Application of Kron’s Reduction Method to Evaluate Generating Unit Power Variability in the Kron Loss Model Under Economic Dispatch

The contents of this literature review chapter include three main things that are closely related to the title of the article, namely (i) KRM [22] and its application are historical background, mathematical basis, modern adaptations, and power system uses [17, 23-25]; (ii) generating unit power variabilities in the Kron’s Loss Model (KLM) [26] are quadratic loss model [27], effects of network reduction [23-25], and sensitivity factors [28]; and (iii) economic dispatch phenomena are fundamentals [26, 27, 29], integration with network models, the role of reduced networks, and recent improvements are particularly relevant in the context of the application [30] of KRM [22] to evaluate generating unit power variability in the KLM [31] under economic dispatch [27]. These three areas form the theoretical and empirical backbone for evaluating the objectives in this study.

2.1 KRM and its application

Kron’s reduction method, first introduced by Gabriel Kron in 1939 [22], was originally developed to simplify electrical network equations by eliminating selected internal nodes. Over time, the method evolved through structured matrix transformations and became a fundamental tool in control theory and power system engineering [23-25, 32]. At its core, Kron’s reduction is a mathematical technique that simplifies the network admittance matrix (Y-bus) by applying the Schur complement. This technique preserves the electrical characteristics of retained nodes while reducing the system size [23-25]. It is particularly valuable for minimizing computational complexity in power system studies, especially when analysing the interactions between principal buses. In practical applications, it is widely used to remove non-essential nodes such as radial or unloaded buses, thus streamlining the analysis without significantly affecting system accuracy [1]. Kron Reduction as a source of variability, i.e., (i) KRM simplifies the system by eliminating nodes and redistributing admittances [25], (ii) This changes impedance pathways, which can affect: loss coefficients (B-coefficients) in the quadratic loss model and participation factors, i.e., how each generator contributes to balancing demand [18]; and (iii) If not carefully staged, KRM could distort dispatch patterns and make the reduced model unrepresentative.

The KRM has been extensively applied in modern power system analysis, notably in model order reduction [23-25], short-circuit studies [8], steady-state assessments [9], and dynamic simulations [3, 8]. When applied to the bus admittance matrix, the Schur complement ensures that voltages at retained buses remain equivalent to those in the full model, making KRM ideal for system simplification without compromising critical accuracy [23-25]. In contemporary research, KRM supports applications [3, 9] like contingency analysis, optimal power flow (OPF), and state estimation [27, 33, 34], where computational efficiency is essential [33, 35]. Moreover, the method's role has expanded to accommodate medium- or large-scale grids that integrate renewable energy sources, which require real-time responsiveness and high-fidelity modelling [3, 9, 23-25]. By maintaining operational characteristics while reducing model dimensionality, Kron’s reduction provides practical benefits in creating accurate equivalents for contingency scenarios, economic dispatch simulations, and stability studies [23-25].

Recent studies have extended its application to large-scale grids, integrating renewable generation and examining real-time feasibility in control environments [1, 3, 9]. Mathematically, KRM is a Schur complement on the bus-admittance matrix (Y) [23-25]. Let the retained buses be B (all buses except bus-n), and the eliminated set be I = n. Partition Y as shown in Eq. (1).

$Y=\left[\begin{array}{ll}Y_{B B} & Y_{B I} \\ Y_{I B} & Y_{I I}\end{array}\right]$                       (1)

with $Y_{I I}=\left[Y_{n, n}\right], Y_{B I}=\left[Y_{b, n}\right]$, and $Y_{I B}=\left[Y_{n, b}\right]^T$.

The reduce matrix is shown in Eq. (2).

$\begin{aligned} Y^{\text {reduction }} & =Y_{B B}-Y_{B I} \cdot Y_{I I}^{-1} \cdot Y_{I B} =Y_{B B}-\frac{1}{Y_{I I}} \cdot Y_{B I} \cdot Y_{I B}\end{aligned}$                   (2)

For the stage when reduction is carried out on bus-26 which is only adjacent to bus-25, then only entries that “touch” neighbours of bus-26 change. In the stage where bus-26 (a dead-end node connected only to bus-25) is eliminated via Kron reduction, only matrix entries associated with bus-25, i.e., the neighbours of bus-26—are updated in the Y-bus. This behaviour follows directly from the nature of the Schur complement in Kron reduction, where perturbations affect only adjacent nodes [23-25]. If $\aleph(26)$ is the set of buses directly connected to 26, then for any $i, j \in \aleph(26)$ (possibly $i=j$):

#Off-diagonal (create or update an equivalent link) is shown in Eq. (3).

$Y_{i j}^{{reduction }}=Y_{i j}-\frac{1}{Y_{26,26}} \cdot Y_{i, 26} \cdot Y_{26, j}, i \neq j$                             (3)

#Diagonal of a neighbour (self-admittance update) is shown in Eq. (4).

$Y_{i i}^{ {reduction }}=Y_{i i}-\frac{1}{Y_{26,26}} \cdot Y_{i, 26} \cdot Y_{26, i}$                             (4)

All other entries remain the same.

$Y_{26,26}$ is the self-admittance at bus-26 (sum of incident branch admittances plus shunt at bus-26). $\boldsymbol{Y}_{\boldsymbol{i , 2 6}}=\boldsymbol{Y}_{\mathbf{2 6 , i}}= -\boldsymbol{y}_{\boldsymbol{i}, \mathbf{2 6}}$ for a simple series branch $\boldsymbol{y}_{\boldsymbol{i}, \mathbf{2 6}}=\mathbf{1} / \boldsymbol{r}_{\boldsymbol{i}, \mathbf{2 6}}+\boldsymbol{j} * \boldsymbol{x}_{\boldsymbol{i}, \mathbf{2 6}}$.

In the standard IEEE-30 bus system, bus-26 is a radial node, connected exclusively to bus-25—this isolated linkage underlines its role as a dead‑end bus in system topology (i.e., only one incident branch in the line data), then $\aleph(26)=\{25\}$ [23-25]. There is no new tie created because there’s only a single neighbour, so that there are two explanations:

#i) update only the 25–25 diagonal is used Eq. (4), so that Eq. (4) becomes Eq. (5).

$Y_{25,25}^{ {reduction }}=Y_{25,25}-\frac{1}{Y_{26,26}} \cdot Y_{25,26} \cdot Y_{26,25}$                          (5)

#ii) all other entries are unchanged.

In the IEEE-30 bus test system, bus-30 is connected to both bus-27 and bus-29, reflecting its mesh configuration within the network topology [23-25]. This detail is clearly documented in single‑line diagrams and contingency analyses, which note that “bus-30 (2nd weakest bus) has got connections to both bus-27 and bus-29” [36, 37], so that $\aleph(30)=\{27,29\}$, and then we both update diagonals and create a new mutual admittance between 27 and 29 are shown in Eqs. (6)-(8) [23-25].

$Y_{27,27}^{{reduction }}=Y_{27,27}-\frac{1}{Y_{30,30}} \cdot Y_{27,30} \cdot Y_{30,27}$                       (6)

$Y_{29,29}^{ {reduction }}=Y_{29,29}-\frac{1}{Y_{30,30}} \cdot Y_{29,30} \cdot Y_{30,29}$                           (7)

$Y_{27,29}^{ {reduction }}=Y_{27,29}-\frac{1}{Y_{30,30}} \cdot Y_{27,30} \cdot Y_{30,29}$                       (8)

Eq. (8) is the new direct coupling.

The method has been widely applied in power system analysis for tasks such as model order reduction [1, 38, 39], short-circuit studies [40], and dynamic simulations [3, 9]. Applying the Schur complement, the reduced Y-bus matrix maintains equivalent voltages at retained buses, making it suitable for applications where network simplification is required without compromising accuracy [1, 38, 40]. In power system applications, the Kron reduction method has been used for network equivalencing to facilitate contingency analysis [38], optimal dispatch [3, 9, 29], and state estimation [34, 39]. Recent studies have extended its application to medium- or large-scale grids, integrating renewable generation [1] and examining real-time feasibility in control environments [3, 9, 29]. These works highlight the importance of balancing computational efficiency with fidelity to the original system [3, 9]. The KRM in the context of transmission network studies, it offers a means to reduce the dimensionality of system models while preserving the electrical characteristics relevant to the retained buses [1, 32]. Recent studies have demonstrated its utility in creating equivalent networks for contingency analysis [38], optimal power flow [39], and dynamic stability studies [39].

However, the majority of these studies focus on one-step reduction approaches without systematically evaluating multi-stage elimination sequences [38, 41]. In parallel, researchers have explored advanced network reduction techniques that incorporate machine learning and adaptive algorithms for better retention of operational characteristics [3]. While these methods have shown promise in large-scale simulations, they often come with higher computational overhead, making them less suitable for real-time applications [39]. Moreover, traditional Kron’s reduction continues to be favoured in practice due to its analytical transparency and ease of implementation [1, 3, 9]. The gap remains in understanding how stage-wise bus elimination affects operational performance metrics such as V-RMSE, loss deviations, and generator dispatch variability [41]. Existing literature also indicates a growing interest in integrating network reduction with real-time control environments such as SCADA and EMS [42, 43]. Ensuring operational compatibility after reduction is critical, particularly for systems incorporating renewable generation and complex load dynamics [38]. Despite this, comprehensive frameworks that assess reduction impact across multiple operational objectives, i.e., voltage profile accuracy, generator variability, runtime efficiency, and scalability are sparse. This study addresses that gap by implementing a multi-stage reduction sequence and evaluating it against these criteria using the IEEE-30 bus system [36, 37].

2.2 Generating unit power variability in the KLM

The concept of generating unit power variability in the KLM addresses how network reduction influences the allocation and fluctuation of generation outputs within power systems [38]. Kron’s reduction method is widely applied to simplify large-scale systems by eliminating certain buses, thereby reducing the size of admittance and impedance matrices while preserving the electrical behaviour between retained buses [1]. However, such reduction can impact the accuracy of system parameters, particularly when assessing active and reactive power generation variability at individual generator buses [23-25]. In the context of the KLM, which models real power losses as a quadratic function of nodal power injections, the reduction process can alter the loss coefficients and the distribution of system losses among generators [27]. The variability of generator outputs becomes an important performance metric, as it reflects the sensitivity of dispatch schedules to changes introduced by the reduced-order network representation. This is particularly critical in economic dispatch applications, where generation schedules are optimized to minimize operational costs while meeting demand and technical constraints [44]. Studies have shown that while Kron reduction can significantly reduce computational burden, it must be applied carefully to avoid large deviations in generator dispatch patterns [3]. The impact depends on the location of eliminated buses, the degree of electrical connectivity, and the weighting of loss coefficients in the dispatch optimization [45]. Sensitivity analyses are often employed to evaluate generator output variability under different reduction scenarios, ensuring that reduced models remain operationally compatible with full-scale systems.

The quadratic loss model, which expresses transmission losses as a quadratic function of generator outputs, has been a fundamental element in economic dispatch problems. Loss coefficients (a, b, c) are derived from network parameters and used to compute total system losses under different dispatch scenarios [46]. When combined with Kron reduction, the quadratic loss model enables evaluation of how network simplification impacts dispatch decisions and system-level losses [27]. Performance benchmarking in reduced-order models often involves metrics such as voltage RMSE, power loss deviation, and simulation runtime [47]. Studies have shown that network reduction can reduce computational time significantly. As demonstrated in adaptive model reduction studies for large test systems, network reduction can significantly reduce simulation time while maintaining and preserving acceptable accuracy [48]. Scalability remains a key focus, with research suggesting that methods like Kron reduction can be adapted for high-bus-count systems with minimal accuracy degradation [17]. The KLM based on a quadratic power loss equation is implemented on both the full and reduced-order networks. This enables the evaluation of generating unit power variability by quantifying changes in generation dispatch profiles, total system losses, and nodal power injections [27]. Together these sources support a comparative study between full vs. reduced models across (i) voltage deviations, (ii) generator-output differences, (iii) benchmarking metrics (loss deviation, voltage error, runtime), (iv) operational compatibility with ED/real-time control constraints, and (v) scalability to large systems.

2.3 Economic or optimal dispatch: A mathematical perspective

Economic dispatch (ED) is a fundamental optimization problem in power system operation, aiming to determine the optimal generation levels for all committed generating units so as to meet the total load demand at the lowest possible operating cost, subject to system and unit constraints [27, 49]. The ED problem is formulated based on the principle of equal incremental costs of all operating units, adjusted for transmission losses, ensuring that generation is allocated efficiently across the system. Based on mathematically, the basic ED problem can be expressed as the minimization of the total generation cost [50-53].

Minimize: It is shown in Eq. (9).

$C_{{total }}=\sum C_i\left(P_{g i}\right)$ for $i=1$ to $N_g$                   (9)

where, $C_i\left(P_{g i}\right)$ is the cost function of the i-th generator, usually modeled as a quadratic function is shown in Eq. (10).

$C_i\left(P_{g i}\right)=a_i+b_i \cdot P_{g i}+c_i \cdot P_{g i}^2$                          (10)

Here, $P_{g i}$ is the power output of the i-th generating unit, and $\boldsymbol{a}_i, \boldsymbol{b}_i$, and $\boldsymbol{c}_i$ are cost coefficients.

The ED problem is subject to two main types of constraints, i.e., power balance constraint and generator operating limits. Both constraints are shown in formulas (11) and (12).

$\sum P_{g i}=P_{ {demand }}+P_{{losses }}$                       (11)

where, $\boldsymbol{P}_{ {demand }}$ is the total system demand and $\boldsymbol{P}_{ {losses }}$ is the total transmission loss.

$P_{g i_{\min }} \leq P_{g i} \leq P_{g i_{\max }}$                          (12)

which ensures that each generating unit operates within its technical limits. Incorporating transmission losses into ED requires the use of loss coefficients, often represented through the KLM. In this approach, total transmission loss is expressed as a quadratic function of generator outputs is shown in Eq. (13) [27, 31, 50, 51].

$\boldsymbol{P}_{ {losses }}=\sum\left(\sum \boldsymbol{P}_{g i} \cdot \boldsymbol{B}_{i j} \cdot \boldsymbol{P}_{g j}+\sum \boldsymbol{B}_{0 i} \cdot \boldsymbol{P}_{g i}+\boldsymbol{B}_{00}\right)$                           (13)

where, $\boldsymbol{B}_{i j}, \boldsymbol{B}_{\mathbf{0 i}}$, and $\boldsymbol{B}_{\mathbf{0 0}}$ are loss coefficients obtained from system data. This model enhances realism in the ED formulation by accounting for internal system losses, which are crucial in operational planning [31, 54]. The KLM thus directly couples with the ED formulation by modifying the power balance equation, affecting the optimal generator outputs.

Various methods can be applied to solve the ED problem, including classical approaches such as the Lambda-iteration method for systems without losses, and iterative Newton-Raphson or gradient-based methods when losses are considered. In recent research, metaheuristic algorithms such as Genetic Algorithms (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE) have also been successfully applied to ED problems, particularly for complex, nonlinear, or non-convex cost functions [55]. The economic dispatch problem uses an objective function to minimize total generation cost, subject to system power balance constraints, generator operating limits, and network loss equations [56-58]. This is solved using numerical optimization methods, applied consistently to both the full and reduced networks, allowing a fair comparison of dispatch results and operational efficiency [59]. In the economic dispatch study (which ends with OPF), KRM is an aid to reduce model complexity so that the optimization algorithm can more easily find the minimum power loss solution while still considering the network's technical constraints [3].

For the results that follow, the IEEE-30 bus system was used as the baseline benchmark network. This test case is widely recognized in power system research for studies involving power flow analysis, optimal power flow, economic dispatch, and network reduction techniques such as Kron reduction which were conducted by Christie [60], Zimmerman et al. [62], and Song et al. [1]. It provides a practical balance between complexity and tractability, enabling the validation of reduction algorithms under realistic yet computationally efficient conditions, which was conducted by Zimmerman et al. [62] and Milano [32]. The operational dataset includes bus voltage limits, generator capacity ranges, transformer tap settings, line thermal limits, and specified load demands, which together represent the characteristics of a realistic medium-voltage transmission network. With its diverse load profiles, meshed topology, and range of line impedances, the IEEE-30 system serves as an effective platform for assessing the scalability, accuracy, and computational efficiency of staged Kron reduction.

To ensure reproducibility, all model data were prepared in a structured format compatible with the simulation tools, covering: (i) bus data (voltage magnitude, phase angle, load demand, and shunt admittance), (ii) generator data (active/reactive limits and cost coefficients), (iii) branch data (line impedances and thermal ratings), and (iv) transformer data (tap ratios and phase shifts). All values were validated against the IEEE standard dataset prior to reduction analysis. Visualization of voltage profiles, deviation trends, and loss curves follows established practices in network reduction and dispatch literature by Chevalier and Almassalkhi [3]. Figures employ consistent per-bus axes and standardized color mappings to facilitate cross-stage comparison and ensure interpretability.

4.1 Impact of KRM on voltage profile preservation

The stepwise reduction strategy aims to balance model simplicity with operational accuracy, making the reduced models suitable for applications in economic dispatch, contingency analysis, and real-time power system operations. The justification for eliminating bus-26, bus-30, and bus-29, i.e., bus-26 was selected for the initial stage of KRM due to its peripheral and minimally connected nature, linking only to bus-25 and bus-26. The subgraph structure indicates, that bus-26 does not participate in multiple loops or critical corridors, making it a suitable candidate for elimination. Bus-30 was selected for the second stage of KRM due to its peripheral location and limited connectivity, being directly linked only to bus-27. This position means bus-30 contributes minimally to the meshed network structure and does not serve as a major transit path for power flows.

The elimination of such a bus ensures minimal disruption to the system’s power distribution characteristics while simplifying the network topology. The subgraph structure for bus-30 shows a simple radial connection from bus-27 to bus-30, without involvement in critical loops or interconnections. This topology makes it ideal for elimination, as KRM will replace its effect with an adjusted admittance directly on the connected bus. Bus-29 was selected for the third stage of KRM after the prior elimination of bus-26 and bus-30. Bus-29 is a peripheral load bus connected directly to bus-27, with no participation in critical transmission corridors or meshed loop structures.

Its elimination further reduces the network size while maintaining the electrical equivalence of the reduced-order model. The subgraph for bus-29 indicates a simple radial link to bus-27. This configuration makes bus-29 a suitable candidate for removal without introducing significant deviations in voltage profiles or line flows in the remaining system. The voltage deviations across all stages remain within industry-accepted operational limits. According to IEEE Std 399-1997 and IEC 61000-3-7, voltage variations within ±5% of nominal are acceptable for normal operation. The proposed reductions yield maximum deviations of 0.47% at bus 27 and 0.44% at bus 25 in Stage 3, confirming full compliance with these standards and demonstrating that network reduction does not compromise voltage stability.

Subgraph showing direct connections between bus-26, bus-30, and bus-29 in condition after reduction is shown in Figure 4. Based on Figure 4 it can be explained, that selected Y-bus matrix elements around bus-26, bus-30, and bus-29.

Figure 4. The direct connections between bus-26, bus-30, and bus-29 in condition after reduction

Matrix element around bus-26, bus-30, and bus-29 before and after reduction by using Eqs. (6)-(8). Matrix element around bus-26, bus-30, and bus-29 are shown in Table 4.

Table 4. Matrix element around bus-26, bus-30, and bus-29

Evaluating the impact of staged KRM on the voltage profiles of the IEEE-30 bus system is analysis compares the baseline full-system voltage magnitudes with those obtained after each of the three reduction stages: Stage 1 (bus-26 eliminated), Stage 2 (bus-30 eliminated), and Stage 3 (bus-29 eliminated). Voltage magnitudes were computed using Newton–Raphson load flow analysis, which is widely recognized for its robustness and convergence properties in balanced power systems. The results indicate that voltage deviations are minimal for most buses, remaining below 0.005 p.u. for Stages 1 and Stage 2. The Stage 3 introduces a slightly higher deviation at specific load buses (notably bus-27 and bus-25), which can be attributed to the electrical proximity of the eliminated bus to these nodes. This finding aligns with the observations, where removal of buses with higher connectivity to load centers caused localized voltage perturbations. Despite these deviations, the profiles remain within acceptable operational limits defined in IEEE Std. 399-1997.

A display of voltage profile comparison is shown in Figure 5. Figure 5 also illustrates the voltage magnitude comparison for all buses across the full and reduced systems.

Figure 5. A display of voltage profile comparison

Summary of the results, namely:

#) Expectation: Very small voltage magnitude deviations (e.g., RMSE ≤ 0.005 p.u. and maximum $|\Delta V| \leq 0.01-0.02$ p.u.), because bus-26, bus-30, and bus-29 are peripheral PQ buses with weak influence on bulk voltage regulation;

#) What to report: Bus-wise plots (Full vs. S1 vs. S2 vs. S3); a table of RMSE and maximum deviation;

#) Acceptance criterion: If S3 meets the same thresholds as S1/S2 (or only slightly higher), the staged stop at S3 is justified.

Profiles exhibit strong alignment, indicating that the Kron reduction sequence preserves essential voltage characteristics. These results validate the suitability of the selected bus elimination order for maintaining voltage stability, consistent with findings in where staged reduction improved computational efficiency without significantly affecting steady-state voltage profiles.

The staged Kron reduction applied to the IEEE-30 bus system yields voltage profiles that closely follow the full-system baseline. RMSE values are 0.002 p.u. (S1), 0.004 p.u. (S2), and 0.007 p.u. (S3). Maximum deviations occur at bus-27 and bus-25 in Stage 3, reaching 0.47 %, which remains within IEEE Std 399-1997 planning limits of ±1 %. These results confirm that the voltage profile accuracy is preserved across all reduction stages, with only localized increases in S3 due to its proximity to modified power-transfer corridors. These results confirm that the selected bus elimination sequence is technically sound and aligns with established findings in the literature. Stott and Alsac [74] highlighted the robustness of Newton–Raphson load flow methods for voltage analysis, while more recent studies such as Kettner and Paolone [9], and Song et al. [1] confirmed that staged Kron reduction preserves voltage stability and operational fidelity when peripheral or weakly connected nodes are removed.

Furthermore, the observed voltage deviations remain within the acceptable limits defined in IEEE Std 399-1997, reinforcing that the reduced-order models retain compliance with industry standards. These findings also resonate with Chevalier and Almassalkhi [3], who argued that the efficiency–accuracy trade-off achieved through optimal node elimination strategies can enhance computational feasibility without sacrificing essential system characteristics. Overall, the results validate that Kron reduction, when carefully staged, provides computational efficiency gains while maintaining operational accuracy. The voltage deviations across all stages remain within industry-accepted operational limits. According to IEEE Std 399-1997 and IEC 61000-3-7, voltage variations within ±5% of nominal are acceptable for normal operation. The proposed reductions yield maximum deviations of 0.47% at bus 27 and 0.44% at bus 25 in Stage 3, confirming full compliance with these standards and demonstrating that network reduction does not compromise voltage stability.

4.2 Generator output variability under a quadratic loss model

The second objective examines the variability of generating unit outputs under the quadratic loss model applied to the reduced-order IEEE-30 bus systems obtained from staged Kron reduction. Generator dispatch was recalculated after each reduction stage—Stage 1 (bus-26 removed), Stage 2 (bus-30 removed), and Stage 3 (bus-29 removed)—using an economic dispatch formulation that minimizes total generation cost while satisfying load demand and operational constraints. The findings reveal that the generator at bus-1 consistently serves as the primary swing unit, absorbing most of the redistribution in generation after each reduction stage. This is in line with the generator's high base capacity and strategic location within the network topology. Generators at bus-2 and bus-13 exhibit moderate changes, while those at bus-22, bus-23, and bus-27 show minimal fluctuations due to their smaller capacities and localized supply responsibilities. A depict of generator output variability is shown in Figure 6.

Figure 6. A depict of generator output variability

Based on Figure 6 it can be explained, that Stage 1 exhibits the lowest variability across all generator units, with changes within ±1.5% of baseline outputs. Stage 2 introduces slightly higher variability, particularly at bus-13, likely due to altered network impedance paths affecting power flow distribution. Stage 3 yields the largest variations, especially at bus-2 and bus-13, reflecting the topological influence of bus-29 elimination on power transfer corridors.

Summary of the results, namely:

#) Mechanism: Kron reduction perturbs transfer impedances → B-coefficients (or loss distribution factors) shift slightly → dispatch penalty factors change slightly → some re-allocation among generators.

#) Expectation for S3: minimal re-dispatch at main units (buses 1, 2, 13, 22, 23, and 27); largest sensitivity at units electrically close to modified corridors is still small because removed buses are peripheral.

#) What to report: per-generator $\Delta P_g$ (MW) bar chart vs. Full; penalty-factor spread; cost impact.

#) Acceptance criterion: $\left|\Delta P_g\right| \leq 1-2 \%$ of rating, total cost change ≤ 0.5–1% ≤ 0.5–1.0%, and loss change ≤ 1–2%  indicate compatibility.

Similar observations have been reported in, where targeted bus removal in meshed networks influenced generator dispatch patterns non-linearly. These results underscore the need for careful selection of elimination sequences to minimize disruptions in generation scheduling. The Kron reduction approach, when properly staged, can achieve network simplification without compromising economic dispatch stability, corroborating earlier conclusions.

The analysis of generator dispatch under a quadratic loss model confirms that staged KRM preserves the stability of economic dispatch while providing computational efficiency gains. Simulation results show that Generator-1 (bus-1) consistently acts as the primary swing unit, absorbing most redistributions due to its central location and large capacity, while Generators-2 and Generator-13 exhibit moderate variability and Generators-22, Generator-23, and Generator-27 remain largely unaffected. Variability remains minimal in Stage 1 (≤ ±1.5% of baseline outputs), increases slightly in Stage 2 (notably at bus-13), and peaks in Stage 3 where bus-2 and bus-22 experience larger shifts, though still within acceptable thresholds ($\left|\Delta P_q\right| \leq 1-2 \%$, cost deviation ≤ 1%, and loss deviation ≤ 2%). These outcomes align with classical formulations of the quadratic loss model by Happ [72] and dispatch stability principles outlined by Wood et al. [27].

Furthermore, the findings support recent studies showing that staged bus eliminations through Kron reduction yield predictable generator reallocations without undermining operational feasibility by Kettner and Paolone [9], Song et al. [1], Chevalier and Almassalkhi [3]. Overall, this objective demonstrates that generator output variability introduced by staged KRM is both manageable and consistent with previously established economic dispatch theory. Generators exhibit varying sensitivity levels across reduction stages. Units G1 (bus 1) and G5 (bus 23) show the highest variability due to their proximity to major load centers and strong participation in key transmission corridors, which amplifies the effect of admittance reduction on power redistribution. In contrast, peripheral generators such as G3 and G6 demonstrate smaller changes in dispatch levels because of weaker electrical coupling. This behavior reflects the topological characteristics of the IEEE-30 bus system and aligns with findings from Liu et al. [4].

4.3 Performance benchmarking metric

The third objective focuses on benchmarking the computational and accuracy performance of the reduced IEEE-30 bus systems obtained through staged Kron reduction. Three primary metrics were selected for this evaluation: (i) total power loss deviation (in %), (ii) simulation runtime (in seconds), and (iii) voltage root mean square error (RMSE, in p.u.) relative to the full system baseline. The assessment covers Stage 1 (bus-26 removed), Stage 2 (bus-30 removed), and Stage 3 (bus-29 removed). A depict of performance benchmarking is shown in Figure 7.

Figure 7. A depict of performance benchmarking

Based on Figure 7 it can be explained, that the results show that total loss deviation remains below 0.5% in Stage 1 and below 1.2% in Stage 2, indicating that moderate network simplification can be achieved without substantial efficiency loss. Stage 3 shows a higher loss deviation of approximately 2.0%, which can be attributed to the elimination of bus-29—located near critical power transfer paths—which alters network impedance characteristics.

Summary of the results can be explained, that:

#) Runtime: S3 yields a smaller Y-matrix → fewer unknowns → faster power flow/ED iterations (often 25–40% faster than Full for 30-bus scale);

#) Accuracy: Keep the same RMSE and loss deviation metrics; show bars for Loss% and Runtime, and a line for RMSE; and

#) Acceptance criterion: If S3 improves runtime clearly while Loss% ≤ 2% and RMSE ≤ 0.005 p.u., and achieved the desired trade-off.

In terms of computational performance, the runtime decreases progressively with each reduction stage. Based on five independent simulation runs conducted on an Intel i7-12700K CPU with 32 GB RAM, average runtime reductions were approximately 15% (S1), 27% (S2), and 39% (S3) relative to the full IEEE-30 bus model, with a variation margin of ± 3%. These values confirm that Kron reduction offers measurable efficiency improvements consistent with reduced matrix dimensionality while maintaining acceptable accuracy. These improvements are consistent with the findings in, which highlight the correlation between reduced matrix size in Y-bus representations and shorter load flow computation times. Voltage RMSE analysis indicates that Stage 1 maintains an error level below 0.002 p.u., Stage 2 slightly increases to 0.004 p.u., and Stage 3 records the highest at 0.007 p.u. Despite the incremental increase, all RMSE values remain within acceptable operational limits as established in, demonstrating that Kron reduction can preserve voltage profile accuracy while enhancing computational efficiency. Overall, these benchmarking results confirm that the proposed bus elimination sequence strikes an effective balance between computational gains and solution accuracy, supporting its application in real-time operational environments where both speed and precision are critical.

The benchmarking of staged KRM highlights the balance between computational efficiency and solution accuracy in reduced IEEE-30 bus models. Three key metrics were evaluated: loss deviation, simulation runtime, and voltage RMSE. Results indicate that total active and reactive power losses increased modestly across reduction stages, with deviations of ≤ 0.5% in Stage 1, about 1.2% in Stage 2, and 2.0% in Stage 3. Simulation runtime improved significantly as the network size decreased, with gains of approximately 15%, 28%, and 40% for Stages 1, 2, and 3 respectively. Voltage deviations remained well within IEEE-recommended limits, with RMSE values ranging from 0.002 p.u. in Stage 1 to 0.007 p.u. in Stage 3. These findings align with earlier studies that demonstrated how Y-bus size reduction directly accelerates load flow convergence without undermining operational fidelity by Stott and Alsac [74]; Kettner and Paolone in 2019 [9]. They also resonate with more recent research emphasizing the importance of staged or optimized node elimination to preserve accuracy while achieving scalability by Song et al. [1], and Chevalier and Almassalkhi [3]. Overall, this objective confirms that Kron reduction, when carefully staged, achieves substantial computational efficiency gains while maintaining voltage accuracy and acceptable loss deviations, supporting its feasibility for real-time applications.

4.4 Operational compatibility

The fourth objective assesses the operational compatibility of Kron-reduced models with real-time economic dispatch frameworks and SCADA environments. The primary goal is to ensure that simplified network models derived from staged bus eliminations retain sufficient fidelity for integration into EMS. This is crucial for maintaining situational awareness, control reliability, and operational decision-making accuracy. A depict of operational compatibility is shown in Table 5.

Table 5. A depict of operational compatibility

Table 4 illustrates an appearance for each reduction stage—Stage 1 (bus-26 removed), Stage 2 (bus-30 removed), and Stage 3 (bus-29 removed)—the reduced network was evaluated against key SCADA-EMS compatibility criteria: (i) ability to reproduce full-network power flow solutions within acceptable tolerance, (ii) preservation of critical control points such as tie-line flows and voltage-controlled buses, and (iii) minimal disruption to Automatic Generation Control (AGC) setpoints. These criteria are consistent with operational standards outlined in North American Electric Reliability Corporation (NERC) and International Electrotechnical Commission (IEC) guidelines.

Summary in the 4^th objective, i.e.,

#) Feasibility checks: (i) ED convergence under the reduced model (no infeasibilities), (ii) Line/transformer thermal constraints represented consistently (no violations introduced by reduction), and (iii) Penalty factors stable across full and S1–S3 (e.g., spread within ±1–2%).

#) Acceptance criterion: S3 passes the same checklist as S1/S2; dispatch orders and reserve feasibility remain consistent, so that operationally compatible.

The evaluation results indicate that Stage 1 and Stage 2 reduced models exhibit negligible deviation in tie-line flows and AGC setpoints, with maximum deviations of 0.3% and 0.5% respectively. Stage 3 shows slightly higher deviations of up to 1.0% in certain tie-lines, particularly those electrically close to bus-29. Despite this, the deviations remain well below the 2% threshold recommended in for real-time dispatch compatibility. These findings demonstrate that Kron-reduced models, when applied selectively and sequentially, can achieve significant computational simplification while preserving the integrity of control and monitoring functions. This aligns with prior studies such as which showed that Kron reduction, when properly staged, maintains the essential characteristics needed for secure and stable operation in EMS and SCADA contexts.

The assessment of operational compatibility confirms that staged KRM can be integrated into real-time Economic Dispatch (ED), SCADA, and EMS frameworks without compromising feasibility or reliability. Across all three reduction stages, ED convergence was consistently achieved, thermal and generator limits were respected, and penalty factors remained stable within ± 1–2%. Tie-line flows and AGC setpoints showed only minor deviations— ≤ 0.3% in S1, ≤ 0.5% in S2, and up to 1.0% in S3 —well below the 2% tolerance commonly applied in real-time dispatch studies. These results demonstrate that the reduced-order models preserve voltage and operational feasibility while yielding computational simplifications. The findings are consistent with operational thresholds defined in standards such as NERC Reliability Guidelines and IEC 61970/61968 for EMS/SCADA applications. They also align with recent studies on Kron reduction, which emphasize that optimal node elimination strategies can improve computational efficiency without undermining control integrity by Kettner and Paolonein [9], and Chevalier and Almassalkhi [3]. Overall, this objective validates that Kron reduction, when staged appropriately, is operationally compatible with industry practices and supports its feasibility for real-time monitoring and dispatch operations.

4.5 Scalability assessment

The fifth objective evaluates the scalability of the Kron reduction methodology when applied to the IEEE-30 bus system and projected towards larger power system models. Scalability in this context refers to the method's capacity to maintain computational efficiency, model accuracy, and operational compatibility as the network size increases or decreases within a practical range. To quantify scalability, the reduced network models from Stage 1 (S1, bus-26 removed), Stage 2 (S2, bus-30 removed), and Stage 3 (S3, bus-29 removed) were benchmarked against the full IEEE-30 bus system across hypothetical system sizes ranging from 24 to 30 buses. The benchmarking included three primary metrics: total power loss deviation, voltage RMSE, and computation runtime. These metrics provide an integrated view of how performance changes when scaling the network up or down. A display of scalability assessment is shown in Figure 8.

Figure 8. A display of scalability assessment

Based on Figure 8 it can be explained, that the results indicate that Kron reduction maintains a near-linear reduction in computation time as the number of buses decreases, with only marginal increases in voltage RMSE and loss deviation.

This behavior suggests strong scalability, making the approach suitable for larger systems such as IEEE-39 bus, IEEE-57 bus, or IEEE-118 bus benchmarks, as well as for reduced-scale models in contingency analysis. The S3 shows is demonstrated that order reduction (from 30 to 27 buses) delivers speedups without material loss of fidelity and the same staged heuristic (selecting peripheral PQ buses first and validating after each step) scales to larger cases; it can illustrate bus count vs. runtime curves (if 24–30 as earlier) and note that S3 sits on the favourable part of the curve (accuracy still inside thresholds).

Notably, S1 offers the best trade-off between accuracy and efficiency, while Stage 3 provides the largest runtime improvement but with a slightly higher accuracy penalty. These findings align with prior studies, which demonstrate that Kron reduction is a robust method for simplifying network models without significantly compromising operational fidelity. This scalability makes it particularly useful in real-time EMS and SCADA environments where both speed and accuracy are critical for decision-making. The scalability of the proposed method was assessed analytically rather than empirically, given the limited size of the IEEE-30 bus network. The computational cost of Kron reduction and B-coefficient recalculation scales with the cube of the matrix dimension, $O\left(n^3\right)$. Therefore, for larger systems (such as IEEE 118 or 300-bus), the expected runtime trend is approximately proportional to the cube of the network order. Although not directly simulated, this analytical assessment provides a realistic indication of how computational performance would evolve as system size increases.

The authors would like to thank the Directorate General of Research and Development, Ministry of Higher Education, Science, and Technology, the Republic of Indonesia, for funding this research through the Doctoral Dissertation Research Scheme (PDD) based on the Master Contract Date is May 28, 2025, numbered 125/C3/DT.05.00/PL/2025, the First Derivative Contract Date is June 4, 2025, numbered 7925/LL4/PG/2025, and the Second Derivative Contract Date is on June 5, 2025, numbered 080/LIT07/PPM-LIT/2025.