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Search Results (2,002)

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21 pages, 3896 KiB  
Article
Optimization Strategy for Integrated Energy Microgrids Based on Shared Energy Storage and Stackelberg Game Theory
by Zhilong Yin, Zhiguo Wang, Feng Yu, Dong Wang and Na Li
Electronics 2024, 13(22), 4506; https://doi.org/10.3390/electronics13224506 (registering DOI) - 16 Nov 2024
Abstract
The implementation of community power generation technology not only increases the flexibility of electricity use but also improves the power system’s load distribution, increases the overall system efficiency, and optimizes energy allocation. This article first outlines the operational context of the system and [...] Read more.
The implementation of community power generation technology not only increases the flexibility of electricity use but also improves the power system’s load distribution, increases the overall system efficiency, and optimizes energy allocation. This article first outlines the operational context of the system and analyzes the roles and missions of the various participants. Subsequently, optimization models are developed for microgrid operators, community power storage facility service providers and load aggregators. Next, the paper explores the game relationship between microgrid operators and load aggregators, proposing a model based on the Stackelberg game theory and proving the presence and singularity of the Stackelberg equilibrium solution. Finally, simulations are conducted using Yalmip tools and the CPLEX solution on the MATLAB R2023a software platform. A combination of heuristic algorithms and solver methods is employed to optimize the strategies of microgrid operators and load aggregators. The research findings show that the proposed framework is not only able to achieve an effective balance of interests between microgrid operators and load aggregators but also creates a win–win situation between load aggregators and shared energy storage operators. Additionally, the solution algorithms used ensure the protection of data privacy. Full article
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<p>Schematic diagram of microgrid framework.</p>
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<p>Different types of constraints.</p>
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<p>Iterative revenue chart for scenarios 3 and 4.</p>
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<p>Electricity and heat pricing chart of the microgrid operator in scenarios 3 and 4.</p>
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<p>Market electricity price and negative load-shifting relationship in scenario 4.</p>
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<p>Optimized electric load results for scenarios 2 and 4.</p>
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<p>Optimized heat load results for scenarios 3 and 4.</p>
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<p>User heat load changes in scenarios 3 and 4.</p>
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22 pages, 4646 KiB  
Article
Concrete Creep Prediction Based on Improved Machine Learning and Game Theory: Modeling and Analysis Methods
by Wenchao Li, Houmin Li, Cai Liu and Kai Min
Buildings 2024, 14(11), 3627; https://doi.org/10.3390/buildings14113627 - 14 Nov 2024
Viewed by 241
Abstract
Understanding the impact of creep on the long-term mechanical features of concrete is crucial, and constructing an accurate prediction model is the key to exploring the development of concrete creep under long-term loads. Therefore, in this study, three machine learning (ML) models, a [...] Read more.
Understanding the impact of creep on the long-term mechanical features of concrete is crucial, and constructing an accurate prediction model is the key to exploring the development of concrete creep under long-term loads. Therefore, in this study, three machine learning (ML) models, a Support Vector Machine (SVM), Random Forest (RF), and Extreme Gradient Boosting Machine (XGBoost), are constructed, and the Hybrid Snake Optimization Algorithm (HSOA) is proposed, which can reduce the risk of the ML model falling into the local optimum while improving its prediction performance. Simultaneously, the contributions of the input features are ranked, and the optimal model’s prediction outcomes are explained through SHapley Additive exPlanations (SHAP). The research results show that the optimized SVM, RF, and XGBoost models increase their accuracies on the test set by 9.927%, 9.58%, and 14.1%, respectively, and the XGBoost has the highest precision in forecasting the concrete creep. The verification results of four scenarios confirm that the optimized model can precisely capture the compliance changes in long-term creep, meeting the requirements for forecasting the nature of concrete creep. Full article
(This article belongs to the Section Building Materials, and Repair & Renovation)
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<p>Histogram of the distribution of input variables.</p>
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<p>Indicators for Model Evaluation.</p>
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<p>Radar charts of the training (<b>a</b>) and testing sets (<b>b</b>) for the HSOA-SVM, HSOA-RF, and HSOA-XGBoost models’ performance.</p>
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<p>The regression results of the test sets for (<b>a</b>) SVM; (<b>b</b>) RF; (<b>c</b>) XGBoost; (<b>d</b>) HSOA-SVM; (<b>e</b>) HSOA-RF; and (<b>f</b>) HSOA-XGBoost.</p>
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<p>The regression results of the test sets for (<b>a</b>) SVM; (<b>b</b>) RF; (<b>c</b>) XGBoost; (<b>d</b>) HSOA-SVM; (<b>e</b>) HSOA-RF; and (<b>f</b>) HSOA-XGBoost.</p>
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<p>Residual analysis plots for (<b>a</b>) HSOA-SVM; (<b>b</b>) HSOA-RF; and (<b>c</b>) HSOA-XGBoost models.</p>
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<p>Residual distribution plot for HSOA-SVM, HSOA-RF, and HSOA-XGBoost.</p>
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<p>Comparing the errors between the testing sets of (<b>a</b>) HSOA-SVM; (<b>b</b>) HSOA-RF; and (<b>c</b>) HSOA-XGBoost.</p>
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<p>Aggregated HSOA-XGBoost-based concrete creep SHAP.</p>
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<p>Global SHAP values using the HSOA-XGBoost model.</p>
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<p>Results of importance analysis using SHAP. (<b>a</b>) Time since loading (days). (<b>b</b>) Cement content (Kg/m<sup>3</sup>). (<b>c</b>) Water–cement ratio. (<b>d</b>) Loading stress (MPa). (<b>e</b>) Compressive strength (MPa).</p>
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<p>SHAP characteristic force diagram. (<b>a</b>) Scenario 1; (<b>b</b>) Scenario 2; (<b>c</b>) Scenario 3.</p>
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<p>Four typical scenarios for predicting creep flexibility: (<b>a</b>) Scenario S1; (<b>b</b>) Scenario S2; (<b>c</b>) Scenario S3; and (<b>d</b>) Scenario S4.</p>
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31 pages, 1865 KiB  
Article
Robustness Analysis of Multilayer Infrastructure Networks Based on Incomplete Information Stackelberg Game: Considering Cascading Failures
by Haitao Li, Lixin Ji, Yingle Li and Shuxin Liu
Entropy 2024, 26(11), 976; https://doi.org/10.3390/e26110976 - 14 Nov 2024
Viewed by 308
Abstract
The growing importance of critical infrastructure systems (CIS) makes maintaining their normal operation against deliberate attacks such as terrorism a significant challenge. Combining game theory and complex network theory provides a framework for analyzing CIS robustness in adversarial scenarios. Most existing studies focus [...] Read more.
The growing importance of critical infrastructure systems (CIS) makes maintaining their normal operation against deliberate attacks such as terrorism a significant challenge. Combining game theory and complex network theory provides a framework for analyzing CIS robustness in adversarial scenarios. Most existing studies focus on single-layer networks, while CIS are better modeled as multilayer networks. Research on multilayer network games is limited, lacking methods for constructing incomplete information through link hiding and neglecting the impact of cascading failures. We propose a multilayer network Stackelberg game model with incomplete information considering cascading failures (MSGM-IICF). First, we describe the multilayer network model and define the multilayer node-weighted degree. Then, we present link hiding rules and a cascading failure model. Finally, we construct MSGM-IICF, providing methods for calculating payoff functions from the different perspectives of attackers and defenders. Experiments on synthetic and real-world networks demonstrate that link hiding improves network robustness without considering cascading failures. However, when cascading failures are considered, they become the primary factor determining network robustness. Dynamic capacity allocation enhances network robustness, while changes in dynamic costs make the network more vulnerable. The proposed method provides a new way of analyzing the robustness of diverse CIS, supporting resilient CIS design. Full article
(This article belongs to the Special Issue Robustness and Resilience of Complex Networks)
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Figure 1
<p>Single-layer networks are coupled into multilayer networks and a false network is constructed through active link hiding: (<b>a</b>) two single-layer networks, each with six nodes; (<b>b</b>) the multilayer networks formed by coupling the single-layer networks, representing the actual network (AN); (<b>c</b>) rule-based link hiding in the multilayer networks; (<b>d</b>) the generated multilayer false network (FN). In subfigures (<b>a</b>–<b>c</b>), the blue circles represent nodes of layer 1, and the orange circles represent nodes of layer 2; gray solid lines represent intra-layer links, green solid lines represent inter-layer links, and red dashed lines represent hiding links. In subfigure (<b>d</b>), the nodes in FN are painted gray.</p>
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<p>Construction of the payoff matrix for the attacker and defender in MSGM-IICF. Let <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. (<b>a</b>) Shows the construction of the defender’s payoff matrix and (<b>b</b>) the construction of the attacker’s payoff matrix. In Step 1, the set of nodes for typical defense and attack strategies is identified. The blue and orange nodes represent the defender’s node selection based on the AN, with blue nodes belonging to layer 1 of the multilayer network and orange nodes belonging to layer 2. The dark gray nodes represent the attacker’s node selection based on the FN. In Step 2, nodes in the AN and FN are removed according to the deletion rule, considering the impact of link hiding. Dashed lines indicate nodes where the attack failed due to link hiding. In Step 3, we consider the set of removed nodes in the AN and FN after accounting for cascading failure effects. In Step 4, the set of nodes identified in Step 3 is removed, obtaining the network after the attack.</p>
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<p>The defender’s equilibrium payoff under different <math display="inline"><semantics> <mi>ε</mi> </semantics></math> for various <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> </mrow> </semantics></math> along with the difference in equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.45</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>: (<b>a</b>) the defender’s equilibrium payoff under different <math display="inline"><semantics> <mi>ε</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; (<b>b</b>) the defender’s equilibrium payoff under different <math display="inline"><semantics> <mi>ε</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>; (<b>c</b>) difference in equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.45</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; (<b>d</b>) difference in equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.45</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>. In subfigures (<b>c</b>,<b>d</b>), dark blue indicates a small difference, and dark red indicates a large difference.</p>
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<p>Comparison of the attacker’s expected equilibrium payoff and actual equilibrium payoff for <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math> when <math display="inline"><semantics> <mi>ε</mi> </semantics></math> takes values of 0, 0.15, 0.3, and 0.45. Blue represents the expected equilibrium payoff, while yellow represents the actual equilibrium payoff.</p>
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<p>For <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, the probability of the defender choosing the HDS and the attacker’s equilibrium strategy choice when <math display="inline"><semantics> <mi>ε</mi> </semantics></math> takes values of 0, 0.15, 0.3, and 0.45. The first row shows the probability of choosing the HDS, where lighter colors indicate higher probabilities of choice, while the second row shows the attacker’s equilibrium strategy choice, with light red representing a high-property attack strategy, light blue a low-property attack strategy, and light green #HAS that the defender’s payoff is the same for both the HAS and LAS.</p>
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<p>Relationship between the actual multilayer node-weighted degree in the AN network and the false multilayer node-weighted degree in the FN network. The <span class="html-italic">x</span>-axis represents the multilayer node-weighted degree in the AN and the <span class="html-italic">y</span>-axis represents the multilayer node-weighted degree in the FN. The blue circles represent the change in multilayer node-weighted degree when <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.15</mn> </mrow> </semantics></math>, green diamonds represent the change when <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, red triangles represent the change when <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.45</mn> </mrow> </semantics></math>, and the dashed line is the bisector of the coordinate axes.</p>
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<p>Comparison of the defender’s payoff under various combinations of cascading failures and link hiding factors. Here, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> represents the defender’s equilibrium payoff without considering cascading failures and link hiding, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> represents the defender’s equilibrium payoff without considering cascading failures and with a link hiding coefficient of 0.3, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>ε</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> represents the defender’s equilibrium payoff considering cascading failures with a tolerance coefficient of 1.5 and a link hiding coefficient of 0.3.</p>
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<p>Tendency graph of the defender’s equilibrium payoff changes when <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.5</mn> <mo>,</mo> <mi>ε</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> under cascading failures. Lighter surface colors indicate higher payoff. The red line on the surface represents the change in the defender’s payoff when attack and defense budget resources are equal <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>The defender’s equilibrium payoff under different tolerance coefficients <math display="inline"><semantics> <mi>λ</mi> </semantics></math> with cascading failures and the difference in defense equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.9</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>: (<b>a</b>) the defender’s equilibrium payoff when <math display="inline"><semantics> <mi>λ</mi> </semantics></math> takes values of 1.1, 1.5, and 1.9; (<b>b</b>) the difference in defense equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.9</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, Dark blue indicates a small difference, and dark red indicates a large difference.</p>
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<p>Comparison of the attacker’s expected equilibrium payoff and the actual equilibrium payoff when the cascading failure tolerance coefficient <math display="inline"><semantics> <mi>λ</mi> </semantics></math> takes values of 1.1, 1.5, and 1.9. Blue represents the attacker’s expected equilibrium payoff, while yellow represents the attacker’s actual equilibrium payoff.</p>
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<p>Probability of the defender choosing the HDS and attacker’s equilibrium strategy choice when the cascading failure tolerance coefficient <math display="inline"><semantics> <mi>λ</mi> </semantics></math> takes values of 1.1, 1.5, and 1.9. The first row shows the probability of choosing HDS, with lighter colors indicating higher probabilities of making that choice; the second row shows the attacker’s equilibrium strategy choice, with light red representing the high-property attack strategy, light blue the low-property attack strategy, and light green #HAS indicating that the defender’s payoff is the same for both the HAS and LAS.</p>
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<p>The effect of the cost sensitivity coefficient <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <mi>q</mi> </mrow> </semantics></math> on the defender’s probability of choosing HDS and the attacker’s choice of equilibrium strategy in SSE: (<b>a</b>) probability of choosing HDS under different <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <mi>q</mi> </mrow> </semantics></math> values without considering cascading failures; (<b>b</b>) probability of choosing HDS under different <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <mi>q</mi> </mrow> </semantics></math> values considering cascading failures (in the grids, colors from dark to light represent increasing probabilities of choosing HDS); (<b>c</b>) attacker’s equilibrium strategy choice under different <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <mi>q</mi> </mrow> </semantics></math> values without considering cascading failures; (<b>d</b>) attacker’s equilibrium strategy choice under different <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>/</mo> <mi>q</mi> </mrow> </semantics></math> values considering cascading failures. Light red represents the high-property attack strategy, light blue represents the low-property attack strategy, and light green #HAS indicates that the defender’s payoff is the same for both the HAS and LAS.</p>
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<p>Effect of the load exponent <math display="inline"><semantics> <mi>θ</mi> </semantics></math> on the defender’s probability of choosing the HDS and the attacker’s choice of equilibrium strategy in SSE: (<b>a</b>) probability of choosing the HDS under different <math display="inline"><semantics> <mi>θ</mi> </semantics></math> values without considering link hiding; (<b>b</b>) probability of choosing the HDS under different <math display="inline"><semantics> <mi>θ</mi> </semantics></math> values considering link hiding (in the grids, colors from dark to light represent increasing probabilities of choosing the HDS); (<b>c</b>) attacker’s equilibrium strategy choice under different <math display="inline"><semantics> <mi>θ</mi> </semantics></math> values without considering link hiding; (<b>d</b>) attacker’s equilibrium strategy choice under different <math display="inline"><semantics> <mi>θ</mi> </semantics></math> values considering link hiding. Light red represents the high-property attack strategy choice, light blue represents the low-property attack strategy choice, and light green #HAS indicates cases where both the defender’s payoff and the attacker’s payoff are equal for choosing either the HAS or LAS.</p>
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<p>The defender’s equilibrium payoff under different <math display="inline"><semantics> <mi>ε</mi> </semantics></math> in the US air transportation network: (<b>a</b>) American–United network, (<b>b</b>) American–Delta network; (<b>c</b>) United–Delta network.</p>
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<p>Probability of the defender choosing HDS and the attacker’s equilibrium strategy choice in the American–United network when <math display="inline"><semantics> <mi>ε</mi> </semantics></math> takes values of 0, 0.15, 0.3, and 0.45. The first row shows the probability of choosing HDS, where lighter color indicates a higher probability of that choice; the second row shows the attacker’s equilibrium strategy choice, where light red represents the high-property attack strategy, light blue represents the low-property attack strategy, and light green #HAS indicates that the defender’s payoff is the same for both the HAS and LAS.</p>
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<p>Defender’s equilibrium payoffs under different link hiding methods in the American–United network: (<b>a</b>) represents link hiding in different layers, where MLH-Am+Un denotes simultaneous hiding in both layers, MLH-Am denotes hiding only in the American network layer, MLH-Un denotes hiding only in the United network layer, and MLH_NO denotes no link hiding; (<b>b</b>) represents link hiding methods based on different rules, where MLH denotes rule-based link hiding, MREC denotes random hiding plus random reconnection, and MLH_NO denotes no link hiding.</p>
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<p>Defender’s equilibrium payoff under different tolerance coefficients <math display="inline"><semantics> <mi>λ</mi> </semantics></math> with cascading failures and the difference in defense equilibrium payoff between <math display="inline"><semantics> <mi>λ</mi> </semantics></math> in the American–United network: (<b>a</b>) defender’s equilibrium payoff when <math display="inline"><semantics> <mi>λ</mi> </semantics></math> takes values of 1.1, 1.5, and 1.9; (<b>b</b>) difference in defense equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>; (<b>c</b>) difference in defense equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.9</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>.</p>
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<p>Defender’s equilibrium payoff under different <math display="inline"><semantics> <mi>α</mi> </semantics></math> and difference in defense payoff between <math display="inline"><semantics> <mi>α</mi> </semantics></math> in the American–United network for <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>: (<b>a</b>) defender’s equilibrium payoff when <math display="inline"><semantics> <mi>α</mi> </semantics></math> takes values of 0.1, 0.5, and 0.9; (<b>b</b>) difference in defense equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>; (<b>c</b>) difference in defense equilibrium payoff between <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p>
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<p>Probability of the defender choosing the HDS and the attacker’s equilibrium strategy choice when the cascading failure tolerance coefficient <math display="inline"><semantics> <mi>λ</mi> </semantics></math> takes values of 1.1, 1.5, and 1.9 in the American–United network. The first row shows the probability of choosing the HDS, where lighter colors indicate a higher probability of that choice. The second row shows the attacker’s equilibrium strategy choice, where light red represents the high-property attack strategy, light blue represents the low-property attack strategy, and light green #HAS indicates that the defender’s payoff is the same for both the HAS and LAS.</p>
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<p>Probability of the defender choosing the HDS and the attacker’s equilibrium strategy choice when <math display="inline"><semantics> <mi>α</mi> </semantics></math> takes values of 0.1, 0.5, and 0.9 in the American–United network. The first row shows the probability of choosing the HDS, where lighter colors indicate a higher probability of that choice. The second row shows the attacker’s equilibrium strategy choice, where light red represents the high-property attack strategy, light blue represents the low-property attack strategy, and light green #HAS indicates that the defender’s payoff is the same for both the HAS and LAS.</p>
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<p>Impact of different edge weights w on network robustness in the American–United network, measured using the size of LMCC: (<b>a</b>) with link hiding and (<b>b</b>) with link hiding and cascading failures.</p>
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<p>Defender’s payoff under different cost adjustment factors <math display="inline"><semantics> <mi>μ</mi> </semantics></math>. The time step for the dynamic cost model is 10.</p>
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<p>Strategy choices of attacker and defender under different cost adjustment factors <math display="inline"><semantics> <mi>μ</mi> </semantics></math>. The time step for the dynamic cost model is 10. The first row shows the probability of choosing the HDS, while the second row shows the attacker’s equilibrium strategy choice.</p>
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<p>Relationship between tolerance coefficients and network robustness: (<b>a</b>) impact of <math display="inline"><semantics> <mi>λ</mi> </semantics></math> on network robustness in SCA and (<b>b</b>) impact of <math display="inline"><semantics> <mi>τ</mi> </semantics></math> on network robustness under different <math display="inline"><semantics> <mi>α</mi> </semantics></math> values in DCA.</p>
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16 pages, 491 KiB  
Article
Optimal Community Energy Storage System Operation in a Multi-Power Consumer System: A Stackelberg Game Theory Approach
by Gyeong Ho Lee, Junghyun Lee, Seong Gon Choi and Jangkyum Kim
Energies 2024, 17(22), 5683; https://doi.org/10.3390/en17225683 - 13 Nov 2024
Viewed by 477
Abstract
The proliferation of community energy storage systems (CESSs) necessitates effective energy management to address financial concerns. This paper presents an efficient energy management scheme for heterogeneous power consumers by analyzing various cost factors relevant to the power system. We propose an authority transaction [...] Read more.
The proliferation of community energy storage systems (CESSs) necessitates effective energy management to address financial concerns. This paper presents an efficient energy management scheme for heterogeneous power consumers by analyzing various cost factors relevant to the power system. We propose an authority transaction model based on a multi-leader multi-follower Stackelberg game, demonstrating the existence of a unique Stackelberg equilibrium to determine optimal bidding prices and allocate authority transactions. Our model shows that implementing a CESS can reduce total electricity costs by 16% compared to the conventional case that does not account for authority transactions among CESS users, highlighting its effectiveness in practical power systems. Full article
(This article belongs to the Special Issue State-of-the-Art Machine Learning Tools for Energy Systems)
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<p>Community energy storage system operation in a multi-power consumer system.</p>
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<p>Operating scheme of a CESS in the energy system.</p>
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<p>Power consumption patterns of three representative factory types.</p>
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<p>Bidding price and RUC variation according to the number of buyers.</p>
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<p>Change in sellers’ sales ratio based on number of buyers.</p>
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<p>Analysis of peak power reduction effect through application of proposed model.</p>
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<p>Additional benefits of participants’ according to proposed RUC transaction model.</p>
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23 pages, 1667 KiB  
Article
Assessment of Water Disaster Resilience in Mountainous Urban Metro Stations by Combination Weighting Method and Extension Cloud Model
by Yiyang Wang, Yunyan Li and Rong Wan
Water 2024, 16(22), 3266; https://doi.org/10.3390/w16223266 - 13 Nov 2024
Viewed by 388
Abstract
Studying the resilience of metro stations in mountainous cities to heavy rain and flooding is of significant importance for enhancing the stability and safety of metro station operations. Considering the topographical and climatic characteristics of mountainous urban areas, this study analyzes the mechanisms [...] Read more.
Studying the resilience of metro stations in mountainous cities to heavy rain and flooding is of significant importance for enhancing the stability and safety of metro station operations. Considering the topographical and climatic characteristics of mountainous urban areas, this study analyzes the mechanisms through which heavy rain and flooding affect metro station resilience. Based on this analysis, 27 factors, influencing metro station resilience, are identified across 4 dimensions: absorptive capacity, resistance capacity, recovery capacity, and adaptive capacity. A water disaster resilience evaluation index system and corresponding rating standards are established for metro stations in mountainous cities. By combining the advantages of objective and subjective weighting, the combination weights of evaluation indicators are calculated using game theory. The extension theory is combined with the cloud model to establish a model for assessing the water disaster resilience of metro stations in mountainous urban areas. The applicability and feasibility of the model are validated through its implementation at Shapingba Station within Chongqing Rail Transit. The evaluation results obtained from the established model indicate a resilience level of IV for Shapingba metro station, reflecting a high level of resilience that aligns with real-world conditions. These findings further validate the proposed evaluation standards and the method for assessing the water disaster resilience of metro stations based on the combination weighting method and extension cloud model. This evaluation method considers the uncertainty in the evaluation process, demonstrating good feasibility and reliability. It offers a new perspective and methodology for assessing and analyzing the resilience of similar metro stations in mountainous cities. Full article
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<p>Conceptual graph for subway station flood safety resilience.</p>
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<p>Evaluation index system of subway station flood safety resilience.</p>
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<p>Evaluation process of the new model for evaluation flood safety resilience of subway stations.</p>
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<p>Geographical location of Shapingba subway station. (Modified according to <a href="https://www.cqmetro.cn/yyxlt.html" target="_blank">https://www.cqmetro.cn/yyxlt.html</a>).</p>
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<p>Weight comparison of subway station flood safety resilience evaluation indexes.</p>
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20 pages, 721 KiB  
Article
Information Theory in a Darwinian Evolution Population Dynamics Model
by Eddy Kwessi
Symmetry 2024, 16(11), 1522; https://doi.org/10.3390/sym16111522 - 13 Nov 2024
Viewed by 373
Abstract
Since Darwin, evolutionary population dynamics has captivated scientists and has applications beyond biology, such as in game theory where economists use it to explore evolution in new ways. This approach has renewed interest in dynamic evolutionary systems. In this paper, we propose an [...] Read more.
Since Darwin, evolutionary population dynamics has captivated scientists and has applications beyond biology, such as in game theory where economists use it to explore evolution in new ways. This approach has renewed interest in dynamic evolutionary systems. In this paper, we propose an information-theoretic method to estimate trait parameters in a Darwinian model for species with single or multiple traits. Using Fisher information, we assess estimation errors and demonstrate the method through simulations. Full article
(This article belongs to the Section Mathematics)
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<p>The blue curve represents the Fisher’s information in Equation (<a href="#FD6-symmetry-16-01522" class="html-disp-formula">6</a>) with its maximum value represented by the red dashed line, for <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>=</mo> <mn>7</mn> <mo>;</mo> <msub> <mi>κ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>;</mo> <mi>u</mi> <mo>=</mo> <mn>0.02</mn> <mo>;</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>b</mi> <mn>0</mn> </msub> <msub> <mi>c</mi> <mn>0</mn> </msub> </mfrac> </mstyle> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
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<p>In (<b>a</b>), represented is the time series of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math>. It shows a convergence to <math display="inline"><semantics> <mrow> <msup> <mi>x</mi> <mo>*</mo> </msup> <mo>≈</mo> <mn>20.789</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math> (blue dashed line). In (<b>b</b>), represented is the time series of <math display="inline"><semantics> <msub> <mi>θ</mi> <mi>t</mi> </msub> </semantics></math>, showing a convergence to <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mo>*</mo> <mo>+</mo> </mrow> </msub> <mo>≈</mo> <mn>6.113</mn> </mrow> </semantics></math> (blue dashed line). Figure (<b>c</b>) represents the time series of the Fisher’s information <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </semantics></math>, showing a convergence to <math display="inline"><semantics> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow> <mo>*</mo> <mo>+</mo> </mrow> </msub> <mo>)</mo> </mrow> <mo>≈</mo> <mn>345.43</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math> (red dashed line). Figure (<b>d</b>) is the plot of <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </semantics></math> versus <math display="inline"><semantics> <msub> <mi>θ</mi> <mi>t</mi> </msub> </semantics></math>, showing that once the fixed point <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <mo>*</mo> <mo>+</mo> </mrow> </msub> </semantics></math> is reached, the Fisher’s information is maximized. This is illustrated by the intersection between the blue and red dashed lines.</p>
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<p>This figure shows <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> <mo>:</mo> <mo>=</mo> <mi>x</mi> <mi>G</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> in blue, <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> in black, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> in red for <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>. The green dots represent the intersection between the vertical <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mi>θ</mi> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </semantics></math> and these curves. We observe that <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> is maximized at the same point <span class="html-italic">x</span> where <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> is minimized (green dots) and vice versa (red dots).</p>
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<p>In (<b>a</b>), represented is the time series of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> in the special case above. It shows a convergence to <math display="inline"><semantics> <mrow> <msup> <mi>x</mi> <mo>*</mo> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (blue dashed line). In (<b>b</b>), represented is the time series of <math display="inline"><semantics> <msub> <mi>θ</mi> <mi>t</mi> </msub> </semantics></math>, showing a convergence to <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mo>*</mo> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math> (blue dashed line). Figure (<b>c</b>) represents the time series of the Fisher’s information <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, showing a convergence to <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <msub> <mi>θ</mi> <mo>*</mo> </msub> <mo>)</mo> <mo>≈</mo> <mn>0.125</mn> </mrow> </semantics></math> (red dashed line) as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>. Figure (<b>d</b>) is the plot of <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </semantics></math> versus <math display="inline"><semantics> <msub> <mi>θ</mi> <mi>t</mi> </msub> </semantics></math>, showing that once the fixed point <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mo>*</mo> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math> is reached, the Fisher’s information is maximized. This is illustrated by the intersection between the blue and red dashed lines.</p>
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<p>Values of <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> (dashed line) for two different starting values of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>, each with their 95% confidence bands (colored-shaded areas). In each case, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> converges to <span class="html-italic">e</span> (light dashed line) as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>. When <span class="html-italic">n</span> is relatively small as (<b>a</b>,<b>b</b>), confidence intervals are relative large. When <span class="html-italic">n</span> is large as in (<b>c</b>,<b>d</b>), <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>n</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> becomes smaller and so is the width of the confidence interval.</p>
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<p>Time series of a Darwinian model when <math display="inline"><semantics> <mi>σ</mi> </semantics></math> is large. We observe that there are oscillations making the critical point unstable (blue dashed line).</p>
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<p>In (<b>a</b>), the solid curve represents the dynamics of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> in the system (<a href="#FD14-symmetry-16-01522" class="html-disp-formula">14</a>) above. The dashed line represents the nontrivial equilibrium point <math display="inline"><semantics> <msub> <mi>x</mi> <mo>*</mo> </msub> </semantics></math>. The blue line represent the value of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>*</mo> </msub> </semantics></math> as given in Equation (<a href="#FD21-symmetry-16-01522" class="html-disp-formula">A3</a>), using the values of <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>∗</mo> </mrow> </msub> <mo>≈</mo> <mn>3.873</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>∗</mo> </mrow> </msub> <mo>≈</mo> <mn>1.873</mn> </mrow> </semantics></math> obtained as nontrivial fixed points from the last two equations in (<a href="#FD14-symmetry-16-01522" class="html-disp-formula">14</a>). That the blue line and the dashed coincide is proof of the first part of the Proposition above. In (<b>b</b>,<b>c</b>), the solid curves represent the dynamics of <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>t</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>t</mi> </mrow> </msub> </semantics></math>, respectively. The dashed lines represent the nontrivial fixed points <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>∗</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>∗</mo> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math>. In (<b>d</b>), the blue curve represents the ellipse given in Equation (<a href="#FD17-symmetry-16-01522" class="html-disp-formula">17</a>) above, with center <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mi>ν</mi> <mo>,</mo> <mo>−</mo> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mo>/</mo> <mi>ν</mi> <mo>)</mo> </mrow> </semantics></math>. The red dot represents the nontrivial fixed <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>∗</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>∗</mo> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math>. This point almost lies on the ellipse (the discrepancy is due to an accumulation of error), which is proof of the second part of Proposition 2 above.</p>
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<p>In (<b>a</b>), represented in black are 100 trajectories of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math> with a starting point selected at random from an exponential distribution with parameter <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The red curve represents their average over time converging to <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mo>∗</mo> </msub> <mo>≈</mo> <mn>1.417</mn> </mrow> </semantics></math>. In (<b>b</b>), represented in light-blue are the 95% confidence bands for the corresponding trajectories of <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>t</mi> </mrow> </msub> </semantics></math>. The red curve represents their average and we verify that they all converge to <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>∗</mo> </mrow> </msub> <mo>≈</mo> <mn>3.083</mn> </mrow> </semantics></math>. We note that these confidence bands are constructed using the Fisher’s information as <math display="inline"><semantics> <mrow> <mover> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>¯</mo> </mover> <mo>±</mo> <mn>1.96</mn> <mo>/</mo> <msqrt> <mrow> <msub> <mi>I</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <msub> <mi mathvariant="normal">Θ</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> </mrow> </semantics></math>, where <math display="inline"><semantics> <mover> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>¯</mo> </mover> </semantics></math> is the average at time <span class="html-italic">t</span>. Similarly in (<b>c</b>), represented in light-blue are the 95% confidence bands for <math display="inline"><semantics> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>t</mi> </mrow> </msub> </semantics></math> and the corresponding sample average in red. They converge to <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>∗</mo> </mrow> </msub> <mo>≈</mo> <mn>1.083</mn> </mrow> </semantics></math>. In (<b>d</b>), represented is the ellipse given in Equation (<a href="#FD17-symmetry-16-01522" class="html-disp-formula">17</a>) above. We verify that the point <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>θ</mi> <mrow> <mn>1</mn> <mo>∗</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow> <mn>2</mn> <mo>∗</mo> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math> is on the ellipse and that the value of <math display="inline"><semantics> <msup> <mi>x</mi> <mo>∗</mo> </msup> </semantics></math> obtained from Equation (<a href="#FD21-symmetry-16-01522" class="html-disp-formula">A3</a>) is the same as convergence value of <math display="inline"><semantics> <msub> <mi>x</mi> <mi>t</mi> </msub> </semantics></math>.</p>
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25 pages, 3232 KiB  
Article
A Framework for Distributed Orchestration of Cyber-Physical Systems: An Energy Trading Case Study
by Kostas Siozios
Technologies 2024, 12(11), 229; https://doi.org/10.3390/technologies12110229 - 13 Nov 2024
Viewed by 444
Abstract
The increasing number of active energy consumers, also known as energy prosumers, is dramatically changing the electricity system. New products and services that adopt the concept of dynamic pricing are available to the market, where demand and price forecasting are applied to determine [...] Read more.
The increasing number of active energy consumers, also known as energy prosumers, is dramatically changing the electricity system. New products and services that adopt the concept of dynamic pricing are available to the market, where demand and price forecasting are applied to determine schedule loads and prices. Throughout this manuscript, a novel framework for energy trading among prosumers is introduced. Rather than solving the problem in a centralized manner, the proposed orchestrator relies on a distributed game theory to determine optimal bids. Experimental results validate the efficiency of proposed solution, since it achieves average energy cost reduction of 2×, as compared to the associated cost from the main grid. Additionally, the hardware implementation of the introduced framework onto a low-cost embedded device achieves near real-time operation with comparable performance to state-of-the-art computational intensive solvers. Full article
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Graphical abstract
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<p>Functionality of a cyber–physical system.</p>
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<p>Template of our case study.</p>
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<p>Simulation framework for supporting the proposed MiL and HiL simulations.</p>
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<p>The proposed energy trading framework. Expected loads per energy prosumer (left part of the figure) are calculated based on [<a href="#B42-technologies-12-00229" class="html-bibr">42</a>].</p>
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<p>Candidate bidding aggressiveness schemes.</p>
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<p>Performance of simultaneous and sequential auctions.</p>
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<p>Efficiency of multiple partial auctions.</p>
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<p>Impact of cluster size on the auction’s outcome.</p>
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<p>VES charge during the 52-week experiment.</p>
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<p>Exploration of maximum number of rounds <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </semantics></math> per auction.</p>
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<p>Demonstration setup for the proposed distributed auction framework.</p>
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<p>Efficiency for energy transactions that are performed (i) at run-time, (ii) a week ahead, and (iii) a day ahead.</p>
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<p>Execution run-times for different numbers of simultaneous games <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </semantics></math> per auction <math display="inline"><semantics> <msub> <mi>a</mi> <mi>i</mi> </msub> </semantics></math>.</p>
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9 pages, 221 KiB  
Article
On Remoteness Functions of k-NIM with k + 1 Piles in Normal and in Misère Versions
by Vladimir Gurvich, Vladislav Maximchuk, Georgy Miheenkov and Mariya Naumova
Games 2024, 15(6), 37; https://doi.org/10.3390/g15060037 - 13 Nov 2024
Viewed by 399
Abstract
Given integer n and k such that 0<kn and n piles of stones, two players alternate turns. On each move, a player is allowed to choose any k piles and remove exactly one stone from each. The player who [...] Read more.
Given integer n and k such that 0<kn and n piles of stones, two players alternate turns. On each move, a player is allowed to choose any k piles and remove exactly one stone from each. The player who has to move but cannot is the loser in the normal version of the game and (s)he is the winner in the misère version. Cases k=1 and k=n are trivial. For k=2, the game was solved for n6. For n4, the Sprague–Grundy function was efficiently computed (for both versions). For n=5,6, a polynomial algorithm computing P-positions was obtained for the normal version. Then, for the case k=n1, a very simple explicit rule that determines the Smith remoteness function was found for the normal version of the game: the player who has to move keeps a pile with the minimum even number of stones; if all piles have an odd number of stones, then (s)he keeps a maximum one, while the n1 remaining piles are reduced by one stone each in accordance with the rules of the game. Computations show that the same rule works efficiently for the misère version too. The exceptions are sparse. We list some. Denote a position by x=(x1,,xn). Due to symmetry, we can assume wlog that x1xn. Our computations partition all exceptions into the following three families: x1 is even, x1=1, and odd x13. In all three cases, we suggest formulas covering all found exceptions, but it is not proven that there are no others. Full article
28 pages, 6286 KiB  
Article
An Evolutionary Game Analysis of China’s Power Battery Export Strategies Under Carbon Barriers
by Chunsheng Li, Xuanyu Ji, Kangye Tan, Yumeng Wu and Fang Xu
Systems 2024, 12(11), 482; https://doi.org/10.3390/systems12110482 - 12 Nov 2024
Viewed by 440
Abstract
With the continuous evolution of international trade, the global market has been steadily expanding while also facing increasing challenges, particularly in relation to the introduction of environmental policies such as carbon barriers. Our research explores how China’s power battery manufacturers can adapt their [...] Read more.
With the continuous evolution of international trade, the global market has been steadily expanding while also facing increasing challenges, particularly in relation to the introduction of environmental policies such as carbon barriers. Our research explores how China’s power battery manufacturers can adapt their export strategies to the EU’s carbon barrier policies. Additionally, we examine the roles of government regulations, research institutions, and manufacturers in either facilitating or hindering compliance with carbon reduction objectives. Using evolutionary game theory, we construct models involving government entities, manufacturers, and research institutions to systematically analyze market evolution, strategic interactions, and outcomes among these stakeholders. Our analysis focuses on understanding the competitive dynamics faced by exporting countries under stringent environmental policies and provides strategic insights to guide export strategies. Taking the EU’s carbon barrier policy as a case study, we explore Chinese battery manufacturers’ adaptive strategies and decision-making processes as they respond to shifting market demands and regulatory environments. The findings not only offer valuable insights into exporting countries but also provide policymakers with information on international trade and industrial policy design. Furthermore, we validate our model through numerical simulations and conduct sensitivity analyses on key parameters. The results underscore the importance of governmental adoption of punitive and incentive policies, revealing their substantial impact on stakeholder behavior. Additionally, the study highlights how participants’ pre-cooperation losses and post-cooperation gains influence participation rates and the speed at which stakeholder consensus is reached. By offering a novel approach with which to address carbon barrier challenges, this research contributes valuable perspectives on environmental regulations’ strategic and policy implications in global trade. Full article
(This article belongs to the Special Issue New Trends in Sustainable Operations and Supply Chain Management)
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<p>A diagram showing participation in the tripartite relationship.</p>
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<p>Trends in the dynamic evolution of government.</p>
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<p>Dynamic evolution trends of power battery manufacturers.</p>
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<p>Dynamic evolution trends of research institutions.</p>
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<p>Initial-stage system evolutionary path.</p>
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<p>The system evolution path in the early development stage.</p>
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<p>The evolutionary path of the system in the later stages of development.</p>
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<p>Mature stage system evolution path.</p>
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<p>The impact of government punishment on the evolution of tripartite behavior.</p>
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<p>The influence of government subsidies on the development of tripartite behaviors.</p>
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<p>Plot of evolutionary trends for R4 = 20.</p>
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<p>Plot of evolutionary trends for R4 = 35.</p>
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<p>Plot of evolutionary trends for R4 = 50.</p>
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<p>Plot of evolutionary trends for S3 = 50.</p>
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<p>Plot of evolutionary trends for S3 = 45.</p>
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<p>Plot of evolutionary trends for S3 = 40.</p>
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24 pages, 1115 KiB  
Article
Cooperative Strategies in Transboundary Water Pollution Control: A Differential Game Approach
by Guoping Tu, Chengyue Yu and Feilong Yu
Water 2024, 16(22), 3239; https://doi.org/10.3390/w16223239 - 11 Nov 2024
Viewed by 350
Abstract
This paper, based on differential game theory, examines governance models and cooperative strategies for managing cross-border water pollution in regions with uneven economic development. To address cross-regional water pollution, three differential game models are constructed under different scenarios: the Nash noncooperative mechanism, the [...] Read more.
This paper, based on differential game theory, examines governance models and cooperative strategies for managing cross-border water pollution in regions with uneven economic development. To address cross-regional water pollution, three differential game models are constructed under different scenarios: the Nash noncooperative mechanism, the pollution control cost compensation mechanism, and the collaborative cooperation mechanism. This study analyzes the dynamic changes in pollution emissions, governance investments, and economic returns within each model. The results indicate that the collaborative cooperation mechanism is the most effective, as it significantly reduces pollution emissions, maximizes overall regional benefits, and achieves Pareto optimality. In comparison, the pollution control cost compensation mechanism is suboptimal under certain conditions, while the Nash noncooperative mechanism is the least efficient, resulting in the highest pollution emissions. Furthermore, the research explores the influence of cooperation costs on the selection of governance models. It finds that high cooperation costs reduce local governments’ willingness to engage in collaborative cooperation. However, an appropriate compensation mechanism can effectively encourage less-developed regions to participate. Numerical analysis confirms the dynamic evolution of pollution stocks and economic returns under different models, and provides corresponding policy recommendations. This paper offers theoretical insights and practical guidance for cross-regional water pollution management, highlighting the importance of regional cooperation and cost-sharing in environmental governance. Full article
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<p>(<b>a</b>) Trajectories of total pollutant stock in Regions A and B under three game models. (<b>b</b>) Individual benefit trajectories of Regions A and B under noncooperative and Stackelberg game models (with compensation mechanism). (<b>c</b>) Total benefit trajectories of Regions A and B under three game models.</p>
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<p>(<b>a</b>) Impact trajectories of discount rate <math display="inline"><semantics> <mi mathvariant="normal">ρ</mi> </semantics></math> on total pollutant stock in Regions A and B under three game models. (<b>b</b>) Impact trajectories of discount rate <math display="inline"><semantics> <mi mathvariant="normal">ρ</mi> </semantics></math> on the individual benefits of Regions A and B under noncooperative and Stackelberg game models (with compensation mechanism). (<b>c</b>) Impact of discount rate <math display="inline"><semantics> <mi mathvariant="normal">ρ</mi> </semantics></math> on the total benefits of Regions A and B under three game models.</p>
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<p>(<b>a</b>) Impact trajectory of natural decomposition rate of pollutants <math display="inline"><semantics> <mi mathvariant="normal">ζ</mi> </semantics></math> on total pollutant stock in Regions A and B under three game models. (<b>b</b>) Impact trajectory of natural decomposition rate of pollutants <math display="inline"><semantics> <mi mathvariant="normal">ζ</mi> </semantics></math> on individual benefits of Regions A and B under noncooperative and Stackelberg game models (with compensation mechanism). (<b>c</b>) Impact trajectory of natural decomposition rate of pollutants <math display="inline"><semantics> <mi mathvariant="normal">ζ</mi> </semantics></math> on total benefits of Regions A and B under three game models.</p>
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51 pages, 554 KiB  
Article
Mean-Field-Type Transformers
by Hamidou Tembine, Manzoor Ahmed Khan and Issa Bamia
Mathematics 2024, 12(22), 3506; https://doi.org/10.3390/math12223506 - 9 Nov 2024
Viewed by 363
Abstract
In this article, we present the mathematical foundations of generative machine intelligence and link them with mean-field-type game theory. The key interaction mechanism is self-attention, which exhibits aggregative properties similar to those found in mean-field-type game theory. It is not necessary to have [...] Read more.
In this article, we present the mathematical foundations of generative machine intelligence and link them with mean-field-type game theory. The key interaction mechanism is self-attention, which exhibits aggregative properties similar to those found in mean-field-type game theory. It is not necessary to have an infinite number of neural units to handle mean-field-type terms. For instance, the variance reduction in error within generative machine intelligence is a mean-field-type problem and does not involve an infinite number of decision-makers. Based on this insight, we construct mean-field-type transformers that operate on data that are not necessarily identically distributed and evolve over several layers using mean-field-type transition kernels. We demonstrate that the outcomes of these mean-field-type transformers correspond exactly to the mean-field-type equilibria of a hierarchical mean-field-type game. Due to the non-convexity of the operators’ composition, gradient-based methods alone are insufficient. To distinguish a global minimum from other extrema—such as local minima, local maxima, global maxima, and saddle points—alternative methods that exploit hidden convexities of anti-derivatives of activation functions are required. We also discuss the integration of blockchain technologies into machine intelligence, facilitating an incentive design loop for all contributors and enabling blockchain token economics for each system participant. This feature is especially relevant to ensuring the integrity of factual data, legislative information, medical records, and scientifically published references that should remain immutable after the application of generative machine intelligence. Full article
(This article belongs to the Section Mathematics and Computer Science)
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<p>Attention mechanism at each layer: it involves a pairwise interaction between <inline-formula><mml:math id="mm1027"><mml:semantics><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1028"><mml:semantics><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> via <inline-formula><mml:math id="mm1029"><mml:semantics><mml:mrow><mml:mo>〈</mml:mo><mml:mi>Q</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>K</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>〉</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, where <italic>Q</italic> is the query linear operator, and <italic>K</italic> is the key linear operator.</p>
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<p>The overall architecture of a classical transformer that follows the encoder–decoder paradigm. <italic>H</italic> is the number of heads of the attention. <inline-formula><mml:math id="mm1030"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mi>M</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> is the masked version of the multi-head attention. Enc-Dec denotes encoder–decoder. QoIs stands for quantities of interest. It takes a sequence of <italic>D</italic> tensors and transform them into another sequence of <italic>D</italic> tensors.</p>
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<p>Mean-field-type transformers. PF describes the push-forward operator. There is no <inline-formula><mml:math id="mm1031"><mml:semantics><mml:mrow><mml:mi>D</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> Only one tensor and a distribution. Input: <inline-formula><mml:math id="mm1032"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>; output: <inline-formula><mml:math id="mm1033"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>Y</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>ν</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
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<p>Mean-field-type transformer block.</p>
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19 pages, 3185 KiB  
Article
Immersive Virtual Reality: A Novel Approach to Second Language Vocabulary Acquisition in K-12 Education
by Mohammed Alfadil
Sensors 2024, 24(22), 7185; https://doi.org/10.3390/s24227185 - 9 Nov 2024
Viewed by 423
Abstract
Today, immersive virtual reality (IVR) is increasing in popularity in a broad range of fields, including science, pedagogy, engineering and so forth. Therefore, this study discusses the Unified Theory of Acceptance and Use of Technology (UTAUT), which can be used to examine the [...] Read more.
Today, immersive virtual reality (IVR) is increasing in popularity in a broad range of fields, including science, pedagogy, engineering and so forth. Therefore, this study discusses the Unified Theory of Acceptance and Use of Technology (UTAUT), which can be used to examine the factors that influence the adoption of immersive VR in the classroom, particularly in second language (L2) vocabulary acquisition. The sample for this study included 32 intermediate students and their teacher. For the purpose of evaluation, the researcher used partial least squares structural equation modelling (PLS-SEM) techniques to analyze the results. The findings of the students’ survey showed that performance expectancy, effort expectancy and social influence were seen to have had a positive impact on the intention to use immersive VR. Likewise, the findings indicated that facilitating conditions were seen to have had a positive impact on the use behavior of actually using immersive VR, whereas behavioral intentions did not. In addition, the teacher’s survey demonstrated a favorable view regarding the potential of immersive VR technology to support teaching L2 vocabulary acquisition. This particular study encouraged educators and educational technologists to utilize immersive VR games as a teaching–learning tool to reduce the challenge of L2 vocabulary acquisition. Full article
(This article belongs to the Section State-of-the-Art Sensors Technologies)
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<p>Summary of virtual reality environment types based on level of immersion and features.</p>
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<p>A modified version of the UTAUT model for understanding factors that are affecting adoption of VR.</p>
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<p>Research process.</p>
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<p>Selection and participation of children.</p>
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<p>Procedural steps and timeline.</p>
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<p>Structural model. Arrows toward the yellow box indicate outer loadings, while arrows pointing to the blue circle represent standardize coefficient effect. R2 is shown inside the blue circle.</p>
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<p>Teacher perception of the effectiveness of VR.</p>
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21 pages, 908 KiB  
Article
Tripartite Evolutionary Game and Policy Simulation: Strategic Governance in the Redevelopment of the Urban Village in Guangzhou
by Dinghuan Yuan, Jiaxin Li, Qiuxiang Li and Yang Fu
Land 2024, 13(11), 1867; https://doi.org/10.3390/land13111867 - 8 Nov 2024
Viewed by 322
Abstract
The scarcity of land drives urban village redevelopment projects, which involve interest redistribution among stakeholders with distinct demands. This paper utilizes evolutionary game theory and simulation methods, constructing a tripartite game model under the institutional arrangement of bottom-up with private developer funding. This [...] Read more.
The scarcity of land drives urban village redevelopment projects, which involve interest redistribution among stakeholders with distinct demands. This paper utilizes evolutionary game theory and simulation methods, constructing a tripartite game model under the institutional arrangement of bottom-up with private developer funding. This study identifies the stable strategies and evolutionary trends of the tripartite interactions under four distinct scenarios and validates these strategies through simulations. The redevelopment of XC village validates the assumptions of the model and theoretical analysis, suggesting that when private developers adopt forced demolition strategies, although villagers ultimately choose to sign the contract of property exchange, it can easily lead to social conflicts. These research findings can enlighten the government to form a tripartite alliance to smooth urban village redevelopment. Full article
(This article belongs to the Section Urban Contexts and Urban-Rural Interactions)
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<p>Dynamic evolutionary game diagram of local government, private developers, and villagers. (<b>a</b>) Scenario 1. (<b>b</b>) Scenario 2. (<b>c</b>) Scenario 3. (<b>d</b>) Scenario 4.</p>
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22 pages, 1545 KiB  
Article
Research on Cooperative Water Pollution Governance Based on Tripartite Evolutionary Game in China’s Yangtze River Basin
by Qing Wang and Chunmei Mao
Water 2024, 16(22), 3166; https://doi.org/10.3390/w16223166 - 5 Nov 2024
Viewed by 443
Abstract
Cooperative governance of water pollution is an effective initiative to implement the strategy for the protection of the Yangtze River Basin. Based on the stakeholder theory, this paper constructs a tripartite evolutionary game model of water pollution in the Yangtze River Basin from [...] Read more.
Cooperative governance of water pollution is an effective initiative to implement the strategy for the protection of the Yangtze River Basin. Based on the stakeholder theory, this paper constructs a tripartite evolutionary game model of water pollution in the Yangtze River Basin from the perspective of “cost–benefit”. This paper analyzes the stability of possible equilibrium points of the evolutionary game system by scenarios and further explores the influence of key factors on the evolution of the cooperative governance system of water pollution in the Yangtze River Basin using numerical simulation. According to the findings, (1) the watershed system comprises three key stakeholders: local governments, enterprises, and the public. Each stakeholder’s behavioral strategy choice is influenced by its unique factors and the behavioral strategy choices of the other two stakeholders. (2) Equilibrium points represent the potential strategic equilibrium presented by each stakeholder. When the net income of a particular behavioral strategy within the set exceeds zero, stakeholders will be more inclined to choose that behavioral strategy. (3) The key influencing factors in the evolutionary game are regulatory costs, reputation loss, material rewards, and violation fines. Therefore, this paper proposes to construct a cooperative governance mechanism for water pollution in the Yangtze River Basin from three aspects: an efficient regulatory mechanism, a dynamic reward and punishment mechanism, and a multi-faceted incentive mechanism, with a view to promoting a higher-quality development of the ecological environment in the Yangtze River Basin. Full article
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<p>Research framework.</p>
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<p>Initial setup.</p>
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<p>Effect of changing C<sub>1</sub> on the evolving system.</p>
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<p>Effect of changing N<sub>1</sub> on the evolving system.</p>
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<p>Effect of changing B on the evolving system.</p>
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<p>Effect of changing F on the evolving system.</p>
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<p>Effect of changing J on the evolutionary system.</p>
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15 pages, 2771 KiB  
Article
Vehicle Lane Changing Game Model Based on Improved SVM Algorithm
by Jian Wang, Hongxiang Wang, Mingzhe Fei and Gang Zhou
World Electr. Veh. J. 2024, 15(11), 505; https://doi.org/10.3390/wevj15110505 - 4 Nov 2024
Viewed by 495
Abstract
In order to improve the autonomous lane-changing performance of unmanned vehicles, this paper aims to solve the problem of inaccurate decision classification in traditional support vector machine (SVM) algorithms applied to the lane-changing decision-making stage of intelligent driving vehicles. By using game theory-related [...] Read more.
In order to improve the autonomous lane-changing performance of unmanned vehicles, this paper aims to solve the problem of inaccurate decision classification in traditional support vector machine (SVM) algorithms applied to the lane-changing decision-making stage of intelligent driving vehicles. By using game theory-related theories and combining the improved support vector machine (SSA-SVM) method, a vehicle autonomous lane-changing strategy based on game theory is established. The optimized SVM method has certain advantages for vehicle lane-changing decision-making with a small sample size in actual production processes. The lane-changing decision judgment accuracy rate of the SSA-SVM algorithm model can reach 93.6% compared with the SVM algorithm model without algorithm optimization; the SSA-SVM algorithm model has obvious advantages in decision performance and running speed. Therefore, the proposed new algorithm can effectively solve the problem of the objective consideration of the payoff function in conventional decision game theory. Full article
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<p>Test sections of the NGSIM dataset.</p>
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<p>A simplified diagram of the test section of the NGSIM dataset.</p>
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<p>Plot of the effect of smoothing for each indicator. (<b>a</b>) Vehicle lateral position information; (<b>b</b>) vehicle longitudinal position information; (<b>c</b>) vehicle lateral velocity information; (<b>d</b>) vehicle lateral acceleration information; (<b>e</b>)vehicle longitudinal velocity information; (<b>f</b>) vehicle longitudinal acceleration information.</p>
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<p>Plot of the effect of smoothing for each indicator. (<b>a</b>) Vehicle lateral position information; (<b>b</b>) vehicle longitudinal position information; (<b>c</b>) vehicle lateral velocity information; (<b>d</b>) vehicle lateral acceleration information; (<b>e</b>)vehicle longitudinal velocity information; (<b>f</b>) vehicle longitudinal acceleration information.</p>
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<p>Information on lane-changing vehicles and their surrounding vehicles. (<b>a</b>) Vehicle location information; (<b>b</b>) primary vehicle lane change; (<b>c</b>) vehicle transverse speed; (<b>d</b>) vehicle longitudinal speed; (<b>e</b>) vehicle lateral acceleration; (<b>f</b>) vehicle longitudinal acceleration.</p>
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<p>Non-cooperative game classification and corresponding equilibrium.</p>
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<p>Free lane change scenario.</p>
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<p>Flowchart based on SSA-SVM.</p>
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<p>Comparison of improved support vector machines based on the sparrow search algorithm.</p>
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<p>Comparison of accuracy graphs of each optimization algorithm. (<b>a</b>) GA-SVM optimization algorithm effect; (<b>b</b>) PSO-SVM optimization algorithm effect; (<b>c</b>) SSA-SVM optimization algorithm effect.</p>
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