Investment Efficiency Analysis and Evaluation of Power Grids in China: A Robust Dynamic DEA Approach Incorporating Time Lag Effects

Abstract

Effective assessment of power grid investment efficiency is crucial for optimizing resource allocation and improving operational performance. However, existing evaluation methods typically fail to account for two critical factors: inherent uncertainties in input–output data and temporal delays in investment returns. To address these limitations, this study introduces an integrated evaluation framework combining robust optimization techniques for uncertain variables with a time-lag Data Envelopment Analysis (DEA) approach to capture the multi-period dynamics and ensure resilience against external shocks and data perturbations. An empirical analysis conducted on panel data from 31 provincial power grid enterprises in China (2015–2023) reveals significant regional disparities in efficiency, particularly between coastal and resource-rich provinces. The findings highlight that excluding time-lag effects leads to systematic underestimation of efficiency and employing robust optimization yields more resilient efficiency scores amidst data uncertainties. The study contributes methodologically by advancing DEA frameworks to better reflect the complexities of power grid investments and empirically provides valuable insights for policymakers seeking to enhance investment strategies and achieve sustainable development goals.

1. Introduction

China has established ambitious strategic goals to achieve carbon peaking by 2030 and carbon neutrality by 2060 to address climate change and promote green, low-carbon development. The realization of these targets critically depends on the development of modern grid infrastructure, which requires substantial investments to enable large-scale deployment and efficient integration of renewable energy [1]. As a fundamental pillar supporting China’s economic and social development, the power sector bears indispensable responsibilities in carbon emissions [2,3]. China’s power grid sector has maintained substantial capital investments in recent years. From 2015 to 2023, the annual nationwide grid investment exceeded CNY 500 billion [4], with projections indicating that this figure will surpass CNY 800 billion by 2025 [5,6].

Notably, China introduced a “cost-plus-revenue” transmission and distribution tariff mechanism as early as 2015, imposing financial constraints on grid enterprises by regulating permitted returns and curbing inefficient investments [7]. Consequently, a systematic and scientific evaluation of grid investment efficiency becomes imperative, considering these financial and regulatory constraints. Effective evaluation is crucial for enhancing investment decision-making by refining the theoretical framework and optimizing resource allocation [8]. Existing electricity industry efficiency evaluation studies predominantly concentrate on power generation, including conventional thermal power generation performance [9,10], carbon emission efficiency in electricity generation [11], renewable energy generation efficiency [12], and renewable energy investment efficiency [13,14]. Recent research has shifted focus to efficiency evaluation in transmission and distribution systems, such as power distribution performance [15,16], environmental evaluation of electricity supply chain systems [3], and electricity distribution network sustainability assessment [17]. However, existing research on grid investment efficiency evaluation remains relatively limited, and it has not yet received sufficient attention and in-depth research from the academic and industrial communities.

The mainstream methods for evaluating power industry efficiency and investment assessment research comprise econometric approaches and Data Envelopment Analysis (DEA). DEA operates as a nonparametric mathematical programming technique that benchmarks decision-making units (DMUs, e.g., firms or projects) by constructing an efficiency frontier formed by best-practice entities, directly computing their relative efficiency on a 0–1 scale. In power industry efficiency and investment evaluation research, DEA has emerged as a predominant methodology due to its distinctive advantages that require neither predefined production functions nor subjective input–output weight assignments. This approach is exemplified in studies such as Bi et al. [10], Azadeh et al. [15], Yadav et al. [16], Tavassoli et al. [17], He et al. [18], Li [19], Sun et al. [20], Zhu and Zhu [21], Peng et al. [22]. A limited number of scholars have also adopted the DEA methodology to assess power grid investment efficiency. For example, He et al. [18] established a multidimensional evaluation framework incorporating economic, technical, and social criteria for assessing the benefits and efficiency of power grid investment projects. Sun et al. [20] applied a three-stage DEA model to assess provincial grid investment efficiency in China. However, a gap in these studies is their failure to account for the time-lag effects inherent in grid investments.

In power industry efficiency and investment assessment research, grid systems exhibit a distinctive investment–construction–revenue cycle characterized by dynamic asset formation processes. Due to extended construction periods, regulatory approvals, and system integration requirements, infrastructure projects typically undergo multi-year transitions from capital deployment to operational returns [23], resulting in significant investment time lags [24]. This temporal characteristic necessitates the adoption of dynamic evaluation frameworks to ensure accurate valuation of long-term investment returns. Conventional static evaluation approach, however, treats investments as instantaneous by assuming a strict contemporaneous relationship between inputs and outputs without time lag effects, potentially leading to long-term efficiency misestimations. Methods commonly employed to address time-lag effects are predominantly based on regression analysis, including multiple regression [25], Almon polynomial distributed lag models [26], and vector autoregression [27]. These methods generally proceed by first determining the number of lag periods based on statistical criteria or empirical assumptions. A regression model is then selected, its parameters estimated using historical data, and new variables are generated to incorporate lagged effects, serving as synthetic indicators that amalgamate past observations. These resulting pseudo-variables are treated as adjusted input or output indicators, forming a revised dataset designed to account for time-lag effects, which is ultimately used within a DEA framework for efficiency assessment. However, they rely on presupposed functional forms (e.g., linear or quadratic), effectively imposing a parametric and assumption-laden structure on the raw output data. Consequently, the resulting values deviate from actually observed outputs, becoming “pseudo-outputs” that are smoothed or artificially adjusted. Incorporating such pseudo-outputs into DEA models means the analyzed data no longer reflect the true production technology, and the resulting production frontier is based on modified data. This may obscure the economic interpretation of efficiency scores and could lead to misleading conclusions. Recent methodological advances have extended DEA to specifically address the characteristic delay effects. Özpeynirci and Köksalan [28] developed a time-lagged (TL) DEA approach that explicitly considers temporal discontinuities between inputs and outputs when assessing decision-making units’ efficiency. This approach captures the dynamic input–output relationship where inputs may yield partial effects across multiple subsequent periods and outputs often reflect cumulative impacts of historical inputs [29,30]. Despite these developments, the integration of time-lag effects into studies on power grid investment efficiency remains limited.

Moreover, power grid efficiency evaluation is also inherently subject to uncertainties that challenge accurate assessment. These uncertainties originate from internal data limitations, including measurement errors and data quality issues, and external environmental volatility such as economic fluctuations, policy changes, and technological disruptions. Such compounded uncertainties significantly distort operational parameters, rendering conventional evaluation methods inadequate. However, current research on the evaluation of power grid efficiency investment failed to adequately account for data uncertainty. For instance, studies by He et al. [18] and Sun et al. [20] employ deterministic DEA methods, which demonstrate particular vulnerability in this context, where minor data perturbations can produce disproportionately large ranking variations [31]. While recent advancements in DEA methodology have incorporated robust optimization techniques to address uncertainty [32], the Robust DEA approach improves parameter resilience by ensuring optimal efficiency scores even in worst-case scenarios and is efficient in all possible realizations of the uncertain parameters without relying on assumptions about probability distributions or statistical properties [33]. The application of these improved frameworks to power grid evaluation remains conspicuously absent in the extant literature.

Investment characteristics in the power industry exhibit two distinct features: (1) pronounced time-lagged effects in investment returns, and (2) substantial uncertainties inherent in the assessment process. However, contemporary DEA frameworks exhibit a critical methodological gap, as none can simultaneously incorporate both the time-lag effects and the uncertainty factors inherent in such investments. This theoretical gap results in systematic biases in assessing the efficiency of power infrastructure investments, particularly in terms of inadequate quantification of long-term investment returns and limited capacity to assess resilience against market risks.

Addressing existing research gaps, this study proposes a novel, robust Dynamic Time-Lagged (DTL) DEA framework to establish a systematic framework for evaluating the investment efficiency of power grid enterprises. The proposed framework constructs a dynamic multi-period evaluation model to identify temporal variations in efficiency and their drivers, further incorporates a time-lagged effects mechanism to capture delayed investment benefit transmissions, and finally employs robust optimization techniques to handle parameter uncertainties. This integrated approach ensures analytical results that are both accurate and robust. To validate the model’s applicability, we conduct an empirical analysis using panel data from 31 provincial grid enterprises in China from 2015 to 2023. The study systematically evaluates regional grid investment efficiency and identifies key drivers of efficiency losses. These findings offer critical quantitative insights for optimizing grid investment structures and enhancing resource allocation efficiency. This study makes three significant contributions to the field of power grid investment efficiency analysis:

(1) A novel robust DTL DEA model is proposed, integrating dynamic DEA with time-lag effects and robust optimization techniques. This approach addresses delayed investment benefits and inherent uncertainties in efficiency assessment, providing a more accurate and resilient alternative to conventional static and deterministic models.

(2) A comprehensive framework is developed for evaluating the efficiency of power grid investments. This framework accounts for both the delayed nature of investment returns and the uncertainties arising from data collection processes and environmental dynamics.

(3) An in-depth analysis of the investment performance of China’s power grids reveals both efficiency deficiencies and the underlying causes, particularly when accounting for operational uncertainties. These findings provide decision-makers with valuable insights into current management practices and offer actionable recommendations for enhancing future investment strategies and resource allocation efficiency.

The remainder of this paper is structured as follows: Section 2 details the problem characteristics and theoretical foundations. Section 3 presents the methodological framework. Section 4 applies the model to China’s provincial power grids, followed by results discussion. Section 5 concludes with policy implications and research directions.

2. Problem Statement

2.1. Power Grid Investment Efficiency Evaluation

With the continuous development of the socio-economic system, the rapid growth in electricity demand, and the large-scale deployment of renewable energy, power grids face increasing requirements for operational stability and reliability, which necessitate rational resource investment to enhance operational efficiency. Given constraints in capital and material resources, there is an urgent need to effectively evaluate power grid investment efficiency and achieve optimal resource allocation [13]. The evaluation of power grid investment efficiency aims to quantitatively assess the actual impact of investments on grid performance by analyzing the relationship between inputs and outcomes, thereby identifying feasible approaches to improving operational efficiency. This evaluation examines the operational performance of grid enterprises, assessing their ability to efficiently utilize resources (e.g., human, physical, finance) to achieve operational objectives (supply reliability, electricity sale volume). It aims to identify potential improvements in resource allocation and operational outcomes, thereby providing a basis for enhancing investment decision-making and overall grid operation quality and efficiency.

2.2. Features of Power Grid Investment

Numerous challenges arise in assessing the efficiency of power grid investments. Primarily, the evaluation process is fraught with a considerable time lag effect on investment returns due to massive investment scales, high capital intensity, and long construction and payback durations, and significant uncertainties caused by policy changes, technology revisions, and the accuracy of data collecting.

2.2.1. Lagged Returns of Power Grid Investments

Power grid construction is a complex system engineering endeavor that demands significant time and resources, encompassing multiple technological domains. As a result, a considerable production lag is inherent in power grid investment. Firstly, the construction period of power grid infrastructure spans several years. Second, there is a waiting period between investment and benefit realization because the power infrastructure must supply the electricity needed for economic and social development beforehand. Thirdly, the benefits of power grid investments depend on the coordinated operation of the entire power system. Lastly, the economic benefits of power grid investments exhibit a cumulative characteristic because these possible advantages progressively transform into measurable output indicators, such as increases in energy production and reliability. Therefore, the power grid investments’ output lag is an unavoidable result of their technical and financial characteristics, making it a crucial element that needs to be specifically taken into account during evaluation procedures.

2.2.2. Efficiency Evaluation Under Uncertainty

In efficiency assessment of power grid investment, data uncertainty frequently arises due to factors such as human error, measurement inaccuracies, or sampling biases during the data collection process. These issues can lead to distortions in the data, thereby affecting the reliability of the efficiency evaluation. Furthermore, it is also significantly impacted by external environmental changes, including economic fluctuations, policy shifts, technological advancements, and unforeseen emergencies. These changes affect the input–output parameters used in system assessments, thereby affecting the accuracy of efficiency evaluations. The cumulative impact of multiple uncertainties amplifies the potential for deviation between static assessment results, which are predicated on deterministic assumptions, and the actual operational reality of the power grid.

3. Methodology

In this section, we introduce a Dynamic Time-Lagged (DTL) DEA model designed to account for delayed returns on investment and robust optimization techniques to handle inherent uncertainties in input and output variables when evaluating electricity grid investments. Accordingly, we develop a novel robust DTL DEA model with dynamic formulations to simultaneously address time lag effects and data uncertainty. Figure 1 illustrates comprehensive methodology for assessing the efficiency of electricity grid investments.

Before establishing models, the following notations need to be introduced in

Table 1. Notations.

Notation	Description
Indexes
j	The j-th DMU, $j=1,2,\dots ,J$
i	The i-th input variable, $i=1,2,\dots ,I$
r	The r-th output variable, $r=1,2,\dots ,R$
t	The t-th period, $t=1,2,\dots ,T$
Parameter
$x_{ijt}$	The i-th input variable of the j-th DMU in the t-th period
$y_{rjt}$	The r-th output variable of the j-th DMU in the t-th period
$\phi ,\xi ,\eta$	Uncertainty parameter vectors
$\Im ,\mathsf{\Xi },\Re$	Uncertainty sets
$\tau ,\lambda ,w$	Auxiliary variable
${\tilde{x}}_{ijt},{\tilde{y}}_{rjt}$	Uncertainty variable
$x_{ijt}^0,y_{rjt}^0$	Nominal values of the uncertainty variables ${\tilde{x}}_{ijt},{\tilde{y}}_{rjt}$
${\alpha }_{ij}$	Parameter for the i-th input uncertainty deviation value of the j-th DMU
${\beta }_{rj}$	Parameter for the r-th output uncertainty deviation value of the j-th DMU
${\mathsf{\alpha }}_j,{\mathsf{\beta }}_j$	Vector of ${\alpha }_{ij},{\beta }_{rj}$
${\widehat{x}}_{ijt}$	Fluctuation value for the i-th input uncertainty of the j-th DMU in period t
${\widehat{y}}_{rjt}$	Fluctuation value for the r-th output uncertainty of the j-th DMU in period t
${\Gamma }_j^{\alpha }$	Robust coefficients for the input variable of the j-th DMU
${\Gamma }_j^{\beta }$	Robust coefficients for the output variable of the j-th DMU
${\theta }_t$	Weight coefficient of efficiency value for the t-th period
Decision variables
$v_{it}$	Reference weights for the i-th input variable associated with DMUj
$u_{rt}$	Reference weights for the r-th output variable associated with DMUj
$z_{jt}^x,z_{jt}^y$	Dual variables associated with input of the j-th DMU in period t
$p_{ijt}^x,p_{rjt}^y$	Dual variables for the i-th input/r-th output associated with DMUj in period t

.

3.1. DEA Model with Time Lag Effects

3.1.1. CCR DEA Model

The CCR DEA model is the classic evaluation model, with the basic assumption of Constant Return to Scale (CRS). Assuming that there are $J(j=1,2,3,\dots ,J)$ DMUs, and each DMU has $I(i=1,2,3,\dots ,I)$ input variables generating $R(r=1,2,3,\dots ,R)$ output variables, where $x_{ij}$ represents the value of $i$th input of the $j$th DMU, $y_{rj}$ denotes the value of $r$th output of the $j$th DMU, $v_i$ and $u_r$ is the weight coefficient for the $i$th input and $r$th output, respectively. The efficiency score of the $O$th DMU is calculated by maximizing the weighted sum of outputs divided by the weighted sum of inputs. The linearized DEA model can be obtained by using Charnes–Cooper variation, as follows:

$$\begin{array}{l}{E}_O=\max {\displaystyle \sum _{r=1}^R{u}_r{y}_{rO}}\\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{r=1}^R{u}_r{y}_{rj}}-{\displaystyle \sum _{i=1}^I{v}_i{x}_{ij}}\le 0 \forall j\\ {\displaystyle \sum _{i=1}^I{v}_i{x}_{iO}}=1\\ v_i\ge 0 \forall i\\ u_r\ge 0 \forall r\end{array}\right.\end{array}$$

(1)

3.1.2. DTL DEA Model

The conventional CCR DEA model is fundamentally characterized as a static framework, operating under the core assumption that a given period’s outputs are exclusively determined by contemporaneous inputs, while inputs in any period solely affect outputs within that same period. However, the assumption fails to hold validity across numerous real-world applications. In practice, a substantial proportion of operational activities exhibit intertemporal effects, thereby making the system inherently multi-period (dynamic) in nature. The dynamic DEA framework explicitly accounts for these persistent activities, where a weighted average is employed to quantify individual periods’ contributions to the overall performance of DMUs.

Furthermore, a prevalent “production time lag” phenomenon exists between resource inputs and output outcomes under the dynamic evaluation framework [24]. Specifically, inputs affect both current outputs and subsequent-period output through sustained carryover effects. Systemically, observed outputs reflect cumulative effects from both contemporaneous and historical inputs (see Figure 2). Therefore, a Dynamic Time-Lag (DTL) DEA approach, proposed by Özpeynici and Köksalan, was introduced to account for time-lag effects. This approach incorporates two modeling methods based on input–output relationships [29], as show in Figure 2.

(1)

Input DTL DEA model

In this model, we assume the input variables have an impact duration spanning D periods, where the time lag period (LP) is set as $LP=D-1$. Consequently, the output in period $t$ is jointly determined by the inputs from the preceding periods $(t-D+1,\dots ,t-1,t)$, as illustrated in Figure 2, where LP = 3 and $D=4$ indicate that current-period outputs are influenced by inputs from four consecutive periods (current period plus three preceding periods). The efficiency is defined as the ratio of single-period output $y_{rjt}$ to multi-period inputs ${\displaystyle {\sum }_{\lambda =0}^{D-1}{x}_{ij(t-\lambda )}}$. Equation (2) calculates the efficiency values of all DMUs in periods $[D,T]$, where $E_{O,t}^{Input}$ denotes the efficiency value of $DM{U}_O$ in period $t$ from Input DTL DEA model.

$$\begin{array}{l}(Input DTL DEA)\\ {\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=D}^T{E}_{O,t}^{Input}}}=\max {\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{y}_{rOt}{u}_{rt}}}}\\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{r=1}^R{y}_{rjt}{u}_{rt}-}{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{x}_{ij(t-\lambda )}{v}_{i(t-\lambda )}}}\le 0 \forall j,t=D,\dots ,T\\ {\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{x}_{iO(t-\lambda )}{v}_{i(t-\lambda )}}}=1 \forall t=D,\dots ,T\\ v_{it}\ge 0,u_{rt}\ge 0 \forall i,r,t\end{array}\right.\end{array}$$

(2)

To account for the relative importance of each period within the entire time horizon, the overall efficiency value is calculated as a weighted sum of individual period efficiencies. After solving Model (2), the optimal solution ${(u_{rt})}^{\ast }$ are obtained to define the period efficiency scores. Below are the formal definitions for both overall and period efficiency values.

Definition 1. 

Overall efficiency value of $DM{U}_O$ from D_th to T_th periods can be defined as

$$E_O^{Input}=\max {\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t{y}_{rOt}{u}_{rt}}}$$

(3)

Definition 2. 

Period efficiency value of $DM{U}_O$ in t_th period can be defined as

$$E_{O,t}^{Input}={\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t{y}_{rOt}{u}_{rt}^{\ast }}}$$

(4)

(2)

Output DTL DEA model

From the output perspective, this model assumes that the input in period $t$ influences outputs over the subsequent D periods $(t,t+1,\dots ,t+D-1)$. The efficiency is measured as the ratio of multi-period outputs ${\displaystyle {\sum }_{\lambda =0}^{D-1}{y}_{rj(t+\lambda )}}$ to single-period input $x_{ijt}$. Denote that Output DTL DEA is a model for the overall efficiency of $DM{U}_O$ for periods $[D,T]$.

$$\begin{array}{l}(Output DTL DEA)\\ E_O^{Output}=\max {\displaystyle \sum _{t=1}^{T-D+1}{\displaystyle \sum _{r=1}^R{\displaystyle \sum _{\lambda =0}^{D-1}{\theta }_t{y}_{rj(t+\lambda )}{u}_{r\lambda }}}}\\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{y}_{rj(t+\lambda )}{u}_{r\lambda }}}-{\displaystyle \sum _{i=1}^I{x}_{ijt}{v}_{it}}\le 0 \forall j,t=1,\dots ,t-D+1\\ {\displaystyle \sum _{i=1}^I{x}_{iOt}{v}_{it}}=1 \forall t=1,\dots ,T-D+1\\ v_{it}\ge 0,u_{rt}\ge 0 \forall i,r,t\end{array}\right.\end{array}$$

(5)

3.2. Robust DEA Model with Time LAG Effects

3.2.1. Robust Optimization Model

Robust optimization aims to find a solution that performs well across all uncertain variables, guaranteeing that the best possible outcome is achieved in the worst-case scenario, thereby ensuring the solution remains feasible and nearly optimal, even in the face of data perturbations [34]. The generalized model with uncertain parameters can be expressed as

$$\begin{array}{l}\underset{\mathrm{x}}{\max } \mathrm{c}(\phi {)}^{⊺}\mathrm{x}\\ s.t. \mathrm{A}(\xi )\mathrm{x}\le \mathrm{b}(\eta ),\forall \phi \in \Im ,\xi \in \mathsf{\Xi },\eta \in \Re \end{array}$$

(6)

where $\mathrm{x}\in {ℝ}^n$ denotes all decision variables, whereas $\mathrm{c}\in {ℝ}^n,\mathrm{A}\in {ℝ}^{m\times n} \mathrm{and} \mathrm{b}\in {ℝ}^m$ are coefficient, $\phi \in {ℝ}^{L1},\xi \in {ℝ}^{L2} \mathrm{and} \eta \in {ℝ}^{L3}$ represent the uncertainty parameter vectors, $\Im ,\mathsf{\Xi }, \mathrm{and} \Re$ are the prespecified uncertainty sets. It is worth noting that the optimization problem involves three types of uncertain coefficients:

(1) Uncertainty in the objective function. To maintain the model’s generality while ensuring computational tractability of dual solutions, we reformulate the problem by introducing an auxiliary variable $\tau$ [35]. Model (6) can be equivalently rewritten as follows:

$$\begin{array}{l}{\displaystyle \underset{\tau ,\mathrm{x}}{\max }} \tau \\ s.t.\left\{\begin{array}{l}\tau \le \mathrm{c}(\phi {)}^{⊺}\mathrm{x},\forall \phi \in \Im \\ \mathrm{A}(\xi )\mathrm{x}\le \mathrm{b}(\eta ),\forall \xi \in \mathsf{\Xi },\eta \in \Re \end{array}\right. \end{array}$$

(7)

(2) Right-hand side uncertainty (RHS). We can rewrite Equation (5) in a generalized form by introducing auxiliary variable vector ${\mathrm{x}}_0=1, {\mathrm{x}}_0\in {ℝ}^m$:

$$\begin{array}{l}\underset{\tau ,\mathrm{x}}{\max } \tau \\ s.t.\left\{\begin{array}{l}\tau \le \mathrm{c}(\phi {)}^{⊺}\mathrm{x},\forall \phi \in \Im \\ \left[\mathrm{A}(\xi )-\mathrm{b}(\eta )\right]{\mathrm{x}}^{\prime }\le 0,\forall \xi \in \mathsf{\Xi },\eta \in \Re \\ {\mathrm{x}}^{\prime }=[\mathrm{x},1]\end{array}\right. \end{array}$$

(8)

(3) Uncertainty in equality constraint. In robust optimization problems, convex optimization techniques are frequently employed to manage uncertainty. Nevertheless, the convexity may be compromised by the equality constraint, leading to overly conservative or unfeasible solutions to the problem. The equality constraint $\mathrm{A}(\xi )\mathrm{x}=\mathrm{b},\forall \xi \in \mathsf{\Xi }$ can be transformed into inequality constraints $\mathrm{A}(\xi )\mathrm{x}\le \mathrm{b} \forall \xi \in \mathsf{\Xi }$. The details of proof refer to the study [32].

The aforementioned three kinds of uncertainties can all be reformulated as left-hand side (LHS) uncertainties, which will be used to illustrate the solution process.

$$\begin{array}{l}\underset{\mathrm{x}}{\max } {\mathrm{c}}^{⊺}\mathrm{x}\\ s.t. \mathrm{A} (\xi )\mathrm{x}\le \mathrm{b},\forall \xi \in \mathsf{\Xi }\end{array}$$

(9)

Robust optimization technique generates an optimal solution for a given uncertainty set. The choice of uncertain sets is essential since they directly affect the final solution’s robustness and conservatism, and better conservatism is frequently linked to uncertain sets’ greater resilience to parameter perturbations. Budget uncertainty sets are more computationally efficient than elliptical uncertainty sets and produce less conservative solutions than the box uncertainty set. The budget uncertainty set can be written as [36]:

$$\mathsf{\Xi }=\left\{\xi \in {ℝ}^L:\xi ={\mu }^0+\mu \delta ,{‖\delta ‖}_{\infty }\le 1,{‖\delta ‖}_1\le \Gamma \right\}$$

(10)

where ${\mu }^0$ represents the nominal value, $\mu$ is the deviation vector from ${\mu }^0$, $\mu \delta$ is the perturbation, $‖\cdot ‖$ represents a generic norm, $\Gamma$ is the protection level of the uncertain parameter. The constraint ${‖\delta ‖}_{\infty }\le 1=\underset{1\le l\le L}{\max }\left|{\delta }_l\right|\le 1$ controls uncertainty vector $\xi$, which follows a symmetric distribution centered at the nominal value ${\mu }^0$ over the interval $\left[{\mu }^0-\mu ,{\mu }^0+\mu \right]$, and it results in a box uncertainty set. Bertsimas and Sim [36] reduce the conservatism of box uncertainty set by constraining the number of uncertain parameters that can deviate from their nominal values simultaneously (${‖\delta ‖}_1\le \Gamma ={\displaystyle \sum _{l=1}^L\left|{\delta }_l\right|}\le \Gamma$). This approach implies that the worst-case scenario occurs when exactly $\Gamma$ uncertain parameters become simultaneously uncertain, with each parameter taking its maximum possible deviation. Optimization problems with budgeted uncertainty sets can be reformulated as linear programming problems through the application of duality theory. Further details can be found in the study [32].

3.2.2. Robust DTL DEA Model

Input and output indicators in DEA model frequently exhibit data uncertainty due to measurement errors during data collection and the dynamic nature of external environmental factors. Here, we use Input DTL DEA model as example to construct robust model. Input DTL DEA Model (2) with uncertain inputs and outputs in the objective functions and equality constraints can be transformed into a general robust optimization framework, as follows:

$$\begin{array}{l}(DTL DEA under \mathit{uncertainty})\\ E{U}_O=\max \tau \\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t{\tilde{y}}_{rOt}{u}_{rt}}}\ge \tau \\ {\displaystyle \sum _{r=1}^R{\tilde{y}}_{rjt}{u}_{rt}-}{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{\tilde{x}}_{ij(t-\lambda )}{v}_{i(t-\lambda )}}}\le 0 \forall j,t=D,\dots ,T \\ {\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{\tilde{x}}_{iO(t-\lambda )}{v}_{i(t-\lambda )}}}\le 1 \forall t=D,\dots ,T\\ v_{it}\ge 0,u_{rt}\ge 0 \forall i,r,t\end{array}\right.\end{array}$$

(11)

Then we construct the budgeted uncertainty set for input and output variables as follows:

$${\tilde{x}}_{ijt}=x_{ijt}^0+{\alpha }_{ij}{\widehat{x}}_{ijt},{‖{\mathsf{\alpha }}_j‖}_{\infty }\le 1,{‖{\mathsf{\alpha }}_j‖}_1\le {\Gamma }_j^{\alpha }$$

(12)

$${\tilde{y}}_{rjt}=y_{rjt}^0+{\beta }_{rj}{\widehat{y}}_{rjt},{‖{\mathsf{\beta }}_j‖}_{\infty }\le 1,{‖{\mathsf{\beta }}_j‖}_1\le {\Gamma }_j^{\beta }$$

(13)

Let $x_{ijt}^0$ and $y_{rjt}^0$ denote the nominal values of the uncertainty variables ${\tilde{x}}_{ij}$ and ${\tilde{y}}_{rj}$ in the period $t$, respectively, indicating the variables fluctuate around the nominal value within the ranges ${\alpha }_{ij}{\widehat{x}}_{ijt}$ and ${\beta }_{rj}{\widehat{y}}_{rjt}$. We assume the data fluctuation remains constant across all periods, meaning the coefficients ${\alpha }_{ij}$ and ${\beta }_{rj}$ maintain identical values throughout all cycles. Therefore, due to the constraints ${‖{\mathsf{\alpha }}_j‖}_{\infty }\le 1$ and ${‖{\mathsf{\beta }}_j‖}_{\infty }\le 1$, the random variables ${\alpha }_{ij}$ and ${\beta }_{rj}$ follows an unknown but symmetric distribution, taking values within $[-1,1]$ for all periods. Consequently, the feasible ranges for ${\tilde{x}}_{ijt}$ and ${\tilde{y}}_{rjt}$ are confined to $[x_{ijt}^0-{\widehat{x}}_{ijt},x_{ijt}^0+{\widehat{x}}_{ijt}]$ and $[y_{rjt}^0-{\widehat{y}}_{rjt},y_{rjt}^0+{\widehat{y}}_{rjt}]$. Therefore, these constraints, ${‖{\mathsf{\alpha }}_j‖}_1\le {\Gamma }_j^{\alpha }$ and ${‖{\mathsf{\beta }}_j‖}_1\le {\Gamma }_j^{\beta }$, are introduced to mitigate excessive conservatism, as the simultaneous occurrence of extreme values for all uncertain parameters is statistically improbable. The robust coefficients ${\Gamma }_j^{\alpha }$ and ${\Gamma }_j^{\beta }$ are constrained within the intervals $[0,\left|I\right|]$ and $[0,\left|R\right|]$, where $\left|\cdot \right|$ represents the cardinal number of the set, respectively. When ${\Gamma }_j^{\alpha }=0 \mathrm{and} {\Gamma }_j^{\beta }=0$, the problem becomes deterministic; it degenerates into box uncertainty sets when ${\Gamma }_j^{\alpha }=\left|I\right| \mathrm{and} {\Gamma }_j^{\beta }=\left|R\right|$. The parameter $\Gamma$ regulates the trade-off between constraint violation probability and its impact on the nominal problem’s objective function. Model (11) with budgeted uncertainty sets (Equations (10) and (11)) can be further transformed into

$$\begin{array}{l}(\mathit{Robust} DTL DEA under \mathit{uncertainty})\\ E{U}_O^{Robust}=\max \tau \\ s.t.\left\{\begin{array}{l}\tau \le {\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t(y_{rOt}^0+{\beta }_{rO}{\widehat{y}}_{rOt})u_{rt}}}\\ {\displaystyle \sum _{r=1}^R(y_{rjt}^0+{\beta }_{rj}{\widehat{y}}_{rjt})u_{rt}-}{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I(x_{ij(t-\lambda )}^0+{\alpha }_{ij}{\widehat{x}}_{ij(t-\lambda )})v_{i(t-\lambda )}}}\le 0 \forall j,t=D,\dots ,T \\ {\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I(x_{iO(t-\lambda )}^0+{\alpha }_{io}{\widehat{x}}_{io(t-\lambda )})v_{i(t-\lambda )}}}\le 1 \forall t=D,\dots ,T\\ v_{it}\ge 0,u_{rt}\ge 0 \forall i,r,t\\ {‖{\mathit{\beta }}_j‖}_{\infty }\le 1,{‖{\mathit{\beta }}_j‖}_1\le {\Gamma }_j^{\beta } \forall j\\ {‖{\mathit{\alpha }}_{jt}‖}_{\infty }\le 1,{‖{\mathit{\alpha }}_{jt}‖}_1\le {\Gamma }_j^{\alpha } \forall j\end{array}\right.\end{array}$$

(14)

The constraint $c\le {\displaystyle {\sum }_{t=D}^t{\displaystyle {\sum }_{r=1}^R{\theta }_t\cdot (y_{rOt}^0+{\beta }_{rO}{\widehat{y}}_{rOt})u_{rt}}}$ means that it is valid for each possibility, so it can be further converted into $c\le {\displaystyle {\sum }_{t=D}^T{\displaystyle {\sum }_{r=1}^R{\theta }_t\cdot y_{rOt}^0{u}_{rt}}}-\underset{{‖{\mathsf{\beta }}_j‖}_{\infty }\le 1,{‖{\mathsf{\beta }}_j‖}_1\le {\Gamma }_j^{\beta } }{\max }{\displaystyle {\sum }_{t=D}^T{\displaystyle {\sum }_{r=1}^R{\theta }_t\cdot {\beta }_{rOt}{\widehat{y}}_{rOt}{u}_{rt}}}$. Similarly, following this principle, Equation (14) is equivalent to

$$\begin{array}{l}(Robust DTL DEA)\\ E{U}_O^{Robust}=\max \tau \\ s.t.\left\{\begin{array}{l}\tau \le {\displaystyle \sum _{t=D}^t{\displaystyle \sum _{r=1}^R{\theta }_t{y}_{rOt}^0{u}_{rT}}}-\underset{{‖{\mathsf{\beta }}_j‖}_{\infty }\le 1,{‖{\mathsf{\beta }}_j‖}_1\le {\Gamma }_j^{\beta } }{\max }{\displaystyle \sum _{t=D}^t{\displaystyle \sum _{r=1}^R{\theta }_t{\beta }_{rO}{\widehat{y}}_{rOt}{u}_{rt}}}\\ \left\{\begin{array}{l}{\displaystyle \sum _{r=1}^R{y}_{rjt}^0{u}_{rt}-}{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{x}_{ij(t-\lambda )}^0{v}_{i(t-\lambda )}}}+\\ \underset{{‖{\mathsf{\beta }}_j‖}_{\infty }\le 1,{‖{\mathsf{\beta }}_j‖}_1\le {\Gamma }_j^{\beta } }{\max }{\displaystyle \sum _{r=1}^R{\beta }_{rj}{\widehat{y}}_{rjt}{u}_{rt}}+\underset{{‖{\mathsf{\alpha }}_j‖}_{\infty }\le 1,{‖{\mathsf{\alpha }}_j‖}_1\le {\Gamma }_j^{\alpha } }{\max }{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{\alpha }_{ijt}{\widehat{x}}_{ij(t-\lambda )}{v}_{i(t-\lambda )}}}\le 0 \forall j,t=D,\dots ,T\end{array}\right.\\ {\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{x}_{iO(t-\lambda )}^0{v}_{i(t-\lambda )}}}+\underset{{‖{\mathsf{\alpha }}_j‖}_{\infty }\le 1,{‖{\mathsf{\alpha }}_j‖}_1\le {\Gamma }_j^{\alpha } }{\max }{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{\alpha }_{iO}{\widehat{x}}_{iO(t-p)}{v}_{i(t-\lambda )}}}\le 1 \forall t=D,\dots ,T\\ v_{it}\ge 0,u_{rt}\ge 0 \forall i,r,t\end{array}\right.\end{array}$$

(15)

In the first constraint, $\underset{{‖{\mathsf{\beta }}_j‖}_{\infty }\le 1,{‖{\mathsf{\beta }}_j‖}_1\le {\Gamma }_j^{\beta } }{\max }{\displaystyle {\sum }_{t=D}^t{\displaystyle {\sum }_{r=1}^R{\theta }_t\cdot {\beta }_{rO}{\widehat{y}}_{rOt}{u}_{rt}}}$ is equal to an objective function of the maximum optimization problem:

$$\begin{array}{l}\max {\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t\cdot {\widehat{y}}_{rOt}\left|u_{rt}\right|\cdot w_{rOt}^y}}\\ s.t.\left\{\begin{array}{l}0\le w_{rOt}^y\le 1 \forall r,t\\ {\displaystyle \sum _{r=1}^R{w}_{rOt}^y\le {\Gamma }_{Ot}^{\beta }} \forall t\end{array}\right.\end{array}$$

(16)

The dual model of Model (16) is as follows:

$$\begin{array}{l}\min {\displaystyle \sum _{t=D}^T{\theta }_t\cdot ({\Gamma }_{Ot}^{\beta }{z}_{Ot}^y+{\displaystyle \sum _{r=1}^R{p}_{rOt}^y})}\\ s.t.\left\{\begin{array}{l}{z}_{ot}^y+p_{rot}^y\ge u_{rt}{\widehat{y}}_{rot} \forall r,t\\ p_{rot}^y\ge 0 \forall r,t\\ z_{ot}^y\ge 0 \forall r,t\\ u_{rt}\ge 0 \forall r,t\end{array}\right.\end{array}$$

(17)

Following the aforementioned duality-based transformation, the robust counterpart of Model (15) reduces to a computationally tractable linear programming problem that can be solved efficiently in polynomial time, as formulated below:

$$\begin{array}{l}(Robust \mathit{counterpart} DTL DEA )\\ E{U}_O^{RC}=\max \tau \\ s.t.\left\{\begin{array}{l}-{\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t{u}_{rt}{y}_{rot}^0}}+{\displaystyle \sum _{t=D}^T{\theta }_t({\Gamma }_{Ot}^{\beta }{z}_{Ot}^y+{\displaystyle \sum _{r=1}^R{p}_{rOt}^y})}+\tau \le 0\\ \left\{\begin{array}{l}{\displaystyle \sum _{r=1}^R{y}_{rjt}^0{u}_{rt}-}{\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{x}_{ij(t-\lambda )}^0{v}_{i(t-\lambda )}}}+{\Gamma }_{jt}^y{z}_{jt}^y+{\displaystyle \sum _{r=1}^R{p}_{rjt}^y}\\ +{\displaystyle \sum _{\lambda =0}^{D-1}({\Gamma }_{j(t-\lambda )}^x{z}_{j(t-\lambda )}^x}+{\displaystyle \sum _{i=1}^I{p}_{ij(t-\lambda )}^x})\le 0 \forall j,\forall t=D,\dots ,T\end{array}\right.\\ {\displaystyle \sum _{\lambda =0}^{D-1}{\displaystyle \sum _{i=1}^I{x}_{iO(t-\lambda )}^0{v}_{i(t-\lambda )}}}+{\displaystyle \sum _{\lambda =0}^{D-1}({\Gamma }_{O(t-\lambda )}^x{z}_{O(t-\lambda )}^x}+{\displaystyle \sum _{i=1}^I{p}_{iO(t-\lambda )}^x})\le 1 \forall t=D,\dots ,T\\ z_{jt}^y+p_{rjt}^y\ge u_{rt}{\widehat{y}}_{rjt} \forall r,j,t\\ z_{jt}^x+p_{ijt}^x\ge v_{it}{\widehat{x}}_{ijt} \forall i,j,t\\ v_{it}\ge 0,u_{rt}\ge 0, p_{rjt}^y\ge 0,p_{ijt}^x\ge 0,z_{jt}^y\ge 0,z_{jt}^x\ge 0 \forall i,r,j,t\end{array}\right.\end{array}$$

(18)

After solving Model (18), overall efficiency of $DM{U}_O$ from $D_{th}$ to $T_{th}$ periods are obtained, which is $\tau$. And the optimal solutions ${(u_{rt})}^{\ast },{(z_{jt}^y)}^{\ast },{(p_{rjt}^y)}^{\ast }$ are obtained to define the period efficiency scores. Below are the formal definitions for both overall and period efficiency values.

Definition 3. 

According to model (18), overall efficiency value of $DM{U}_O$ from D_th to T_th periods can be defined as

$$E{U}_O^{RC}=\max {\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t{u}_{rt}{y}_{rot}^0}}-{\displaystyle \sum _{t=D}^T{\theta }_t({\Gamma }_{Ot}^{\beta }{z}_{Ot}^y+{\displaystyle \sum _{r=1}^R{p}_{rOt}^y})}$$

(19)

Definition 4. 

According to model (18), period efficiency value of $DM{U}_O$ in t_th period can be defined as

$$E{U}_{O,t}^{RC}=\max \left({\displaystyle \sum _{t=D}^T{\displaystyle \sum _{r=1}^R{\theta }_t{u}_{rt}^{\ast }{y}_{rot}^0}}-{\displaystyle \sum _{t=D}^T{\theta }_t({\Gamma }_{Ot}^{\beta }{z}_{Ot}^{y \ast }+{\displaystyle \sum _{r=1}^R{p}_{rOt}^{y \ast }})}\right)$$

(20)

4. Case Study: Investment Efficiency Evaluation of Power Grid in China

4.1. Overview of China’s Power Grids

The power grid, which serves as the foundation of China’s energy transmission and distribution network, is strategically essential for promoting socioeconomic growth, guaranteeing energy security, and easing the integration of renewable energy sources. In 2015, China implemented a landmark policy, the Opinions on Further Deepening the Reform of the Electric Power System, which prioritized the reform of transmission and distribution tariffs by establishing a new regulatory mechanism based on “permitted costs plus reasonable returns”. The reform effectively broke traditional monopoly structures and introduced competitive market mechanisms. In this context, the reform implemented an earnings regulation mechanism that imposes investment capacity constraints on grid enterprises, thereby elevating the requirements for their investment efficiency.

Therefore, conducting a scientific assessment of provincial-level power grid investment efficiency in China carries significant theoretical and practical implications. The evaluation can identify optimization potential in regional grid investments, providing policymakers with evidence-based support for formulating targeted and differentiated regulatory measures. Additionally, the evaluation results can guide grid companies in optimizing resource allocation and enhancing operational efficiency, thereby promoting high-quality development and facilitating the achievement of carbon peak and carbon neutrality goals.

4.1.1. Inputs and Outputs of the Evaluation System

The selection of input and output indicators constitutes a critical step in evaluating power grid investment efficiency. Power grid investment efficiency not only reflects the comprehensive performance of investment activities across economic, social, and environmental dimensions within a given period but also reveals the intrinsic relationship between resource inputs and value outputs. A scientifically constructed evaluation index system must balance comprehensiveness and practicality, encompassing both direct factors (e.g., capital investment and economic benefits) and externalities (e.g., social and environmental impacts). Selection of indicators was informed by established studies in the field of power grid investment efficiency (He et al. [18], Sun et al. [20]), while also considering the integrity of the overall evaluation system and data availability.

Input indicators were chosen to quantify the core resources consumed by power grid enterprises to achieve their outputs, covering both financial and physical dimensions of assets. The annual investment in grid construction serves as the most direct measure of capital input, reflecting both the scale and intensity of funding. The length of transmission lines at 35 kV and above indicates the geographical coverage and power transmission capability of the grid, representing a key measure of infrastructure stock. Similarly, the transformer capacity of substations at 35 kV and above reflects the load capacity and power distribution functionality of critical nodes, significantly influencing the scale of transmission and distribution within the system. Collectively, these three indicators offer a comprehensive portrayal of grid investments from three distinct aspects: capital flow (investment amount), line asset stock (length), and substation asset stock (capacity).

Output indicators are designed to evaluate the benefits and outcomes of power grid investments across multiple dimensions. Electricity sales volume represents the primary financial and economic performance metric for grid enterprises, directly determining operational revenue and reflecting the efficiency of investments in generating economic returns. Power supply reliability serves as a key performance indicator (KPI) for assessing the technical proficiency and service quality of the power grid. It reflects the grid structure, equipment condition, and maintenance standards, demonstrating the role of investments in enhancing supply security, service performance, and social benefits. The line loss rate measures operational efficiency and managerial refinement, with a lower rate indicating higher efficiency. Renewable energy accommodation underscores the grid’s role in facilitating energy transition and mitigating environmental impacts, constituting a critical measure of the sustainable development capacity of modern power systems. Collectively, these indicators balance economic performance with social responsibility, thereby transcending a narrow economic focus to comprehensively capture the multifaceted value of power grid investments.

Therefore, this study establishes a multidimensional evaluation framework. Input indicators focus on capital investment, including the annual investment in grid construction (million CNY), length of transmission lines at 35 kV and above (KM), and transformer capacity of substations at 35 kV and above (MVA). Output indicators measure performance from economic, technical, and sustainability perspectives, comprising electricity sales volume (economic output, GWh), power supply reliability (service quality, %), line loss rate (operational efficiency, %), and renewable energy accommodation (environmental benefit, GWh). The units of these indicators primarily comply with the standards established by the National Bureau of Statistics, with certain units being adjusted based on their data magnitude to facilitate analytical coherence. Figure 3 demonstrates all the inputs and outputs and the evaluation process of the grid investment efficiency.

4.1.2. Data Source and Processing

The study focuses on provincial-level grid companies as they represent the core operational units of China’s power infrastructure, ensuring comprehensive geographic and operational coverage. These companies collectively account for over 90% of the national electricity distribution, making them highly representative for efficiency analysis. There are 34 administrative provinces in China. Due to differences in power grid management systems, this study excludes Hong Kong, Macao, and Taiwan, resulting in 31 provincial-level units for analysis. The evaluation period spans from 2015 to 2023, which aligns with the availability of standardized data following the implementation of key national electricity market reforms in 2015. This period also avoids anomalies such as the pre-2015 regulatory heterogeneity and captures recent trends up to the latest available data. Data for 2024 were excluded as official statistics were not yet available at the time of research. All raw data of each indicator were obtained from the China Electric Power Yearbook and official statistical reports published by the National Energy Administration.

Table 2. Descriptive statistics (Source: Author’s own calculation based on collected data).

Variables (Unit)	Min	Median	Mean	Max	St. Dev.
Input indicator
Annual investment (Million CNY)	2102.88	60,237.63	15,204.50	12,464.44	10,896.73
Length ≥ 35 kv (KM)	5290	810,200	241,201.25	189,550	171,969.57
Transformer capacity ≥ 35 kv (MVA)	8944	139,916	64,211.31	66,676	32,558.43
Output indicator
Electricity sales volume (GWh)	114.97	714,000	165,061.52	132,600	145,032.89
Power supply reliability (%)	99.23	100	99.86	99.86	0.11
Renewable energy accommodation (GWh)	2600	282,036	66,547.29	54,585.90	54,309.90
Undesirable output indicator
Line loss rate (%)	1.42	15.64	5.69	5.47	2.19

presents the descriptive statistics of input and output variables.

The study employs a panel data analysis framework, incorporating complete datasets from 31 DMUs $J=31$ across 9 observation periods ($T=9$). The established model includes 3 input indicators ($I=3$) and 4 output indicators ($R=4$), with line loss rate treated as an undesirable output and inversely incorporated as an input variable. To ensure model robustness, specific robustness coefficients were set at ${\Gamma }^{\alpha }=3$ and ${\Gamma }^{\beta }=4$ across all variables and periods, with comparative analyses conducted using various time lags. The selection of the robustness coefficient is governed by a probabilistic constraint. According to reference [36], maintaining a constraint violation probability below 1% mandates that the robustness coefficient be selected at its maximum permissible value. And we consider the efficiency value of each period to be equally important to the overall efficiency value, hence ${\theta }_t=1/(T-LP)$. The collected input and output variables are initially set at their nominal values ($x_{ijt}^0,y_{rjt}^0$). Perturbation values are then applied as ${\widehat{x}}_{ijt}=x_{ijt}^0\cdot 10\%, {\widehat{y}}_{rjt}=y_{rjt}^0\cdot 10\%$. For instance, regarding the three input indicators for Anhui Province in 2015, the recorded input values were CNY 15,616 million, 181,710 km, and 60,257 MVA, with associated fluctuations of CNY 1561.6 million, 18,171.0 km, and 6025.7 MVA, respectively. The corresponding uncertainty intervals are defined as (14,054.4, 17,177.6) CNY million, (163,539.0, 199,881.0) KM, and (54,231.3, 66,282.7) MVA. This implies that in the model, the values of these three input indicators are set to the extreme values of the uncertainty set, either the maximum or minimum values, depending on the constraints.

4.2. Model Solution and Implementation Process

This subsection primarily describes the solution process of the proposed model, summarizes the data sources and key assumptions, and conducts several multiple scenarios. Figure 4 outlines the step-by-step solution and implementation procedure.

• Step 1: Model implementation

Model (18) is formulated as a linear programming problem aimed at evaluating the efficiency value of $DM{U}_O$. The corresponding code for solving the model was implemented in Python 3.13, utilizing the Gurobi solver.

• Step 2: Incorporation of model assumptions and data

Entering the model parameters, including the number of decision-making units DMUs $J=31$, the time periods $T=9$, the number of inputs $I=3$, and the number of outputs to $R=4$. Uniform robustness coefficients ${\Gamma }^{\alpha }=3$ and ${\Gamma }^{\beta }=4$ were applied across all variables and periods. The time-lag parameter LP was assigned values in the range [0, 1, 2, …, 6], and ${\theta }_t=1/(T-LP)$. For uncertain inputs and outputs, the nominal values ($x_{ijt}^0,y_{rjt}^0$) were initially assigned based on collected data. Perturbations were then introduced in the form of ±10% deviations of nominal values, namely ${\widehat{x}}_{ijt}=x_{ijt}^0\cdot 10\%, {\widehat{y}}_{rjt}=y_{rjt}^0\cdot 10\%$. All input and output data for the DMUs were obtained from the China Electric Power Yearbook and official statistical reports published by the National Energy Administration, ensuring the reliability and authority of the data.

• Step 3: Efficiency calculation and extraction

The input–output data of each $DM{U}_O(O=1,2,\dots ,31)$ were sequentially incorporated into the model. By solving the linear programming problem, the optimal values of the decision variables $(u_{rt}^{\ast },z_{jt}^{y \ast },p_{rjt}^{y \ast })$ were obtained. In accordance with Definition 3 and Definition 4, both the overall efficiency and the period-specific efficiency values for each DMU were computed, resulting in a complete set of efficiency measures for all 31 DMUs.

• Step 4: Parameter adjustment and comparative analysis

To facilitate comparative analysis of the model under multiple scenarios, the following procedures were conducted.

(1) The time-lag parameter LP was varied from 0 to 6, and the efficiency values of all DMUs for each value of LP were solved and recorded.

(2) The parameters ${\Gamma }^{\alpha }$ and ${\Gamma }^{\beta }$ were set to various values to analyze the efficiency values under various protection level. And when ${\Gamma }^{\alpha }=0$ and ${\Gamma }^{\beta }=0$, it simulates a deterministic environment. Under this condition, the values of LP were adjusted again to analyze the influence of different time lags on both overall and periodic efficiency values.

4.3. Results and Discussion

The proposed robust counterpart DTL DEA model (18) was implemented for China’s 31 provincial power grids in Python, generating the following results: (1) overall and period-specific efficiency scores under various lag lengths condition. Furthermore, two scenarios were implemented for comparative analysis: (2) baseline efficiency evaluation without time lag effects, and (3) efficiency assessment under deterministic data conditions.

4.3.1. Efficiency Patterns Under Varying Lag Periods

To assess the influence of time lag effects on power grid investment performance, we examined two distinct lagged approaches. The first approach excluded time lags (LP = 0), positing that grid investments exhibit no temporal latency effects, whereby annual output depends exclusively on contemporaneous investment. The second approach explicitly incorporated investment time lags, acknowledging that investments in a given year affect output both in the current year and in subsequent periods. We calculated six lag-period scenarios (LP = 1–6) and a no time lag scenario (LP = 0) and obtained seven sets of overall efficiency values, as presented in Figure 5.

The results demonstrate that the overall efficiency scores of all DMUs remain below 1, confirming that none achieve production frontier optimality. This empirically confirms that no DMU achieves production frontier optimality, indicating systemic inefficiency manifested through both input excesses and suboptimal output generation. Significant efficiency disparities exist across DMUs, with scores ranging from 0.3942 (Xinjiang, LP = 0) to 0.8188 (Tibet, LP = 6).

Figure 6 depicts the average overall efficiency scores of all the DMUs and their growth rate of scores across lag periods, from no time lag (LP = 0) to a 6-year lag (LP = 6). Comparative analysis of the seven lag-period scenarios reveals a consistent upward trend in efficiency, with increasing from 0.6404 (LP = 0) to 0.7174 (LP = 6) as the lag period extends, reflecting a 12.0% overall improvement. Specifically, Shandong and Xinjiang exhibited the most substantial efficiency gains, increasing by 30.2% and 24.7%, respectively, while Yunnan and Hainan demonstrated the most marginal improvements with growth rates of merely 0.9% and 1.3%. The analysis reveals a statistically significant divergence between dynamic models that incorporate time-lag adjustments and traditional static models. Notably, the efficiency values of each region when considering time-lag effects consistently exceed those without such consideration. The analysis reveals a pronounced positive time-lag effect in the investment efficiency of China’s provincial power grid enterprises. As the lag period (LP) extends, the efficiency values of the majority of DMUs exhibit progressive improvement.

From the perspective of the growth rate of average overall efficiency values, it reveals that incorporating time lags generates peak improvement (3.7798% gain from LP = 0 to LP = 1), followed by progressively diminishing returns (1.1378% at LP = 6). During a 3-year to 5-year lag, the growth rate gains stabilize within 1.5–1.6% (1.5863%, 1.5887%, 1.5006%, respectively). Static models (LP = 0) systematically underestimate efficiency by 7.42% (LP = 3), 9.12% (LP = 4), and 10.76% (LP = 5) compared to their dynamic counterparts. This pattern strongly suggests that 3–5 years constitutes the optimal latency window for power grid investment returns.

Two key conclusions can be drawn from these results. Firstly, time lag exclusion systematically underestimates power grid investment efficiency, and lag-period adjustment is essential for assessment accuracy as it mitigates significant biases. This systematic underestimation highlights the limitations of static DEA models in accurately reflecting the true efficiency of power grid investments, particularly in systems where inputs yield outputs over extended periods. Without such consideration, policymakers and grid operators may misallocate resources and miss opportunities for optimizing long-term investment strategies. Therefore, the inclusion of time-lag effects in efficiency assessments provides a more comprehensive and reliable framework for decision-making in the power grid sector. Secondly, stabilized growth rate gains of efficiency during 3–5-year lags indicate an archetypal 3–5-year investment lag due to power grid infrastructure development periods (2–3 years) and operational impact realization periods (1–2 years). When the lag period exceeds 5 years (LP > 5), the growth rate significantly decreases, proving that excessively extending the lag period consideration in performance evaluation has no practical value.

4.3.2. Temporal and Spatial Efficiency Dynamics

Subsequently, we enable systematic examination of temporal and spatial analysis of power grid investment performance based on three lag period specifications: (1) three-period lag, (LP = 3), measuring overall and period-year efficiency scores for 2018–2023; (2) four-period lag (LP = 4), deriving efficiency scores for 2019–2023; (3) five-period lag (LP = 5), calculating efficiency scores for 2020–2023.

Table 3. Overall and period efficiency scores when LP = 3 (Source: Author’s own calculation based on experimental data).

DMU	Overall Efficiency	Period Efficiency
		2018	2019	2020	2021	2022	2023
Anhui	0.5013	0.4962	0.4609	0.4783	0.4792	0.5679	0.5254
Beijing	0.7463	0.7419	0.7498	0.7498	0.7498	0.7461	0.7403
Fujian	0.7389	0.7378	0.7380	0.7390	0.7391	0.7392	0.7401
Gansu	0.7139	0.7426	0.7429	0.7409	0.7382	0.6434	0.6756
Guangdong	0.7666	0.7810	0.7810	0.7810	0.7523	0.7523	0.7523
Guangxi	0.6527	0.7107	0.6319	0.6405	0.6448	0.6387	0.6495
Guizhou	0.7139	0.7395	0.7395	0.7388	0.6721	0.7154	0.6782
Hainan	0.7513	0.7539	0.7539	0.7539	0.7539	0.7461	0.7461
Hebei	0.7452	0.7517	0.7517	0.7420	0.7420	0.7420	0.7420
Henan	0.5071	0.5039	0.4770	0.4679	0.5038	0.5602	0.5300
Heilongjiang	0.7437	0.7463	0.7463	0.7477	0.7382	0.7382	0.7457
Hubei	0.5235	0.4872	0.4469	0.4927	0.5613	0.5619	0.5914
Hunan	0.5183	0.5882	0.5006	0.5203	0.5118	0.5191	0.4695
Jiling	0.7426	0.7403	0.7412	0.7409	0.7409	0.7393	0.7528
Jiangsu	0.7428	0.7420	0.7420	0.7432	0.7427	0.7433	0.7433
Jiangxi	0.6103	0.5605	0.5501	0.6294	0.6322	0.6379	0.6515
Niaoning	0.6714	0.7372	0.7372	0.6139	0.5726	0.6841	0.6832
Neimenggu	0.7020	0.6332	0.6685	0.7074	0.7282	0.7363	0.7386
Ningxia	0.7833	0.8026	0.8026	0.7735	0.7735	0.7735	0.7743
Qinghai	0.7771	0.8184	0.8184	0.7565	0.7565	0.7564	0.7564
Shandong	0.6271	0.5369	0.5219	0.5254	0.7255	0.7255	0.7275
Shanxi	0.6429	0.6454	0.6517	0.6500	0.6032	0.6588	0.6480
Shaanxi	0.7171	0.6180	0.7352	0.7350	0.7350	0.7408	0.7387
Shanghai	0.7835	0.7441	0.7441	0.8033	0.8031	0.8031	0.8031
Sichuan	0.7517	0.7521	0.7521	0.7574	0.7493	0.7493	0.7499
Tianjing	0.7375	0.7390	0.7426	0.7432	0.7335	0.7326	0.7338
Tibet	0.7931	0.8188	0.8188	0.8188	0.8188	0.7417	0.7417
Xinjiang	0.4388	0.4294	0.4702	0.4251	0.4430	0.4364	0.4287
Yunan	0.7463	0.7492	0.7492	0.7471	0.7471	0.7426	0.7426
Zhejiang	0.7481	0.7383	0.7465	0.7465	0.7463	0.7463	0.7644
Chongqing	0.6871	0.5641	0.7000	0.7387	0.7385	0.6863	0.6949

,

Table 4. Overall and period efficiency scores when LP = 4 (Source: Author’s own calculation based on experimental data).

DMU	Overall Efficiency	Period Efficiency
		2019	2020	2021	2022	2023
Anhui	0.5235	0.5009	0.4589	0.5059	0.5789	0.5728
Beijing	0.7497	0.7504	0.7504	0.7504	0.7486	0.7484
Fujian	0.7386	0.7388	0.7392	0.7384	0.7384	0.7384
Gansu	0.7367	0.7384	0.7392	0.7392	0.7325	0.7345
Guangdong	0.7756	0.7810	0.7810	0.7810	0.7676	0.7676
Guangxi	0.6703	0.7468	0.6488	0.6534	0.6408	0.6616
Guizhou	0.7226	0.7415	0.7415	0.7268	0.7291	0.6740
Hainan	0.7517	0.7539	0.7539	0.7539	0.7539	0.7428
Hebei	0.7448	0.7529	0.7529	0.7392	0.7397	0.7397
Henan	0.5154	0.4975	0.4734	0.4824	0.5740	0.5497
Heilongjiang	0.7477	0.7482	0.7482	0.7479	0.7454	0.7487
Hubei	0.5443	0.4939	0.5049	0.5727	0.5667	0.5832
Hunan	0.5341	0.6056	0.5236	0.5174	0.5456	0.4781
Jiling	0.7459	0.7422	0.7435	0.7435	0.7435	0.7569
Jiangsu	0.7432	0.7420	0.7455	0.7454	0.7416	0.7416
Jiangxi	0.6248	0.5623	0.5790	0.6393	0.6414	0.7019
Niaoning	0.6914	0.7372	0.7041	0.6210	0.7000	0.6947
Neimenggu	0.7184	0.6981	0.7162	0.7238	0.7238	0.7301
Ningxia	0.7763	0.7661	0.8008	0.7711	0.7711	0.7727
Qinghai	0.7768	0.8159	0.8203	0.7493	0.7493	0.7493
Shandong	0.6576	0.5177	0.6051	0.7167	0.7243	0.7243
Shanxi	0.6849	0.7040	0.7041	0.6574	0.6807	0.6783
Shaanxi	0.7540	0.7495	0.7554	0.7554	0.7554	0.7545
Shanghai	0.7853	0.7380	0.7972	0.7972	0.7972	0.7972
Sichuan	0.7508	0.7548	0.7548	0.7553	0.7432	0.7458
Tianjing	0.7413	0.7410	0.7450	0.7383	0.7383	0.7441
Tibet	0.8024	0.8188	0.8188	0.8188	0.8188	0.7366
Xinjiang	0.4495	0.4787	0.4805	0.4280	0.4276	0.4330
Yunan	0.7488	0.7485	0.7485	0.7475	0.7475	0.7518
Zhejiang	0.7469	0.7469	0.7469	0.7469	0.7469	0.7469
Chongqing	0.7109	0.7083	0.7420	0.7389	0.7044	0.6611

and

Table 5. Overall and period efficiency scores when LP = 5 (Source: Author’s own calculation based on experimental data).

DMU	Overall Efficiency	Period Efficiency
		2020	2021	2022	2023
Anhui	0.5480	0.5111	0.5246	0.5816	0.5744
Beijing	0.7501	0.7500	0.7500	0.7507	0.7498
Fujian	0.7404	0.7405	0.7405	0.7403	0.7403
Gansu	0.7443	0.7452	0.7452	0.7452	0.7419
Guangdong	0.7711	0.7810	0.7810	0.7810	0.7414
Guangxi	0.6774	0.7385	0.6671	0.6406	0.6635
Guizhou	0.7215	0.7392	0.7388	0.7394	0.6685
Hainan	0.7540	0.7618	0.7618	0.7461	0.7461
Hebei	0.7459	0.7533	0.7533	0.7374	0.7398
Henan	0.5378	0.5376	0.5206	0.5542	0.5390
Heilongjiang	0.7518	0.7526	0.7526	0.7526	0.7492
Hubei	0.5610	0.5067	0.5633	0.5892	0.5848
Hunan	0.5401	0.6023	0.5196	0.5399	0.4988
Jiling	0.7470	0.7438	0.7438	0.7438	0.7567
Jiangsu	0.7442	0.7453	0.7453	0.7453	0.7410
Jiangxi	0.6406	0.6258	0.6440	0.6458	0.6469
Niaoning	0.7224	0.7369	0.7369	0.7078	0.7080
Neimenggu	0.7357	0.7273	0.7383	0.7385	0.7385
Ningxia	0.7867	0.7917	0.7917	0.7814	0.7818
Qinghai	0.7781	0.8201	0.8201	0.7361	0.7361
Shandong	0.6979	0.6035	0.7234	0.7324	0.7324
Shanxi	0.7088	0.7024	0.7292	0.7091	0.6943
Shaanxi	0.7675	0.7773	0.7773	0.7577	0.7577
Shanghai	0.8033	0.8034	0.8034	0.8032	0.8032
Sichuan	0.7545	0.7553	0.7553	0.7553	0.7521
Tianjing	0.7477	0.7452	0.7452	0.7452	0.7552
Tibet	0.8188	0.8188	0.8188	0.8188	0.8188
Xinjiang	0.4677	0.4748	0.4953	0.4495	0.4510
Yunan	0.7482	0.7490	0.7490	0.7473	0.7473
Zhejiang	0.7514	0.7452	0.7452	0.7452	0.7702
Chongqing	0.7254	0.7478	0.7478	0.7231	0.6829

present detailed results for both overall and period efficiency values of three lag periods.

(1)

Temporal efficiency dynamics analysis

The analysis of individual period efficiency scores reveals a consistent pattern that all efficiency values consistently fall below 1, with a range spanning from 0.4251 to 0.8203. Furthermore, in accordance with Definition 3 and Definition 4, the calculated efficiency values for each period demonstrate that the average efficiency score of each period closely corresponds to the overall efficiency. This result confirms that each period contributes equally to the overall efficiency, aligning perfectly with our initial hypothesis. The consistency and alignment of these findings provide robust validation for the accuracy of the calculations, ensuring that the results are free from errors.

The temporal efficiency dynamics of China’s provincial power grid investment from 2018 to 2023 reveal a clear upward trend with notable fluctuations. The efficiency scores started at 0.6823 in 2018, gradually improving to 0.6979 in 2020, peaking at 0.7022 in 2021, and slightly declining to 0.6982 in 2023. The initial years (2019–2021) witnessed moderate growth primarily driven by early-stage regulatory reforms and policy harmonization. This period further benefited from accelerated technological advancements, enhanced renewable energy grid integration, and the comprehensive implementation of transmission and distribution tariff mechanisms. The decline in 2022–2023 was attributed to economic slowdowns, policy adjustments, and environmental constraints, though overall efficiency remained robust compared to the baseline. This temporal analysis highlights the dynamic nature of grid investment efficiency, emphasizing the importance of continuous policy refinement and technological innovation to sustain high performance and support long-term development goals.

(2)

Spatial Efficiency Dynamics Analysis

We conducted an analysis of regional disparities based on the average overall efficiency derived from the LP = 3, 4, 5. It unveils significant regional disparities in the investment efficiency of provincial power grids across China. Tibet was the top-performing region with an efficiency score of 0.8048, in sharp contrast to Xinjiang, which ranked the lowest at 0.4520. The efficiency distribution can be stratified into four distinct tiers. By comparing the input–output data of high-efficiency provinces (e.g., Tibet, Shanghai, Qinghai) with that of low-efficiency regions, this study focuses on analyzing input redundancy and output shortfalls in underperforming areas.

The first tier (average overall efficiency < 0.6) includes five inland provinces: Xinjiang, Henan, Anhui, Hunan, and Hubei. Xinjiang exhibited the lowest efficiency. Although one of its output indicators is nearly 50% higher than that of Qinghai, and other outputs are similar, its inputs and undesirable outputs are approximately twice those of Qinghai. This indicates that Qinghai achieves higher efficiency through a “low-input, low-output” paradigm, a pattern also observed in Tibet. When compared with Shanghai, Xinjiang shows comparable output levels, yet its input indicators, such as power capacity, are nine times higher, revealing significant input redundancy. This is largely attributable to Xinjiang’s geographical conditions, including its vast territory and sparse population, which lead to considerably higher grid construction and operational costs. Among other provinces, Anhui exhibits a relatively low level of renewable energy absorption, resulting in underperformance in output. In contrast, Henan, Hubei, and Hunan have made substantial investments, particularly in transformer capacity, which exceeds that of Shanghai by more than sevenfold. However, the resulting outputs in these provinces remain largely comparable to those of Shanghai and Qinghai, indicating disproportionately low returns relative to the scale of investment. The findings suggest that inefficient regions generally suffer from excessive resource input, resulting in resource misallocation and over-investment.

The second tier (average overall efficiency between 0.6 and 0.7) comprises Jiangxi, Shandong, Guangxi, Shanxi, and Liaoning provinces. The efficiency scores of these provinces are generally lower, mainly attributable to their inadequate renewable energy accommodation, which falls significantly behind that of high-performing regions such as Qinghai and Guangdong. Moreover, these provinces exhibit notably higher line loss rates compared to the latter. To illustrate, provinces like Shandong and Liaoning have invested substantially in grid infrastructure, including transmission line length and transformer capacity, yet their electricity sales volume remains comparatively low, resulting in underutilization of resources. Most of these less efficient provinces are traditional fossil fuel bases, including Shandong, Shanxi, and Liaoning. Their energy systems remain heavily dependent on coal power, characterized by limited flexibility, thereby constraining overall operational efficiency.

The third tier (average overall efficiency between 0.7 and 0.75) encompasses the largest number of provinces (14), representing the intermediate efficiency level of China’s power grid system. These provinces exhibit certain deficiencies in input–output allocation, resulting in moderate efficiency performance. For instance, Jilin and Heilongjiang experience line loss rates that exceed the provincial average by more than 20%. Meanwhile, Beijing and Tianjin demonstrate notably inadequate clean energy integration capacity, with renewable energy absorption levels reaching only 20.41% and 27.88% of the average, respectively. Conversely, Jiangsu has made substantial investments in grid infrastructure, such as transmission line length and transformer capacity, yet its electricity sales volume remains relatively limited, leading to suboptimal resource utilization.

The fourth tier (average overall efficiency > 0.75) includes seven top-performing provinces, which are Sichuan, Hainan, Guangdong, Qinghai, Ningxia, Shanghai, and Tibet. Among these, Sichuan and Guangdong exhibit exceptional performance in renewable energy integration, with annual renewable energy accommodation exceeding 210,000 GWh, significantly surpassing the national average of 66,500 GWh. Nevertheless, Sichuan is constrained by a relatively high line loss rate, and Guangdong has invested more substantially in both physical infrastructures (length of transmission lines and transformer capacity) and financial resources (annual investment). In contrast, Ningxia and Hainan have achieved high efficiency scores through relatively low investment coupled with comparatively high output. To further enhance efficiency, Ningxia could focus on expanding its electricity sales volume, while Hainan has potential for improvement in both sales volume and renewable energy accommodation. Despite exhibiting the highest line loss rate and the lowest electricity sales volume, Tibet achieved the highest efficiency score of 0.8048. This may be explained by its relatively high efficiency within a low-input–low-output operational model.

Furthermore, analysis across varying time lags reveals distinct provincial trajectories that Anhui and Shandong demonstrate significant efficiency gains of 9.3% and 11.3%, respectively, potentially attributable to recent large-scale infrastructure investments. Beijing and Jiangsu maintain remarkable stability in efficiency metrics, reflecting the maturity of their established grid systems, while Xinjiang exhibits the most marginal improvement (4.52%), suggesting exceptionally long investment return periods in this region. The result analysis demonstrates that the lag period for power grid investment efficiency necessitates region-specific calibration, accounting for disparities in regional resource endowments and development constraints. For instance, when evaluating the efficiency value of Xinjiang, we can moderately expand its lag period because its investment payback period is longer than other provinces

Based on these findings, we conclude that the efficiency performance of provincial power grids in China exhibits notable heterogeneity. Accordingly, future policy-making should be tailored to local conditions. High-efficiency provinces should focus on addressing specific deficiencies, such as reducing line losses, optimizing returns on investment. Meanwhile, other provinces can draw lessons from various high-efficiency models to seek breakthroughs in expanding renewable energy integration and enhancing operational leanness.

4.3.3. Comparative Analysis Under Diverse Protection Level

To examine the impact of the protection level (i.e., robustness coefficient $\Gamma$) of uncertain parameters on efficiency values and their sensitivity, this study employs multiple protection level configurations across different lag periods (LP = 3, 4, 5), specifically $({\Gamma }^{\alpha },{\Gamma }^{\beta })=(1,1),(2,2),(3,3),(3,4)$. The results are illustrated in Figure 7, with subfigures (a), (b), and (c) representing the efficiency values corresponding to lag periods of 3, 4, and 5, respectively.

As depicted in the figure, efficiency values follow a consistent trend across all lag periods, which gradually decrease as the protection level $\Gamma$ increases. Notably, the lowest efficiency occurs under the configuration $({\Gamma }^{\alpha },{\Gamma }^{\beta })=(3,4)$. This can be attributed to the fact that the protection level represents the maximum number of uncertain parameters allowed to simultaneously reach their extreme values within predefined uncertainty sets. Specifically, ${\Gamma }^{\alpha }=3$ permits three input indicators to concurrently attain their upper or lower bounds, while ${\Gamma }^{\beta }=4$ allows four output indicators to simultaneously assume extreme values. The simultaneous occurrence of such a “worst-case” scenario significantly increases the system’s requirement to resist uncertainty, thereby leading to a notable decline in efficiency. In contrast, when only a limited number of parameters approach extreme values (e.g., $({\Gamma }^{\alpha },{\Gamma }^{\beta })=(1,1)$), the decline in efficiency remains moderate. This indicates that the system retains a certain degree of robustness against fluctuations in a subset of parameters, reflecting lower sensitivity to such perturbations. These findings underscore the comprehensive effect of global protection levels on efficiency within robust optimization frameworks and reaffirm the inherent trade-off between robustness and efficiency across various parameter configurations. Nonetheless, the effect of protection level on efficiency remains consistent throughout different LP settings, demonstrating the reliability of this conclusion across multiple time horizons.

4.3.4. Comparative Analysis of Deterministic Scenarios

In this subsection, we implemented the model under deterministic conditions, specifically with the robust coefficients ${\Gamma }^{\alpha }=0$ and ${\Gamma }^{\beta }=0$, while considering time-lag effects for LP = 3, 4, 5, and without time-lag effects. The results are illustrated in Figure 8.

The observed patterns in efficiency dynamics align with those under uncertainty, with the average overall efficiency increasing from 0.8765 to 0.9450 across the three lag scenarios. This analysis underscores that the deterministic models, when time-lag effects are excluded, systematically underestimate efficiency. Specifically, the exclusion of time-lag considerations leads to an underestimation of efficiency by an average of 8.9%, highlighting the critical role of accounting for the inherent time-lagged nature of grid investments. These findings further reinforce the necessity of incorporating time-lag effects in efficiency assessments to accurately reflect the true performance of power grid investments.

The empirical results present compelling evidence regarding the critical importance of accounting for parameter uncertainty in efficiency evaluation. As illustrated in Figure 6, the conventional deterministic approach that ignores input–output uncertainty systematically overestimates efficiency scores across all temporal scenarios. Particularly striking is the observation that mean efficiency values under various time-lag conditions consistently exceed 0.9, which clearly demonstrates the methodological bias inherent in traditional DEA models.

This overestimation phenomenon is further substantiated by the comparative analysis of

Table 6. Results analysis of deterministic scenarios (Source: Author’s own calculation based on experimental data).

Overall Efficiency	LP = 0	LP = 3	LP = 4	LP = 5
Min	0.8765	0.9209	0.9342	0.9450
Mean	0.5397	0.5933	0.6103	0.6327
Max	1	1	1	1
St. dev.	0.1306	0.1210	0.1137	0.1056
Number of efficient DMUs	6	16	18	19

, which reveals that all DMUs exhibit inflated efficiency values in deterministic settings, with relative overestimation ranging from 12.7% to 28.3% compared to robust measurements. The number of DMUs reaching the efficiency frontier shows dramatic variation, with six frontier DMUs when LP = 0 and 16, 17, 18 when LP = 3, 4, 5, respectively. These findings carry significant theoretical and practical implications. More than 200% increase in frontier-reaching DMUs suggests that traditional models fail to capture approximately 60–75% of achievable efficiency potential due to their inability to account for temporal dynamics and data uncertainty. The consistently exaggerated efficiency scores (mean > 0.9) in deterministic models indicate they may produce mathematically feasible but economically meaningless results, potentially leading to overoptimistic investment planning and inaccurate regulatory benchmarking.

4.3.5. Managerial Insights

The empirical findings of this study provide actionable managerial insights for policymakers, grid operators, and investors in the power sector, addressing critical factors that influence grid investment efficiency.

(1) It is imperative to consider time-lagged effects when evaluating the efficiency of large-scale infrastructure investments like power grids. Ignoring time-lagged effects can lead to underestimating true efficiency, since investment benefits typically materialize through multi-period accumulation, thereby distorting policy decisions and resource allocation. For accurate efficiency evaluations, a dynamic DEA model with time lag effects employing LP = 3/4/5 lag parameters is recommended to capture temporal evolution, particularly in region-specific contexts. Empirically justified adjustments to the lag period may enhance evaluation robustness. For accurate efficiency evaluations of grid investment, a dynamic DEA model incorporating time-lag effects with LP = 3/4/5 parameters is recommended to capture temporal evolution. In region-specific contexts, evidence-based adjustments to the lag period may further enhance evaluation robustness.

(2) The investment strategy for the power grid should be deeply integrated with the regional resources situation and development status. Policymakers need to coordinate regional differences and strategic priorities, and develop differentiated investment strategies and resource allocation plans. For instance, coastal regions can leverage their geographical advantages to foster technological innovation; provinces abundant in clean energy resources should accelerate their green transition while enhancing grid integration and absorption capacity; and in regions constrained by resources, technology, or geography, precise resource allocation becomes essential to optimize investment returns, thus providing a replicable framework for similar regions.

(3) Uncertainty in input–output parameters must be incorporated into the efficiency assessment process. Evaluation models typically rely on predefined parameters and assumptions, however, inherent uncertainties abound in reality, such as measurement errors and data gaps during collection. Neglecting parameter uncertainty will lead to overoptimistic efficiency assessments. This assessment bias can severely misguide decision-making. On the one hand, it may overestimate project benefits or underestimate implementation risks, fostering unsustainable investment planning; on the other hand, operational decisions based on distorted assessments may fail when confronted with the real world’s complexity and volatility, potentially triggering cascading systemic risks. Therefore, it is imperative to account for uncertainties in efficiency evaluations. This practice enhances the robustness and resilience of assessment outcomes, thereby laying a solid foundation for formulating more adaptive and forward-looking policies and investment strategies.

5. Summary and Conclusions

This study presents a novel robust DTL-DEA framework to address the challenges of assessing power grid investment efficiency in the presence of data uncertainty and time-lag effects. The framework was applied to evaluate power grid investment efficiency across 31 Chinese provinces from 2015 to 2023. We calculated both overall and period-specific efficiency values under various time-lag scenarios (LP = 1, 2, …, 6) and conducted comparative analyses with no time-lag effects and deterministic scenarios. The empirical results yield several key conclusions. Firstly, time-lag effects are fundamental to accurate efficiency assessment. Empirical results confirm that grid investments exhibit a 3–5-year return lag, and neglecting this lag systematically underestimates efficiency by 12.0%. Secondly, robust DEA generates more resilient efficiency values, ensuring their validity under data fluctuations. In contrast, deterministic models systematically overestimated efficiency levels, inflating the count of “efficient” Decision-Making Units (DMUs) by 200–300%. Thirdly, Thirdly, there is a significant inter-provincial difference in the investment efficiency of provinces, with the efficiency values of the highest and lowest provinces differing by nearly two times. This efficiency gap is mainly due to imbalances in economic development stages, resource endowment conditions, and power grid infrastructure among different regions. Therefore, it is urgent to develop more regionally targeted investment optimization strategies to enhance the overall efficiency of resource allocation.

In the evaluation of power grid efficiency, there may be differences in the time lag effects of different input and output indicators. Future research could extend this framework to the field of dynamic modeling, constructing models with different lag periods between indicators.