A Novel Method for Optimizing the Robustness and Carbon Emissions of Anchored Slopes

Abstract

Anchored slopes have been widely adopted globally, yet their design faces two critical challenges: the unreliable characterization of geomaterial variability and the urgent need for low-carbon. To address the first challenge, we propose a novel sensitivity index of variability (SIV) formulated through Monte Carlo Simulation (MCS), which systematically quantifies robustness within reliability-based design. Meanwhile, in response to global low-carbon initiatives, carbon emission metrics are innovatively established as the optimization objective for robust geotechnical design (RGD), replacing conventional cost-centric approaches. Building on these methodological advancements, we develop a framework that simultaneously optimizes engineering robustness and environmental sustainability. The operational efficacy of this framework is demonstrated through a case study of a rock slope at the tailwater tunnel outlet of a hydropower station in southwest China. Comparative analysis with the initial design solution reveals that our framework achieves higher robustness while reducing carbon emissions, demonstrating the superior performance of this framework. This study offers a systematic approach to advance both safety and sustainability in geotechnical infrastructure. The proposed methodology is readily adaptable to other earth structures facing similar reliability–environmental trade-offs.

1. Introduction

In slope engineering, the “anchor cable” is a structural component designed to stabilize unstable slopes or active landslides, with the core function of effectively transferring the sliding force of unstable rock layers or soil masses to stable layers, thereby significantly improving the overall stability of the supported geological structure [1,2,3]. Since the application of this technology, anchor cable structures have been widely used in slope stability enhancement and landslide prevention and control projects. They improve the safety of engineering during service [4]. For example, in many high slopes of reservoir areas in southwestern China, the use of anchor/anchor cable reinforcement has significantly modified slope safety [5,6,7].

Geomaterials (such as soil and/or rock) and structural surfaces, as components of rock slopes, have geological properties determined by geological evolution and engineering activities [8,9,10]. These factors are beyond the control of engineers through design. Due to incomplete information about geological actions and stress history, the geological properties of a site cannot be precisely determined prior to site investigation. Furthermore, due to cost and technical limitations, in engineering projects, only a limited number of laboratory/field tests can be conducted at discrete points. Consequently, the geological properties of slopes can only be reflected by discrete test data. The geological properties of areas outside the discrete test set cannot be directly determined and can only be estimated based on test data. However, due to the inherent spatial variability of geological properties and the limitations of test data, uncertainties exist in the geological properties of the site and their statistical parameters. Therefore, whether using safety factor assessment based on deterministic methods or reliability assessment based on probabilistic methods, the safety of slopes cannot be precisely represented by a single fixed value. To address this challenge, robust design methods, which have been widely applied in quality and industrial engineering [11,12,13], have been introduced into geotechnical design in recent years [14,15,16,17,18,19,20,21]. Within the framework of robust geotechnical design (RGD), robustness is defined as the ability of a system’s response (here referred to as reliability) to resist uncertainty (specifically defined here as statistical parameters of the mechanical properties of geological materials) introduced by noise factors. This approach thereby addresses the trade-off dilemma between over-design for safety and under-design for cost reduction.

Juang and Wang first developed the standard deviation of failure probability as the design robustness [14]. However, due to the limitations of geological characteristic data, the uncertainty of the standard deviation/coefficient of variation is difficult to quantify, which presents significant challenges for the representation of robustness [22,23]. To address this, Gong et al. proposed using signal-to-noise ratio (SNR) [21] and sensitivity index (SI) [24] to measure design robustness, while Zhang et al. attempted to apply SI for evaluating the performance of rock tunnel support systems under multiple failure modes [25]. However, these approaches only reflect robustness characteristics in terms of safety factors and fail to characterize the robustness performance of the reliability index, thereby falling short of the requirements for reliability-based RGD. In contrast, the sensitivity index of variability (SIV) developed by Fan et al. [26] quantifies the potential impact capability through which uncertainties in statistics of geomaterial parameters propagate to reliability indices. This index can measure the robustness of different designs even in the absence of statistical parameters for geological properties. This robustness index requires reliability algorithms to additionally provide the checkpoint and sensitivity coefficient vector, thereby depending on the choice of reliability algorithms. Furthermore, for engineering-oriented robust optimization design, these methods are somewhat complex and not user-friendly for designers. These factors have hindered the widespread adoption of robust optimization design. In response to these challenges, the Monte Carlo Simulation (MCS)-based approach has shown certain advantages. The MCS is simple in principle but precise in computation, stable, and reliable, often serving as the “true value” for verifying reliability algorithms [27,28]. Additionally, its computational efficiency is not affected by system complexity, making it easier for designers to accept and apply. If a method based on MCS for implementing SIV can be proposed, it could not only improve the practicality and efficiency of the RGD but also provide more sustainable, optimized solutions for engineering design.

In recent years, extreme climate issues caused by carbon emissions have gradually become a major global topic [29]. The Paris Agreement, signed in 2015, called for the establishment of clear decarbonization goals worldwide. In response, more than 150 countries have successively declared their emission reduction commitments [30]. Ecological civilization construction now takes precedence over short-term economic gains in central and local government planning in China. Drastic reduction of carbon emissions from construction projects and ecological environment improvement have emerged as industry priorities. Therefore, against this global backdrop, the optimization design of anchored slopes needs to shift from focusing on both safety and cost to considering reliability, robustness, and low-carbon performance simultaneously, aligning with the sustainable development philosophy of human society. To achieve this, anchored slope design must integrate knowledge from geotechnical engineering, structural engineering, and low-carbon environmental protection to optimize the balance of reliability, robustness, and low-carbon performance throughout the entire life cycle of the project.

To sum up, this paper proposes a new MCS-based method for implementing SIV in situations where the statistical characteristics of statistical parameters are lacking. It also establishes a bi-objective optimization design framework that comprehensively considers robustness and carbon emissions with reliability constraints. The paper is organized as follows: first, the optimization design process for anchored slope is established based on the new framework; then, taking the tailwater tunnel outlet slope of the Wudongde hydropower station in southwest China as a practical case study, the robust design framework in rock slope anchor engineering is detailed, and the advantages of this method in achieving design robustness and reducing carbon emissions are analyzed through the case study, thereby providing effective implementation methods for ensuring the safety and sustainability of geotechnical engineering constructions.

2. Methodology

2.1. Design Robustness Analysis and MCS Implementation

In the reliability design of an anchor–slope system, the failure probability assessment function P_f is generally composed of the statistical characteristics θ of the noise factor (i.e., the mechanical parameters of geomaterials in this paper) and the design factor d:

$$P_f\left(\mathit{\theta },\mathit{d}\right)=\mathrm{Pr}\left[Fs\left(\mathit{\kappa },\mathit{d}\right)<1\right]={\displaystyle \underset{Fs\left(\mathit{\kappa },\mathit{d}\right)<1}{\int }f\left(\mathit{\kappa }\right)\mathbf{d}\mathit{\kappa }}={\displaystyle {\int }_{-\infty }^{0}f\left(Fs\right)dFs}$$

(1)

$$Fs\left(\mathit{\kappa },\mathit{d}\right)={\displaystyle \frac{R\left(\mathit{\kappa },\mathit{d}\right)}{L\left(\mathit{\kappa },\mathit{d}\right)}}$$

(2)

$$\beta =-{\Phi }^{-1}\left({\mathbf{P}}_f\right)$$

(3)

where $Fs\left(\mathit{\kappa },\mathit{d}\right)$ represents the stability function of the anchored slope system. In engineering applications, the form of Equation (2) is adopted, where the resistance term is divided by the load term. Generally, $Fs\left(\mathit{\kappa },\mathit{d}\right)$ should be greater than 1 to ensure the safety of the anchored slope system. $f\left(\mathit{\kappa }\right)$ is the probability density function (PDF) of the noise factor κ, and $f\left(Fs\right)$ is the PDF of the safety factor. In engineering practice, the reliability index β is typically used to represent reliability, with the relationship shown in Equation (3), where $\Phi \left(·\right)$ is the cumulative distribution function (CDF) of the standard normal variable. In traditional reliability design, to ensure the feasibility of the final design solution, a target reliability index β_T or a target failure probability P_fT is usually specified (β_T is used in this paper). If the calculated reliability index is greater than the target reliability or the corresponding failure probability is less than the target failure probability, it indicates that the design solution is safe.

It is important to note that, while the concept of reliability is straightforward and its core formula is easy to apply, the implementation of reliability methods in engineering practice is challenging. As mentioned in the introduction, the most significant obstacle to reliability-based design lies in the fact that geomaterials are natural, rather than man-made. Their mechanical properties exhibit a higher degree of uncertainty compared to artificial materials, coupled with limited experimental data. Although the accuracy of the mean values of the mechanical properties of geomaterials can be ensured, the coefficient of variation δ is difficult to accurately characterize [31,32]. Under such circumstances, if the reliability method employs inaccurate or assumed statistical information, the design results may not be “reliable”. To minimize the adverse effects of this issue on the design solution, the concept of “sensitivity index of variability” [26] is introduced for RGD. In the robust design of anchored slope systems, the robustness R of the reliability of anchored slope to the uncertainty of input geomaterial mechanics parameters’ statistical characteristics δ is characterized using Equation (4).

$$SIV=‖S_{\delta }‖=\sqrt{{\displaystyle \frac{\partial P_f}{\partial \mathit{\delta }}}{{\displaystyle \frac{\partial P_f}{\partial \mathit{\delta }}}}^T}$$

(4)

where $S_{\delta }=\left(S_{\delta 1},S_{\delta 2},\dots ,S_{\delta M}\right)$ is the sensitivity vector for the variation coefficients of M noise factors. As can be seen from Equation (4), the core purpose of SIV is to solve the partial derivatives of the failure probability with respect to the variation coefficients of each geotechnical parameter, thereby reflecting the sensitivity of the failure probability to fluctuations in their variation coefficients. A smaller SIV value indicates lower variability in the reliability of the anchored slope (i.e., a weaker response to uncertainty in input parameters), which in turn results in higher robustness R of the anchored slope design, as shown in Figure 1.

The implementation of SIV in MCS can be categorized into single failure mode and multiple failure mode, both of which share the same computational principle. This section primarily focuses on a single failure mode, with multiple failure modes following similar logic.

According to the definitions of failure probability and the coefficient of variation sensitivity, it follows that

$${\mathit{S}}_{\delta }={\displaystyle \frac{\partial P_f}{\partial \mathit{\delta }}}={\displaystyle \underset{Fs\left(\mathit{\kappa },\mathit{d}\right)<1}{\int }\frac{\partial f\left(\mathit{\kappa }\right)}{\partial \mathit{\delta }}d\mathit{\kappa }}$$

(5)

Equation (5) indicates that calculating the coefficient of variation sensitivity involves evaluating the integral in the equation. MCS can address this by transforming the integral and employing numerical simulation methods for evaluation. By performing the following transformation on Equation (6), the coefficient of variation sensitivity can be expressed in terms of mathematical expectation:

$$\begin{array}{cc}\hfill {\mathit{S}}_{\delta }& {\displaystyle =\underset{Fs\left(\mathit{\kappa },\mathit{d}\right)<1}{\int }\frac{\partial f\left(\mathit{\kappa }\right)}{\partial \mathit{\delta }}d\mathit{\kappa }}\hfill \\ & ={\displaystyle \underset{F}{\int }{I}_F\left(\mathit{\kappa }\right)\frac{\partial f\left(\mathit{\kappa }\right)}{\partial \mathit{\delta }}\frac{1}{f\left(\mathit{\kappa }\right)}d\mathit{\kappa }}\hfill \\ & =E\left[{\displaystyle \frac{I_F\left(\mathit{\kappa }\right)}{f\left(\mathit{\kappa }\right)}}{\displaystyle \frac{\partial f\left(\mathit{\kappa }\right)}{\partial \mathit{\delta }}}\right]\hfill \end{array}$$

(6)

F represents the failure region defined by $Fs\left(\mathit{\kappa },\mathit{d}\right)<1$, $I_F\left(\mathit{\kappa }\right)=\left\{\begin{array}{c}1,\mathit{\kappa }\in F\\ 0,\mathit{\kappa }\notin F\end{array}\right.$ denotes the indicator function of the failure region, and $E\left[·\right]$ signifies the mathematical expectation operator. To estimate the coefficient of variation sensitivity expressed in the form of mathematical expectation in Equation (6) while simulating with MCS, the sample mean can be used to approximate the population mean. Specifically, the sensitivity can be estimated using the average value of the sample function, as shown in the following expression:

$${\mathit{S}}_{\delta }\approx {\displaystyle \frac{1}{\mathrm{N}}}{\left.{\displaystyle \sum _{i=1}^{\mathrm{N}}\frac{I_F\left(\mathit{\kappa }\right)}{f\left(\mathit{\kappa }\right)}\frac{\partial f\left(\mathit{\kappa }\right)}{\partial \mathit{\delta }}}\right|}_{\mathit{\kappa }={\mathit{\kappa }}_i}$$

(7)

where ${\mathit{\kappa }}_i$ is the i-th sample drawn from the joint PDF $f\left(\mathit{\kappa }\right)$ among N total samples. Subsequently, the sensitivity index value (SIV) based on MCS can be determined using Equation (4).

As indicated by the aforementioned computational formulas, the robustness index proposed in this study represents a natural extension of reliability calculation. During the reliability analysis, SIV emerges as a by-product and can be directly obtained, rendering the proposed design framework fully compatible with traditional reliability-based design methodologies. The coupling design between robustness assessment and reliability analysis during computations does not significantly increase the computational burden, with the reduction in computational efficiency primarily stemming from the bi-objective optimization process in Equation (14). The key advantage of this design framework lies in its ability to comprehensively balance the robustness and carbon emissions of the design.

2.2. Methods of Calculating Carbon Emissions for Anchored Slopes

The anchored slopes process can generally be divided into four phases: the production phase, transportation phase, construction phase, and service phase [33]. All four phases involve carbon emission processes with multiple sources and complex compositions. While scholars and researchers globally have yet to establish a unified framework for accounting theoretical models, life cycle assessment (LCA) has emerged as a widely adopted methodology [34]. The LCA method quantifies carbon emissions across the whole life cycle of building materials, encompassing production, transportation, construction, and service phases. This methodology systematically integrates carbon emissions from all phases. Given the inherently cyclical nature of anchored slope processes, which exhibit distinct life cycle characteristics, the LCA method is recommended for comprehensive carbon emission estimation throughout the entirety of anchored slope projects.

The carbon emissions at each phase can be quantified using the carbon emission factor method [35]. This approach is defined as the calculation of greenhouse gas emissions generated per unit of resource utilization during a specific process. The total carbon emissions are determined by multiplying activity data by their corresponding carbon emission factors [36,37], thereby deriving an estimated value for emissions, as shown in Equation (8).

$$C=G\times E$$

(8)

where C represents the carbon emission; G is the activity data, which refers to the production of energy or resources that generate carbon dioxide within a defined temporal and spatial scope; and E denotes the carbon emission factor, corresponding to the activity data, defined as the quantity of greenhouse gases generated per unit mass of material produced or consumed.

Equation (8) constitutes a universal theoretical framework that assumes distinct formulations depending on the specific phase under analysis. When calculating carbon emissions during the production phase, Equation (8) can be adapted to the following form:

$$C_{e,m}={\displaystyle \sum _{i=1}^{\mathrm{N}}{M}_i}\times E_{i,m}$$

(9)

where C_e,m denotes the carbon emissions of producing building materials; M_i represents the quantity of the i-th category of building material consumed; and E_i,m corresponds to the carbon emission factor associated with the i-th category of building material.

When calculating carbon emissions during the transportation phase, Equation (8) can be adapted to the following form:

$$C_{e,t}={\displaystyle \sum _{i=1}^{\mathrm{N}}{M}_i}\times E_{i,t}\times D_i$$

(10)

where C_e,t is the carbon emissions of transporting building materials; E_i,t represents the carbon emission factors for the i-th category of energy; and D_i signifies the transportation distance for the i-th category of building materials.

When calculating carbon emissions during the construction phase, Equation (8) can be adapted to the following form:

$$C_{e,c}={\displaystyle \sum _{j=1}^{\mathrm{Q}}C{E}_j}\times E_{j,c}\times F_j$$

(11)

where C_e,c represents the carbon emissions of the construction activities; CE_j denotes the energy consumption per shift for the j-th type of construction equipment; E_j,c signifies the carbon emission factor of the energy source utilized; and F_j corresponds to the number of operational shifts for the j-th type of construction equipment.

When calculating carbon emissions during the service phase, Equation (8) can be adapted to the following form:

$$C_{e,s}={\displaystyle \sum _{x=1}^X{M}_x}{E}_{x,m}+{\displaystyle \sum _{x=1}^X{M}_x}{E}_{x,t}{D}_x+{\displaystyle \sum _{z=1}^{\mathrm{Z}}C{E}_z}{E}_{z,c}{F}_z+{\displaystyle \sum _{p=1}^{\mathrm{P}}DM{E}_p}{E}_{p,s}-C_s\cdot S\cdot Y+{\displaystyle \sum _{r=1}^R{C}_{r,l}\cdot S_r\cdot Y}$$

(12)

where C_e,s represents the repaired material consumption during the service phase; E_x,m, E_x,t, and E_z,c denote the carbon emission factors for material production, transportation, and construction during the operational phase, respectively; DME_p signifies the energy consumption of the p-th inspection/monitoring equipment; E_p,s corresponds to the carbon emission factor of the energy source; C_s represents the annual carbon sequestration capacity of vegetation; S denotes the vegetated area; and Y indicates the design service life of the engineering project. Anchored slope construction typically involves large-scale earthwork excavation, which inevitably damages existing vegetation, resulting in the loss of original vegetation carbon sinks. This loss is quantified as equivalent carbon emissions, where ${\displaystyle \sum _{r=1}^R{C}_{r,l}\cdot S_r\cdot Y}$. Here, C_r,l denotes the annual carbon sequestration capacity of the r-th vegetation type, and S_r represents the area of the r-th vegetation type.

In summary, the integrated calculation model for life cycle carbon emissions in geotechnical engineering is formulated as follows:

$$C_e=C_{e,m}+C_{e,t}+C_{e,c}+C_{e,s}$$

(13)

where C_e represents the total carbon emissions for the life cycle of geotechnical engineering.

2.3. Framework for Robust Optimization Design

RGD can be implemented through the Taguchi method, Bayesian optimization, bi-objective optimization, or simplified single-objective optimization approaches [38]. Among these, bi-objective optimization methods are well-established in terms of theoretical foundation and have a broad range of applications; they are frequently employed in reliability-based geotechnical engineering for robust design. RGD consists of a set of objective functions and associated constraints. The optimization framework that incorporates robustness, as illustrated in Equation (14), provides a systematic approach for achieving this design.

$$\begin{array}{cc}\mathrm{Find}:\hfill & \mathrm{Supporting} \mathrm{parameters}, \mathit{d}\hfill \\ \mathrm{Subject}\mathrm{to}:\hfill & \mathrm{Safety} \mathrm{objective} \beta \left(\mathit{\kappa },\mathit{d}\right)\ge {\beta }_T, \mathrm{or} P_f\left(\mathit{\kappa },\mathit{d}\right)\le P_{fT}\hfill \\ & \mathrm{Design} \mathrm{space} \left(DS\right) \mathit{d}=\left[d_1,d_2,\dots ,d_q,\dots ,d_Q\right],d_{\mathrm{q}}\in \left[d_{q\_\min },d_{q\_\max }\right]\hfill \\ \mathrm{Objectives}:\hfill & \mathrm{Maximizing} \mathrm{design} \mathrm{robustness} R\hfill \\ & \mathrm{Minimizing} \mathrm{carbon} \mathrm{emissions} C_e\hfill \end{array}$$

(14)

The objective of Equation (14) is to search for an optimal anchor design solution within the design space (DS) that satisfies the target reliability of the anchored slope while simultaneously optimizing the design robustness R and carbon emissions C_e. This dual objectives optimization aims to enhance robustness and reduce carbon emissions. In this framework, d represents the Q-dimensional decision variables, which should correspond to anchor design parameters that are easily controllable by designers. DS is the set of candidate anchor design solutions, which can be determined based on engineering experience and local conditions. The target reliability β_T is a mandatory requirement for the safety of the anchored slope. It can be specified by owners or clients according to relevant design codes or based on the importance of the project and the consequences of failure, but it must meet code requirements. Design robustness R can be evaluated using SIV implemented through the MCS approach. Carbon emissions, C_e, refer to the carbon dioxide emissions generated during the whole life cycle of the anchored slope, which have an adverse impact on the environment. As shown in Figure 2, the primary steps for this paper are illustrated as follows:

(1)

Construct the model of interest prior to design optimization. In this step, the probability computation for anchored slopes is established with physical and reliability models; the carbon emission phases and boundaries are demarcated, and the carbon emission models of various phases for anchored slopes are established.

(2)

Determine the diverse parameters and specify the design domain in computational analyses. The noise factors and their stochastic properties are identified; the typical ranges of the design parameters are specified according to the particular design situation at hand; these design parameters are modeled as discrete variables in the DS; and quantify the phase-specific calculation parameters such as building material consumed, energy consumption and emission factors.

(3)

Obtain the objective function and constraint function values for each design solution. The reliability index β can be computed by MCS; the SIV can be evaluated by the proposed method in the Section 2.1 which is more accurate than the SIV based on the first-order reliability method (FORM) according to the MCS theory; and the carbon emissions can be assessed based on the LCA and the carbon emission factor methods.

(4)

Implement the bi-objective optimization using the non-dominated sorting genetic algorithm NSGA-II [39]. After applying Equation (14), numerous feasible design solutions are generated. Furthermore, by employing the NSGA-II [39], we obtain the warm-colored diamond-shaped scatter points as illustrated in Figure 3. During this process, the ideal scenario is often envisioned as a “utopian” design solution d₀ (as shown in Figure 3) that simultaneously achieves optimal design robustness R and minimal carbon emissions C_e. However, in reality, maximizing design robustness R (i.e., minimizing SIV) and minimizing carbon emissions C_e are inherently conflicting objectives. Consequently, it is impossible to achieve a single optimal design solution that simultaneously fulfills both objectives. Instead, the optimization process yields a set of non-dominated design solutions. These solutions are such that, between the two objectives, no solution is simultaneously superior or inferior to another; rather, they are collectively superior to other feasible designs within the solution space d_i. For example, as shown in Figure 3, while the non-dominated design d₂ offers higher carbon emissions, another non-dominated design d₃ achieves a lower SIV value (indicating better robustness R), and vice versa. These non-dominated designs collectively form a Pareto front, which reveals the trade-offs between the two design objectives [23,39]. As mentioned above, the Pareto front can be obtained using the NSGA-II algorithm [39]. Although the optimization algorithm has been widely applied in fields such as industrial and electrical engineering, its application in geotechnical engineering, particularly in anchor design, differs due to the unique nature of the discipline.

(5)

Select the optimal design using the normal boundary intersection (NBI) method [40]. After identifying the Pareto front via NSGA-II, the algorithm can assist designers in making informed decisions tailored to specific project requirements. For instance, within DS, one may opt for a design solution that operates below a predefined carbon emission threshold C_e,P while minimizing SIV (as exemplified by design d₄ in Figure 3) or select a solution that achieves a specified robustness level SIV_P at the lowest level of carbon emissions C_e (as illustrated by design d₃ in Figure 3). Such choices are typically made in response to the preferences of stakeholders or clients. However, in the absence of predefined priorities, the knee point of the Pareto front (as shown by design d₅ in Figure 3) is often regarded as the most optimal solution, as it strikes the best balance between the two objectives. As Figure 3 demonstrates, selecting a design to the left of the knee point d₅ on the Pareto front results in only marginal reductions in carbon emissions C_e, but significantly compromises robustness R, which is undesirable. Conversely, choosing a design to the right of the knee point achieves only modest improvements in robustness R but at the expense of substantially higher carbon emissions C_e, which is also impractical. Therefore, when no specific preferences are articulated by the stakeholders or clients, the knee point design serves as the most favorable option within the DS. The NBI method [40] is employed to identify the knee point design following the extraction of the Pareto front in geotechnical engineering.

3. Illustrative Example for a Framework with an Anchored Slope

To demonstrate the applicability and effectiveness of the MCS-based SIV and corresponding robust optimization design framework, the 1# tailwater tunnel outlet slope of the Wudongde hydropower station in southwestern China was selected as the engineering case study. This case not only features complex geological conditions and stringent design requirements but also serves as an ideal example for evaluating the practical significance of the proposed method in terms of improving engineering safety and reducing the environmental impact.

3.1. Engineering Overview

The 1# tailwater tunnel outlet slope, ranging from an elevation of 820 m to 940 m, represents a steep slope with a grade of 65–67° and a slope height of 110 m. The section from an elevation of 940 m to 1050 m forms a bench terrain with an average slope grade of 35°. Elevations above 1050 m are characterized by steep terrain with an average slope grade of 55°. The stratigraphic lithology of the 1# tailwater tunnel outlet slope consists of thin-bedded intercalated limestone from the 8th member of the Luoxue Formation (Pt_2l⁸), gray thin-bedded limestone from the 9th member of the Luoxue Formation (Pt_2l⁹), and thin-bedded intercalated gray limestone from the 10th member of the Luoxue Formation (Pt_2l¹⁰). The overall rock stratigraphic orientation is near EW, S∠75–85°. The non-conforming contacts generally exhibit hard contact characteristics without weak layers, and the bonding is tight, showing favorable properties. Figure 4 provides a generalized engineering geological cross-section of the 1# tailwater tunnel axis engineering in Wudongde hydropower station. Since this study focuses on the reinforcement and stabilization of the slope to improve its overall stability, only those joints influencing the local stability of the slope surface are simplified in the cross-section. From Figure 4, it can be observed that the slope is a typical bedding-controlled slope. However, due to the presence of a bedding plane dipping at 73°, potential issues related to bedding-controlled stability may arise after slope excavation. The primary failure mode is sliding along the rock layer surfaces, with the most critical potential slip surface located at a maximum horizontal depth of approximately 19 m, situated near an elevation of 850 m.

The excavated slope was reinforced with integrated support measures, including shotcrete, systematic rock bolts, and prestressed anchor cables. The specific implementation scheme was as follows: (1) Two rows of systematic rock bolts (28 mm diameter, 12 m length) were installed at each slope tier crest. Above 850 m elevation, bolts were spaced at 2 m × 2 m intervals and inclined downward at 5°. Below 850 m elevation, identical spacing was maintained with a 10° downward inclination. (2) Five rows of systematic rock bolts (25 mm diameter, 6 m length) were positioned at mid-height of each slope tier. For sections above 850 m elevation, bolts were arranged in 2 m × 2 m grids at a 5° downward inclination. Below this elevation, identical spacing and 5° inclination were applied. (3) above an elevation of 850.50 m, each slope level is equipped with two rows of anchor cables with T = 2000 kN, L = 30 m @ 4.5 m × 4.5 m; the anchorage segment is 8 m in length, installed at a downward inclination of 10°. Below an elevation of 850.50 m, the cut slope is reinforced with three rows of anchor cables of the same specification; the first row of anchor cables is placed at an elevation of 848.75 m, with an 8 m anchorage segment, also installed at a 10° downward inclination [41].

In this case study, the experimental data available for research at the dam site of the hydropower station are limited, with sampling points spaced at considerable distances, far from sufficient to obtain accurate variation coefficients. As a result, the variation coefficients of parameters such as the cohesion and friction angle of the discontinuity surfaces exhibit significant uncertainty. Therefore, when calculating the robustness of the reliability in anchored slope designs, these data must be taken into account. Although the yield strength of anchor steel and the bond strength of grouting materials are also random parameters, they are artificial materials that have undergone extensive mechanical testing before being incorporated into the slope. Consequently, the uncertainty in their statistical parameters can be considered negligible and thus need not be included in robustness calculations compared to geomaterials. It is important to note that all the aforementioned parameters must still be treated as random variables in reliability analysis, with their specific values provided in

Table 1. Statistical information of the related random variables (or uncertain input parameters).

Random Variables		Mean	COV	Distribution
Rock mechanical parameters
c	Layer cohesion, /MPa	0.205	0.146	Normal
φ	Layer internal friction angle, /°	38	0.085	Normal
Anchorage structure performance
σu	Yield strength of anchor cable strands, /MPa	1860	0.05	Lognormal
τq	Bond strength between anchor cable and grout, /MPa	2.92	0.12	Lognormal
τs	Bond strength between grout and bore wall, /MPa	1.54	0.2	Lognormal

[41,42,43]. To simplify the problem, it is permissible to assume that these random variables are mutually independent, which is not a strict requirement for robustness optimization design [24,26].

3.2. Reliability Analysis

Although the excavated slope is supported by shotcrete, systematic rock bolts and prestressed anchor cables, the prestressed anchor cables essentially play a key anti-sliding role in the overall sliding. Therefore, the reliability analysis of the anchored slope on the 1# tailwater outlet only considers the prestressed anchor cables, as shown in Figure 5. It should be noted that in the design solution proposed in this paper, the anchor segments are only required to extend at least 0.5 m below the potential slip surface. Additionally, the anchor cables on each slope level should be staggered in the depth direction to prevent through cracks from forming in the rock mass at the anchor ends due to tensile stress. Shukla and Hossain [44] proposed a model that simplifies anchor forces into a total resistance force, T, using the limit equilibrium method to determine the safety factor. However, their model is limited by its consideration of only a unit width of the slope and disregard for the horizontal distance S_H of anchors, which does not align with practical conditions. Accordingly, following Jiang’s idea [45], this study employs the rigid block limit equilibrium method to derive the safety factor calculation Equation (15) for the anchored slope. Importantly, the vertical distance S_V of anchors dictates the number of anchors M in a single row, significantly influencing the overall stability of the slope. This approach provides a more comprehensive understanding, accounting for the critical factors omitted in earlier models.

$$F_S={\displaystyle \frac{\left(cl\right)S_H+\left(W\cos \psi S_H+{\displaystyle \sum _{i=1}^M{T}_i\sin \left({\epsilon }_i+\psi \right)}\right)\tan \phi }{W\sin \psi S_H-{\displaystyle \sum _{i=1}^M{T}_i\cos \left({\epsilon }_i+\psi \right)}}}$$

(15)

where c and φ are the parameters of shear strength for rock joints; l represents the length of the sliding body’s base; W is the weight of the sliding body; $\psi$ is the dip angle of the sliding surface; and T_i is the resistance force of the i-th anchor. In the above analysis, although the anchor resistance is uniformly represented by T_i in the calculation of the safety factor, the specific form of Equation (15) varies depending on the failure mode of the anchors that lead to overall slope instability. This study primarily considers three anchor failure modes [46]: (1) debonding failure between the grout in the anchorage segment and the surrounding rock (denoted as failure mode 1), (2) debonding failure between the grout and the anchor itself (denoted as failure mode 2), and (3) tensile failure of the free segment of the anchor (denoted as failure mode 3).

(1)

Failure mode 1

Debonding failure often occurs between the grout in the anchorage segment and the surrounding rock. Under such circumstances, the anchorage resistance T^I provided by the anchor is primarily determined as follows:

$$T^I=\pi D_Z{L}_a{\tau }_s$$

(16)

where τ_s is the bond strength between grout and bore wall, L_a is the length of the anchorage segment, and D_Z is the equivalent diameter of the grout in the anchorage segment (shown as Figure 5b), which can be determined based on the Code for design of prestressed anchorage for hydropower projects (NB/T 10802-2021) [47] and the literature [48].

(2)

Failure mode 2

Debonding failure may occur between the grout and the anchor. Under such circumstances, the anchorage resistance T^II provided by the anchor is primarily determined as follows:

$$T^{\mathrm{II}}=\pi D_M{L}_a{\tau }_q$$

(17)

where τ_q is the bond strength between grout and bore wall, and D_M is the equivalent diameter of the anchor in the anchorage segment (shown as Figure 5b), which can be determined based on the Code for design of prestressed anchorage for hydropower projects (NB/T 10802-2021) [47] and the literature [48].

(3)

Failure mode 3

The free segment of the anchor exhibits potential tensile failure when subjected to axial force, which can result in the rupture of steel strands. At this stage, the maximum axial force T^III provided by the anchor is as follows:

$$T^{\mathrm{III}}=n{\displaystyle \frac{\pi }{4}}{D}^2{\sigma }_u$$

(18)

where n is the number of anchor strands and D is the diameter of a single anchor strand (shown as Figure 5b), and σ_u is the yield strength of anchor cable strands.

As demonstrated in the previous section, the limit state equations for the slope subjected to the three failure modes of anchors are presented in

Table 2. Limit state functions and their physical interpretation for the illustrative example.

Limitation State Function	Interpretation
g1 = FSI − 1	Block sliding due to failure mode 1
g2 = FSII − 1	Block sliding due to failure mode 2
g3 = FSIII − 1	Block sliding due to failure mode 3

. An anchored slope exhibits multiple potential failure modes, and the occurrence of any single failure mode results in the slope becoming unstable, which is a series system reliability problem.

Thus, regarding the reliability index of a series system for anchored slopes, the calculation formula is as follows:

$$\beta =-{\Phi }^{-1}\left(P\left\{\min \left[g_1,g_2,g_3\right]\le 0\right\}\right)$$

(19)

To ensure computational accuracy, this study employs a direct MCS method. The required number of samples is dependent on the reliability index; as the reliability index increases, the computational accuracy increases, necessitating a higher number of samples. The necessary number of samples can be determined using Equation (20) as follows [49]:

$$N>{\displaystyle \frac{\ln \left(1-\alpha \right)}{{\Phi }^{-1}\left(P_f\right)}}$$

(20)

where α is the confidence level, which is adopted as 98% herein.

3.3. Determining the Design Space

Input parameters are categorized into design parameters and noise parameters (i.e., geomaterial properties) in RGD. Design parameters refer to the geometric and physical properties of the support structure that can be easily controlled during engineering design. These parameters influence the reliability and robustness of the anchor–slope system, and they are generally treated as deterministic variables. In this case study, the high slope of the tailwater tunnel outlet requires prestressed anchor support, whose structure is illustrated in Figure 5b (taking the 100 t anchor as an example).

Table 3. Basic parameters of prestressed anchor cable.

Design Resistance Td/t	Geometrical Parameter of Anchor Cable			Elasticity Modulus E/GPa	Yield Strength σu/MPa	Bond Strength with Grout τq/kPa
	Strand Amount and Diameter D/mm	Length L/m	Length of Anchorage Segment La/m
100	6Φ15.24	40~120	8~12	180	1860	2920
150	9Φ15.24
200	12Φ15.24
250	16Φ15.24
300	19Φ15.24
350	23Φ15.24
400	26Φ15.24

presents the specifications and mechanical parameters of anchors at various levels applied in engineering practice.

The resistance of a single anchor is primarily determined by the number and diameter of the steel strands used, which, for calculation convenience, is represented by the equivalent diameter D_e of the free segment to signify different levels of prestressed anchors. The resistance provided by the anchor is also related to the anchorage length L_a, which typically ranges from 8 m to 12 m in practice. Prestressed anchors are generally embedded in a grid pattern on the slope face. The spacing between anchors, including the vertical distance S_V and horizontal distance S_H, determines the density of anchor installation and thus influences the stability of the reinforced slope. These spacings are typically set between 3 m and 6 m. Additionally, the anchorage angle ε, which is typically between 10° and 25° based on engineering experience, is another factor affecting slope stability. Therefore, in this robust design, the equivalent diameter D_e of the free segment, anchorage length L_a, vertical distance S_V, horizontal distance S_H, and anchorage angle ε are defined as controllable design parameters. To facilitate construction, the above parameters are discretized, with their design domains listed in

Table 4. Design space DS selected in the illustrative example.

Design Parameters	Design Pool
Equivalent diameter of anchor cable, De/m	{0.037, 0.046, 0.053, 0.061, 0.066, 0.073, 0.078}
Length of anchorage segment, La/m	{8, 9, 10, 11, 12}
Vertical distance, SV/m	{3, 3.5, 4, 4.5, 5, 5.5, 6}
Horizontal distance, SH/m	{3, 3.5, 4, 4.5, 5, 5.5, 6}
Anchorage inclination, ε/°	{10, 15, 20, 25}

. All possible combinations of these parameters total 6860 sets, which serve as input parameters for the robust-based geotechnical design. The optimal combination will be determined from these designs.

3.4. Carbon Emission Calculation

According to the mathematical model of robustness design, in addition to ensuring the robustness of the designs, reducing carbon emissions is also one of the optimization objectives for this anchored slope engineering project. The carbon emissions associated with individual phases of the entire life cycle of anchored slopes can be systematically quantified following the methodological framework outlined below.

(1)

Calculating the carbon emissions of producing building materials

In accordance with the construction organization regulations and field experience for anchored slope engineering, the primary building materials for anchored slope projects include concrete, cement mortar, steel strands, and reinforcement bars. Carbon emission factors in anchored slope engineering exhibit significant variability due to regional differences, sector-specific practices, and production techniques. Therefore, to ensure regional relevance and data credibility, carbon emission factors are predominantly sourced from databases and the peer-reviewed literature published by local research institutions [50,51,52,53]. The carbon emission factors for primary building materials, as collated in this study, are summarized in

Table 5. Carbon emission factors and transportation distances for primary building materials.

Types of Building Materials	Carbon Emission Factor	Unit	Transportation Distance/km
Concrete	240	kg/m3	20
Cement mortar	790	kg/t	21
Steel strand	2150	kg/t	24
Reinforcement bar	2150	kg/t	24

. Using Equation (9), the carbon emissions during the production phase of primary materials are estimated for different design solutions.

(2)

Calculating the carbon emissions of transporting building materials

This project employs heavy-duty diesel trucks for material transportation, with a carbon emission factor of 1.61 × 10⁻⁴ kg/(kg/km). The transportation distances for primary building materials are detailed in Table 5. Using Equation (10), the carbon emissions during the transportation phase of primary materials for the anchored slope engineering project are estimated.

(3)

Calculating the carbon emissions during the construction phase

Anchored slope engineering is primarily categorized into three construction components: prestressed anchor cable support, systematic rock bolt support, and shotcrete application. This study delineates the key construction processes for these three components, establishing the energy consumption per shift for each type of construction machinery based on construction quota standards and empirical field practices. The carbon emissions per operational shift are calculated using energy-specific emission factors, as detailed in

Table 6. Carbon emissions per operational shift for key construction processes in the illustrative example.

Key Construction Process	Construction Machinery	Energy Consumption per Shift		Carbon Emissions per Shift/kg
		Diesel/kg	Electric Energy/kW·h
Prestressed anchor cable support	Rebar cutting machine		32.1	3.98
	Multi-functional anchoring drilling trolley	106.5		390.86
	Electric single-drum slow-speed winch		126	15.61
	Mortar mixer		8.61	1.07
	Extrusion-type mortar conveying pump		23.7	2.95
	Grout pump		32.5	4.03
	Prestressed anchor cable tensioning machine		54.25	6.72
Systematic rock bolt support	Rebar cutting machine		32.1	3.98
	Multi-functional anchoring drilling trolley	106.5		390.86
	Electric single-drum slow-speed winch		126	15.61
	Mortar mixer		8.61	1.07
	Extrusion-type mortar conveying pump		23.7	2.95
	Grout pump		32.5	4.03
Shotcrete support	Rebar cutting machine		32.1	3.98
	Electric single-drum slow-speed winch		126	15.61
	Concrete mixer		34.1	4.22
	Concrete conveying pump		243.46	30.15
	Wet-mix shotcrete machine		15.4	1.91

. Finally, by integrating the anchor cable design solutions to determine the total number of shifts, the carbon emissions during the construction phase of the anchored slope project are quantified using Equation (11).

(4)

Calculating the carbon emissions during the service phase

During the service phase of anchored engineering, the primary tasks involve monitoring anchor cable axial forces and slope displacements using anchor cable load cells and multipoint extensometers, respectively. The number of monitoring points depends on the installed anchor quantity and the slope support scale. For analytical simplification, it is assumed that 10% of the total anchor cables are instrumented with load cells within the unit cross-section, and two multipoint extensometers are deployed on each level of the slope. The monitoring process primarily consumes electrical energy; however, the associated energy consumption is negligible compared to that of construction-phase equipment. Thus, carbon emissions from monitoring activities are excluded from this study. Additionally, over the 100-year service life of the hydropower project, carbon emissions generated by periodic concrete repairs must be considered. Based on field engineering experience, the annual maintenance material consumption is preliminarily assumed to be 1% of the initial building material usage. Given that the anchored slope employs shotcrete slope protection, vegetation-related carbon sequestration on the slope surface is not considered. Furthermore, this study focuses exclusively on the anchored slope, thereby excluding carbon sink losses from pre-existing vegetation disruption due to excavation. The carbon emissions during the service phase of the anchored slope are estimated using Equation (12).

It should be emphasized that what causes the variation in carbon emissions is the difference in the anchorage design parameter d, which only occurs in the material production phase, the transportation phase, and the construction phase. Systematic rock bolt support and shotcrete support, and the consumption during the service phase, are considered to have essentially unchanged carbon emissions across different design solutions. Although discrepancies may exist between the estimated and actual carbon emissions due to uncertainties in material production efficiency, transportation logistics, or operational energy sources, these estimated and actual carbon emissions are positively correlated. This relationship does not compromise the validity of the bi-objective optimization process or the selection of final design solutions.

3.5. Bi-Objective Optimization Model for the Anchored Slope

The robust design of prestressed anchor-reinforced slopes aims to achieve carbon emission and robustness benefits while ensuring safety. This is accomplished by adjusting controllable design parameters to identify an optimal design. The goal is to maximize the design robustness R (i.e., minimize SIV) of the design reliability against uncontrollable noise factors, such as uncertainties in the statistical characteristics of rock joint shear strength parameters. At the same time, the design must prioritize carbon emissions, ultimately advancing the overarching objective of enhancing engineering safety and sustainability.

From this, it can be seen that the robust design of prestressed anchor-reinforced slopes can be transformed into a bi-objective optimization decision-making process. In this paper, the reliability is adopted as the safety constraint boundary for robust optimization design. According to the current standard of the People’s Republic of China, Unified Standard for Reliability Design of Hydraulic Engineering Structures (GB 50199-2013) [54], plane sliding failure of anchored high slopes on 1# tailwater outlet should be regarded as a type of secondary ductile failure. Under natural conditions, the target reliability index β_T is set to 3.2; that is to say, the reliability index of all alternative designs should be larger than 3.2. For accidental conditions (i.e., rainfall conditions), the target reliability index β_T is allowed to be one level lower than that under natural conditions, specifically 2.7. SIV is used as the robustness index for the anchored slope, and the carbon emissions of the design are calculated using Equations (9)–(13). Both of these are treated as objective functions in bi-objective optimization decision-making. At this point, the bi-objective optimization model for the robust design of prestressed anchor-reinforced slopes is as shown in Equation (21).

$$\begin{array}{cc}\mathrm{Find}:\hfill & \left[\mathit{d}\right]\in \left\{D,L,S_V,S_H,\epsilon \right\}\hfill \\ & \mathit{D}\mathit{S}\hfill \\ & \mathrm{Reliability} \mathrm{index} \mathrm{of} \mathrm{anchored} \mathrm{slope} \mathrm{under} \mathrm{natural} \mathrm{conditions}:\hfill \\ \mathrm{Subject} \mathrm{to}:\hfill & \beta \ge {\beta }_T=3.2\hfill \\ & \mathrm{Reliability} \mathrm{index} \mathrm{of} \mathrm{anchored} \mathrm{slope} \mathrm{under} \mathrm{rainfall} \mathrm{conditions}:\hfill \\ & \beta \ge {\beta }_T=2.7\hfill \\ \mathrm{Objectives}:\hfill & \begin{array}{c}\mathrm{Min} SIV\hfill \\ \mathrm{Min} C_e\hfill \end{array}\hfill \end{array}$$

(21)

3.6. Results and Analysis

This study begins with an analysis under natural conditions. By employing the corresponding algorithm, the design robustness R (SIV, herein) and carbon emissions C_e of all candidate design solutions are calculated. These values are plotted in Figure 6, where approximately 5547 candidate designs meet the target reliability. Using the NSGA-II algorithm, 75 non-dominated solutions are identified for robust design. These non-dominated designs collectively form the Pareto front, as indicated by the red solid pentagrams in Figure 6. On the Pareto front, the non-dominated designs exhibit a clear negative correlation between SIV and carbon emissions C_e; as carbon emissions C_e increase, SIV decreases, and vice versa. Therefore, it is not possible to identify a single design solution on the Pareto front that simultaneously achieves optimal robustness and low carbon emissions. To determine the optimal design, the knee point theory is applied.

In this study, the knee point design was determined using the normal-boundary intersection (NBI) method. It is important to note that prior to applying the NBI method, the performance of all non-dominated designs was normalized. The calculation results are presented in Figure 7. The orthogonal boundary line passes through the points representing the maximum SIV (minimum C_e) and minimum SIV (maximum C_e). While there is no clear dominance hierarchy among the non-dominated designs on the Pareto front, the distances between these designs and the orthogonal boundary line vary. The knee point is identified by comparing these distances, with the solution furthest from the orthogonal boundary line selected as the knee point. As shown in Figure 7, the black pentagram has the maximum distance from the orthogonal boundary line, thereby identifying Design 44 as the knee point on the Pareto front, representing the optimal design. This design employs 100 t prestressed anchors (D_e = 0.037 m), with an anchorage segment length L_a of 8 m, a vertical distance S_V of 3.5 m, a horizontal distance S_H of 3 m, and an anchorage inclination ε of 10°. The reliability β is 4.19, the design robustness SIV is 1.39 × 10⁻³, and the carbon emission C_e is 2.50 × 10⁴ kg per width.

Following the same methodology, the Pareto front and optimal design under rainfall conditions were determined, which are presented in Figure 8.

The optimal designs for robustness optimization of the reinforced slope under natural and rainfall conditions are identified as RGD-1 and RGD-2. The respective design parameters and performance metrics for these designs are presented in

Table 7. Performance comparison of knee point designs under various conditions vs. the initial design.

Condition (βT)	Design Solution	Anchor Design Parameters d					Design Safety	Carbon Emissions	Design Robustness, R
		Td (t)	L (m)	SV (m)	SH (m)	ε (°)	β	Ce (kg)	SIV
Natural (3.2)	RGD-1	100	8	3.5	3	10	4.19	2.50 × 104	1.39 × 10−3
	RGD-2	150	8	3.5	3.5	10	4.51	2.63 × 104	2.79 × 10−4
	Initial	200	8	4.5	4.5	10	3.46	2.52 × 104	1.85 × 10−2
Rainfall (2.7)	RGD-1	100	8	3.5	3	10	3.33	2.50 × 104	4.73 × 10−2
	RGD-2	150	8	3.5	3.5	10	3.87	2.63 × 104	6.06 × 10−3
	Initial	200	8	4.5	4.5	10	2.66	2.52 × 104	0.297

.

The calculation results demonstrate that under natural conditions, the reliability of the robust designs RGD-1 and RGD-2 is 4.19 and 4.51, respectively, both of which are higher than the initial design reliability of 3.46. Under rainfall conditions, the reliability of RGD-1 and RGD-2 is 3.33 and 3.87, respectively, exceeding the initial design reliability of 2.66. From the perspective of SIV, under natural conditions, the SIV of RGD-1 and RGD-2 is 1.39 × 10⁻³ and 2.79 × 10⁻⁴, respectively, both of which are lower than the initial design SIV of 1.85 × 10⁻². Under rainfall conditions, the SIV of RGD-1 and RGD-2 is 4.73 × 10⁻² and 6.06 × 10⁻³, respectively, both of which are lower than the initial design SIV of 0.297. These results clearly indicate that the RGD method can significantly improve the reliability of the reinforced slope and reduce the impact of statistical uncertainty in rock shear strength on the reliability results. Additionally, as shown in Table 7, RGD-1 not only surpasses the initial design in terms of both reliability and robustness but also achieves a carbon emission of 2.50 × 10⁴ kg per unit width, a reduction of 200 kg compared to the initial design. Meanwhile, RGD-2 requires only an additional 110 kg compared to the initial design while significantly enhancing the reliability and robustness of the reinforced slope. Considering engineering sustainability, RGD-1 (the bolded row in Table 7) is more suitable as the optimal anchorage support design, where the 1# tailwater tunnel outlet slope is reinforced with 100 t prestressed anchors, with an anchor segment length of 8 m, vertical and horizontal distance of 3.5 m and 3 m, respectively, and an inclination angle of 10°. Proper drainage measures are also required. To ensure the long-term safety of the project during both the construction and operational phases, monitoring points should be installed on the exposed segments of the potential slip surface, at select anchor heads, and at the slope crest to perform long-term monitoring of the reinforced slope.

Through the aforementioned comparative analysis, it is evident that the knee point design obtained through a modified robust design framework exhibits superiority over the initial design in terms of reliability, robustness, and carbon emissions. This demonstrates that the modified robust design framework presented in this paper possesses distinct advantages in the practical implementation of safety and sustainability concepts.

4. Validation and Discussion

4.1. Validation of the SIV Based on MCS

To validate the effectiveness of the SIV derived from the MCS method, a comparative study is required to investigate the relationship between the SIV calculated via FORM for different design solutions and the proposed computational approach. For clarity, the former is denoted as SIV_MCS, and the latter as SIV_FORM, where SIV_FORM can be derived using Equation (22) [26].

$${\mathit{S}}_{\mathit{\delta }}={\displaystyle \frac{\partial P_f}{\partial \beta }}{\displaystyle \frac{\partial \beta }{\partial {\mathit{y}}^{*}}}{\displaystyle \frac{\partial {\mathit{y}}^{*}}{\partial \mathit{\delta }}}\mathit{e}=[-\phi (\beta ){\alpha }_1{\displaystyle \frac{x_1^{*}-{\mu }_1}{{\mu }_1{\delta }_1^2}},\dots ,-\phi (\beta ){\alpha }_i{\displaystyle \frac{x_i^{*}-{\mu }_i}{{\mu }_i{\delta }_i^2}},\dots ,-\phi (\beta ){\alpha }_n{\displaystyle \frac{x_n^{*}-{\mu }_n}{{\mu }_n{\delta }_n^2}}]$$

(22)

where μ_i and δ_i are the mean value and the COV of the i-th noise factor, respectively, and they are test statistics; φ(·) is the standard norm PDF; β is the reliability index; α is the coefficient of sensitivity; and x^* is the checkpoint in the original norm space. All can be derived from reliability computations based on FORM. If the random variables do not obey the normal distribution, the equivalent normalization must first be performed [55].

Utilizing the case study data presented in this paper, we calculated the SIV_FORM values for all design solutions satisfying the prescribed constraints under two conditions. These SIV_FORM values were plotted on the abscissa (x-axis), while the corresponding SIV_MCS values were plotted on the ordinate (y-axis), resulting in a comparative scatter plot to visualize the relationship between the two metrics.

As illustrated in Figure 9, although the scatter distribution of SIV_MCS versus SIV_FORM under rainfall conditions exhibits greater dispersion compared to that under natural conditions, both scenarios demonstrate a high degree of linear correlation between SIV_MCS and SIV_FORM, with data points closely aligned to the reference line y = x. Furthermore, the relative deviations of SIV values for the two conditions are merely 2% and 4.64%, respectively. These findings collectively indicate strong consistency between SIV_MCS and SIV_FORM. The mutual validation of FORM and MCS underscores the feasibility and reliability of employing MCS for SIV calculations in this study.

4.2. Limitation of the Proposed Framework

Despite the promising potential demonstrated by this framework in practical applications, further modification is still needed in several areas:

(1)

Computational trade-offs between Monte Carlo simulation precision and runtime efficiency

The SIV calculation in this paper is a by-product of reliability computation using MCS. Therefore, its limitations depend on the shortcomings of MCS for reliability, including primarily three challenges: First, the efficiency bottleneck in low failure probability scenarios. For example, in this study’s engineering case, over 10⁷ simulations were required to ensure result accuracy, consuming several days, constituting a significant cost barrier. Second, insufficient exploration in high-dimensional spaces. For other complex systems with high dimensions, dimensionality growth causes sample sparsity, limiting precise modeling capability. Third, statistical bias in rare events. Uniform sampling causes missing tail failure samples, leading to increased estimation variance or even non-convergence. Therefore, to overcome these limitations, future frameworks should focus on the following improvement paths during reliability analysis: First, targeted sampling strategies, adopting importance sampling or subset simulation to concentrate computational resources on the failure domain, enhancing sampling efficiency for low-probability events. Second, surrogate model acceleration, using adaptive Kriging or neural network surrogate models to replace high-cost physical simulations, dynamically approximating the limit state function boundary. Third, dimensionality reduction techniques, combining global sensitivity analysis to identify key variables and compress the stochastic space dimension. Such methods are expected to systematically resolve the efficiency-accuracy trade-off dilemma while preserving the generality of MCS

(2)

Regional constraints of emission factors applied to global contexts

This paper’s carbon emission calculation is based on LCA theory and the carbon emission factor method, providing fundamental data for case study accounting. However, it has significant regional limitations. Carbon emission factors heavily depend on local infrastructure and regulatory policies, struggling to reflect technologically lagging regions or areas with inferior fuel quality. Direct application of this study’s fundamental data would cause systematic distortion in local emission accounting. Additionally, carbon emission factors have long update cycles, failing to capture rapid technological transitions in emerging economies, causing temporal prediction offsets. Thus, future improvements for carbon accounting accuracy can focus on two solutions. First, climate-economic zoning calibration: establishing spatial stratification based on technological level and climate zones, generating localized carbon emission factor databases through UAV sampling and ground station measurements. Second, multi-source data assimilation: integrating satellite remote sensing and IoT sensor networks to build carbon emission factor update algorithms, achieving monthly-scale bias correction.

In summary, although this study has room for improvement in reliability analysis, SIV assessment, and carbon emission quantification, its fundamental contribution lies in shifting the traditional cost-oriented design framework towards a systematic engineering decision-making framework integrating dual-objective collaborative optimization of “robustness-carbon emission”. This shift overcomes the industry’s long-standing path dependence on economic indicators, providing a trade-off tool for engineering fields worldwide that balances engineering safety and environmental sustainability.

5. Conclusions

In this study, we developed a MCS-based method for calculating the SIV and established a reliability-constrained bi-objective optimization framework integrating robustness and carbon emissions for anchored slopes. This work directly addresses two persistent challenges in international RGD research: (1) the computational limitations of existing robustness quantification methods [14,21,24,25,26], particularly their dependency on specific reliability algorithms; and (2) the prevailing focus on cost-safety trade-offs that overlooks decarbonization imperatives aligned with the Paris Agreement [29,30]. By implementing SIV through MCS—a globally recognized benchmark for reliability computation [27,28]—our approach achieves algorithm-independent robustness assessment, and it offers a more universally applicable solution for geotechnical projects in regions with limited geotechnical data. On the other hand, the framework explicitly incorporates low-carbon objectives, thereby advancing RGD toward sustainable infrastructure paradigms emerging worldwide.

Validated through a hydropower slope case study in Southwest China, the framework demonstrates the potential for international practice. Beyond confirming compatibility with conventional design codes, the derived Pareto-optimal solution set empowers designers worldwide to balance carbon reduction against robustness requirements under reliability constraints. The solution reduced carbon emissions while enhancing design robustness compared to the initial design, outperforming traditional methods. The framework effectively bridges the gap between geotechnical design and UN Sustainable Development Goals [30].

This study contributes to global geotechnical practice by establishing a decision-making architecture that harmonizes reliability, robustness, and sustainability. Its modular nature supports rapid integration of regional parameters, offering a unified tool for infrastructure decarbonization from alpine tunnel projects to coastal landslide mitigation. We anticipate this framework will catalyze international collaborations to refine region-specific carbon databases and uncertainty models, ultimately setting new standards for geotechnical engineering design.