection and capacity determination results under different indices.

Index	Site	Capacity Ratio (%)
Upper bound	[15, 16, 17, 20]	[17.42, 5.63, 19.66, 12.30]
FSI	[15, 16, 18, 25]	[12.65, 12.05, 16.90, 13.41]
SCR	[15, 16, 17, 27]	[17.01, 15.28, 6.46, 16.25]

and Figure 13 present the site selection and capacity determination under the three indices. Table 6 lists that the proposed index selects buses 15, 16, 17, 20 with capacity shares 17.42%, 5.63%, 19.66%, 12.30%, more concentrated on key buses. During the optimization process, capacity under the proposed index converges quickly to the selected buses with clear concentration on buses 17 and 20. Under FSI and SCR the allocation is more even and the convergence pattern is gentler. Therefore, the proposed index under the same settings achieves a more coordinated spatial siting and capacity allocation compared to FSI and SCR.

As shown in Figure 14, the three index curves during the iteration process verify the effectiveness of the adopted optimization algorithm for different evaluation systems (the SCR curve is negative because its maximization objective is transformed into a minimization problem for solving). Although the FSI exhibits the fastest convergence speed, the reduction in its objective function value is highly limited. In contrast, the objective function value of the proposed upper bound index achieves the most significant reduction (from approximately 0.131 to 0.116). This indicates that the proposed upper bound index, by directly accounting for the evolution of the global network topology, can guide the algorithm to suppress the frequency deviation baseline under the worst-case conditions.

5.2.3. Discussion on Results Considering Topological Structure Changes

The previous section evaluated the proposed index and planning under a fixed (same condition) topology, demonstrating their effectiveness. Frequency stability in renewable-rich grid-connected systems is particularly sensitive to topology changes. To rigorously verify the robustness and fault tolerance of the proposed method against such structural deteriorations, this section therefore systematically evaluates the optimization performance of the proposed index and planning scheme under various extreme topology change scenarios.

Typical scenarios include selected line outages (e.g., backbone and corridor links) and generator outages. Under each scenario, the WOA-optimized wind farm configuration is compared with the unoptimized (uniform) configuration in terms of algebraic connectivity ${\alpha }_{\mathrm{AC}}$, average maximum frequency deviation, and RoCoF.

The topological change scenarios used in this section are presented in

Table 7. Topology change scenario settings.

Scenario Type	Number	Faulty Component	Impact Characteristics
Line Outrage	L1	Bus 16–17	load center
Line Outrage	L2	Bus 21–22	generator unit and load area
Line Outrage	L3	Bus 26–27	southern region and load center
Line Outrage	L4	Bus 16–17 Bus 21–22	dual line simultaneous outrage
Generator Outage	Gen1	Bus 32	load support capacity decreases
Generator Outage	Gen2	Bus 35	inertia drops sharply
Generator Outage	Gen3	Bus 37	decrease in reserve consumption
Generator Outage	Gen4	Bus 32 Bus 37	system capacity decreases significantly

. To cover representative fault types, we selected four types of line faults and four types of generator outages, including load center interconnections, unit load corridors, regional interconnections, and scenarios involving double-line or double-unit outages. The table also lists the fault components and impact characteristics for each scenario.

It should be pointed out that the fault scenarios are utilized as multi-scenario testing to verify the applicability and robustness of the planning method under topological changes, rather than pre-known inputs that alter the optimization results. Such topological changes sever synchronous paths or remove local frequency regulation sources, leading to a decrease in ${\alpha }_{\mathrm{AC}}$. This structural degradation diminishes the network’s mutual active-power support capability and amplifies local frequency deviations.

Table 8. Topology change performance index statistics.

Scenario	${∆\mathit{\alpha }}_{\mathit{A}\mathit{C}}$	Deviation	RoCoF
L1	−12.97	5.58	4.08
L2	−14.09	6.51	5.02
L3	−14.71	7.47	5.95
L4	−16.78	8.22	6.84
Gen1	−13.26	5.77	4.09
Gen2	−14.60	6.71	5.03
Gen3	−14.15	7.51	5.95
Gen4	−17.04	8.46	6.86

$\Delta {\alpha }_{\mathrm{AC}}=({\alpha }_{\mathrm{AC}}^{\mathrm{WOA}}-{\alpha }_{\mathrm{AC}}^{\mathrm{Unopt}})/{\alpha }_{\mathrm{AC}}^{\mathrm{Unopt}}\times 100\%$. For the average maximum frequency deviation and maximum RoCoF, $\mathrm{Imp}(\%)=(X^{\mathrm{Unopt}}-X^{\mathrm{WOA}})/X^{\mathrm{Unopt}}\times 100\%,X\in \{\mathrm{Deviation},\mathrm{RoCoF}\}$ indicates improvement percentage. The unoptimized reference: ${\alpha }_{\mathrm{AC}}=2.1636$, average maximum frequency deviation = 0.1653 Hz and maximum RoCoF = 0.1980 Hz/s.

summarizes the changes in key performance indices under the previously mentioned fault scenarios. For each scenario, the table shows the performance difference between the optimized and unoptimized configurations. A negative $\Delta {\alpha }_{\mathrm{AC}}$ entry indicates that ${\alpha }_{\mathrm{AC}}$ under the WOA-optimized configuration is lower than under the uniform unoptimized scheme on that contingency topology. Meanwhile, positive deviation and RoCoF values indicate improved WOA performance (smaller deviation and RoCoF values indicate better frequency response performance). It can be seen that the proposed planning method remains effective under the selected topology-change scenarios.

Figure 15 presents the frequency response under three cases: (a) baseline topology, (b) worst line outage scenario, and (c) worst generator outage scenario. After topology change, the spatial and temporal spread of the response increases and the worst-case deviation rises. The proposed optimized planning configuration reduces the deviation and narrows the spread in both line outage and generator outage scenarios. The figure illustrates that the proposed planning improves frequency stability not only under the baseline topology but also after topology degradation.

Figure 16 presents the bus connectivity distribution under topology change. Each subplot shows the spatial distribution of connectivity over the 39-bus network (e.g., the participation or correlation measure of the algebraic connectivity of buses). Topology change weakens network connectivity and reduces ${\alpha }_{\mathrm{AC}}$. The distribution becomes more uneven and weak links become more pronounced. Depending on the scenario, the site selection and capacity provided by the optimized planning method adjust accordingly. This figure illustrates how the planning results adapt to different scenarios. The figure also links topology degradation to connectivity loss and supports the use of ${\alpha }_{\mathrm{AC}}$ as an evaluation indicator for planning under topology change.

The results above demonstrate that the proposed upper bound index and WOA-based planning remain effective under line and generator faults: the optimized configuration improves ${\alpha }_{\mathrm{AC}}$ and frequency response in the considered topological variation scenarios, and the planning exhibits good adaptability to the fault set established in this section.

6. Discussion

6.1. Physical Mechanism of Topological Enhancement

Mathematically, algebraic connectivity quantifies the overall topological strength of a network. In the physical context of power systems, the embedded Laplacian matrix represents the synchronizing power coefficient matrix. Therefore, an improvement in algebraic connectivity physically translates to a stronger electrical coupling among generators and inverter-based resources. When a local active-power disturbance occurs, a system with higher algebraic connectivity exhibits lower spatial impedance to power flow. This allows the disturbance energy to be rapidly and evenly shared among all available inertial resources system-wide, thereby preventing severe local frequency limit violations and enhancing overall system stability.

6.2. Validity Under Complex Disturbances

The proposed topological boundary remains strictly valid under complex scenarios. Based on the superposition principle of linear systems, the worst-case response to multiple simultaneous disturbances is mathematically bounded by the maximum singular value of the MIMO transfer function matrix. Furthermore, since the $H_{\infty }$ norm defines the maximum induced gain for any finite energy input signal, the derived envelope intrinsically covers arbitrary time-varying disturbances, rather than being limited to simple step changes.

6.3. Computational Efficiency and Robust Control

As highlighted in the Introduction, recent advancements in robust frequency control primarily focus on real-time operational resilience. Compared to these complex dynamic control methods, the proposed framework provides a complementary planning-stage solution. By mapping the worst-case frequency response to a static topological index, our framework avoids the heavy computational burden of simulating complex dynamic controls. Consequently, during the iterative optimization process (e.g., using WOA), the fitness evaluation only requires matrix eigenvalue decomposition rather than time domain ODE solving. This dimensional reduction keeps the computational complexity minimal and ensures strong algorithmic robustness.

6.4. Scalability and Limitations in Realistic Grids

To demonstrate the scalability of the proposed framework, an extended case study is conducted on a modified IEEE 118-bus system. As shown in

Table 9. N-1 Contingency Screening and Validity Verification for IEEE 118-bus System.

Rank	Outrage Line	${\mathit{\alpha }}_{\mathit{A}\mathit{C}}$	Actual Max Dev (Hz)	Upper Bound (Hz)
/	Base case	6.6157	0.1210	0.1270
1	Line 126	2.8012	0.1946	0.3000
2	Line 127	2.8162	0.1921	0.2984
3	Line 104	3.0361	0.2153	0.2768
4	Line 96	3.4520	0.2351	0.2434
5	Line 54	4.0209	0.2320	0.2460
…	…	…	…	…
176	Line 140	6.6147	0.1213	0.1271
177	Line 59	6.6123	0.1216	0.1272

, the proposed upper bound index accurately evaluates the impact of various N-1 line outage. The index ranks the severity of these contingencies, identifying critical lines that severely weaken global connectivity and increase the worst-case frequency deviation. Furthermore, the proposed upper bound index consistently covers the actual maximum frequency deviation across all scenarios, maintaining a safe margin ranging from 1.04 to 1.55. For remaining validity validation data, please refer to the Supplementary Materials.

Unlike traditional robust planning relying on time-consuming ODE simulations, calculating the sparse Laplacian eigenvalues for the 118-bus system averages only $8\times {10}^{-4}$ s per evaluation. Considering the 118-bus system consists of 186 branches, a complete N-1 contingency scan can be performed with minimal computational cost. It is worth mentioning that 9 out of the 186 branches are radial lines whose outages lead to system islanding (${\alpha }_{\mathrm{AC}}$), therefore the ranking focuses on the 177 contingencies that maintain global connectivity.

It is important to acknowledge the practical considerations when extending this framework to ultra-large realistic grids. First, as the network scale increases, the discrete search space for optimal siting expands exponentially. While our analytical index significantly accelerates fitness evaluations, applying metaheuristics directly to tens of thousands of buses may require prior regional decomposition. Second, the analytical bound is derived from a linearized model, which neglects nonlinear deadbands and complex inverter controls. Although this guarantees a safely conservative margin for worst-case planning, it introduces a degree of conservativeness. Ultimately, the proposed framework bridges the gap between network topology and frequency stability, establishing a computationally tractable and theoretically rigorous foundation for early-stage planning.

7. Conclusions

To address the problem that existing wind farm planning methods fail to fully account for the spatial frequency response differences under uncertain disturbances, this paper establishes a theoretical upper bound evaluation index of frequency response directly correlated with network topology. Based on this index, an optimal wind farm siting and sizing planning method, using the frequency response upper bound as a security constraint, is proposed. The main research conclusions are as follows:

• The proposed upper bound index can effectively quantify the worst-case frequency deviation under uncertain disturbances. Simulation results show that this theoretical upper bound can strictly cover the actual dynamic frequency responses under different disturbance locations. The relative error between the estimated value and the actual maximum frequency deviation is only about 3.2%, providing an evaluation basis for preventing local frequency limit violations.
• A spatial frequency response evaluation method based on algebraic connectivity is proposed. Based on graph theory and control theory, this method integrates the physical topological characteristics of the network with the dynamic parameters of the system, quantifying the impact of spatial uncertain disturbances on frequency dynamics from a global structural perspective. The proposed method effectively overcomes the limitations of traditional single-scenario evaluations, easily identifying weak links in the network that are prone to inducing local frequency limit violations, and providing an evaluation basis for wind farm spatial layout optimization.
• The proposed wind farm optimal planning method can effectively avoid local frequency limit violation risks under uncertain disturbances, thereby fundamentally enhancing the overall frequency security baseline of the system from the network structural perspective. Simulation results show that, when a 1.2 p.u. active-power disturbance is applied at each bus individually, the average maximum frequency deviation is reduced by 36.78% relative to the unoptimized siting-and-sizing scheme, with optimized values in 0.0173–0.1174 Hz. Furthermore, when facing preset topology-change scenarios, the maximum frequency deviation achieves an average reduction of 7.03%, significantly enhancing the adaptability of the system.

Despite the effectiveness of the proposed framework, it has certain limitations. The linear analytical boundary provides a conservative safety margin but neglects the complex nonlinear control dynamics of modern wind turbines, and the heuristic optimization faces dimensionality challenges in ultra-large realistic grids. Future work will focus on integrating data-driven methods to account for nonlinear converter controls and developing distributed optimization algorithms to further enhance the scalability for large-scale realistic grid planning.