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Edit | Sugo
Edit | Sugo
●—≺ Optimization model of bridge network resilience≻—●
Once a failed bridge has been identified, it needs to be repaired. Since the restoration resources that can be invested in the short term after the earthquake are limited, the repair efficiency of the system is generally improved by adjusting the repair sequence, and the resilience of the system is improved.
In the figure below, S1 and S2 are the network performance curves of Solution 1 and Solution 2, respectively. In the face of the same seismic action, different restoration solutions will obtain different resilience indexes. Clearly, with limited resources, restoration options have a direct impact on bridge network resilience.
System performance curves for two different repair scenarios
This problem can be translated into determining the order of bridge repair under D2 conditions under resource-limited conditions, and ultimately maximizing the resilience index R. The number of bridges is [B1, B2,...,BD].
The optimization model is:
maxxR(x|e)=∫tr(x)teQ(x,t|e)dtQ(t0)(tr(x)−te)maxxR(x|e)=∫tetr(x)Q(x,t|e)dtQ(t0)(tr(x)-te) (5)
x∈X (6)
where x is a repair scheme, B=(⋯,bi,⋯)B=(⋯,bi,⋯), bi is the bridge number ordered i, tr(x) is the moment when the bridge network returns to normal when scheme x is adopted, Q(x,t|e) is the average efficiency of the bridge network at the time t when the earthquake e occurs using option x, and X is the set of alternatives.
Equation (5) is the objective function, indicating that the toughness index of the bridge network is the largest when an earthquake E occurs, and Equation (6) is the constraint.
When the number of failed bridges in the network is small and there are fewer alternatives, the exhaustive method can be used to compare the various options to arrive at the optimal solution, but as the number of failed bridges increases, the number of repair options increases exponentially.
For example, if 8 bridges fail in a bridge network, the number of alternatives is 8!=40320 to determine the order of repair of these 8 bridges. Faced with this huge number of calculations, the author uses a genetic algorithm with powerful computing power to solve it.
Bridge network layout
●—≺ Case Analysis ≻—●
The bridge network is shown in Figure 3, consisting of 13 nodes and 19 bridges, bridge types include multi-span continuous concrete girder bridge (MSCcon), multi-span simple support concrete girder bridge (MSSScon), multi-span continuous steel girder bridge (MSCsteel), bridge network information is shown in the literature, the vulnerable curves of the four types of bridges are shown in the literature, and the use of MATLAB programming.
Under normal operating conditions, the average efficiency of the network is 0.2490.
The figure below shows the vulnerability curves of four types of bridges under moderate damage conditions.
Figure 4: Vulnerability curves of four bridges under medium damage conditions
Simulate a scenario
Assuming that bridges 1, 7, and 9 in the above network fail, and only one repair team participates in the rescue, you need to select the appropriate repair plan, and there are six repair solutions, as shown in the following table.
Bridge network resilience index under different repair schemes
| Repair scenarios | Repair order | Resilience index |
| S1 | 9-7-1 | 0.48144 |
| S2 | 9-1-7 | 0.48308 |
| S3 | 7-9-1 | 0.48261 |
| S4 | 7-1-9 | 0.48582 |
| S5 | 1-9-7 | 0.48414 |
| S6 | 1-7-9 | 0.48571 |
Resilience assessment
Assuming that the repair time required for each failed bridge is the same, and the repair time is 10 hours, the average efficiency of different repair schemes is calculated by formula (1), the performance curve of the bridge network is drawn, and the toughness index of the bridge network is calculated by the integral of the segmentation function by formula (4), and the calculation results are shown in Table 2.
Average efficiency performance curves for networks with different remediation scenarios
The importance of bridges 1, 7 and 9 is 0.2411, 0.2432 and 0.2451, respectively, and the average efficiency performance curve of the network with different repair schemes after failure is shown in Figure 5, and the toughness index of scheme S4:7-1-9 is the largest and the best repair scheme is the optimal repair scheme, which proves that the bridge repair scheme in order of bridge importance makes the bridge network the most resilient.
Most bridge simulation scenarios
Assuming that a 6.5 magnitude earthquake occurs at time t, and the epicenter is located 20 km away from the bridge network, the median PGA improvement coefficient corresponding to slight, moderate, severe and collapse of the bridge is shown in the literature. Suppose that only one team participates in rescue and emergency repairs after the earthquake, and only one bridge can be repaired at a time.
According to Equation (1), the damage class Li of the bridge in the network is calculated, and the calculation result is shown in the following table.
Statistics on the damage level of bridges after the earthquake
| Degree of injury | Number of bridges | Bridge number |
| 3<Li≤4 | 2 | 3、12 |
| 2<Li≤3 | 6 | 1、6、8、9、10 |
| 0≤Li≤2 | 11 | 2、4、5、7、11、12、13、14、15、16、18、19 |
According to Equation (3), the average efficiency importance of the bridge with damage class Li2 is calculated, and the results are shown in the table below.
Importance of damaged bridges
| Bridge number | Importance |
| 1 | 0.2411 |
| 3 | 0.2430 |
| 6 | 0.2429 |
| 8 | 0.2437 |
| 9 | 0.2451 |
| 10 | 0.2432 |
| 13 | 0.2362 |
Resilience assessment
The order of repair according to the importance of the bridge is 9-8-10-3-6-1-13, and the corresponding toughness index is 0.8444, and the performance curve of the bridge network under this repair order is shown in the figure below.
Average network efficiency curve under different remediation strategies
The order of random repair is 1-3-6-8-9-10-13, and the corresponding toughness index is 0.8344, and the resilience curve of the bridge network under the repair sequence. The order in which the optimal repair scheme was adopted was 8-13-3-10-1-6-9, and the corresponding toughness index was 0.8741, as shown in the table below.
The impact of remediation strategies on bridge network resilience
| serial number | Remediation policy | Bridge repair sequence | Resilience index |
| 1 | Random fixes | 1Wingdings3b@@3Wingdings3b@@6Wingdings3b@@8Wingdings3b@@9Wingdings3b@@10Wingdings3b@@13 | 0.8344 |
| 2 | Based on bridges of importance Prioritize repairs | 9Wingdings3b@@8Wingdings3b@@10Wingdings3b@@3Wingdings3b@@6Wingdings3b@@1Wingdings3b@@13 | 0.8444 |
| 3 | Optimal remediation | 8Wingdings3b@@13Wingdings3b@@3Wingdings3b@@10Wingdings3b@@1Wingdings3b@@6Wingdings3b@@9 | 0.8741 |
The results show that the network resilience obtained by the optimal repair scheme is 4.76% and 3.51% higher than that based on bridge importance priority repair strategy and random repair strategy, respectively.
The author combines resilience assessment with repair plan decision-making, proposes to take the average efficiency of the bridge network as the performance index of the bridge network, constructs a bridge network resilience assessment model based on average efficiency, discusses the impact of different repair schemes on the average efficiency of the network after the failure of some bridges in the post-earthquake bridge network, and adopts the scheme with the largest resilience index as the optimal scheme, so that the bridge network can quickly restore to normal operation to mitigate the impact of earthquakes on network interruption. The analysis led to the following conclusions:
(1) Through the resilience assessment of simulation scenario 1, in the case of a small number of bridge failures, all possible schemes are listed by exhaustive method, and the repair schemes sorted by bridge importance make the bridge network resilience index the largest;
(2) Through the toughness assessment of simulation scenario 2, the toughness index of the optimal repair scheme is 4.76% and 3.51% higher than that of the priority repair plan and the random repair scheme in the case of most bridge failures, respectively, and the optimal repair scheme makes the bridge network toughness reach the optimal.
(3) The resilience assessment model based on average efficiency proposed by the author is suitable for the resilience assessment of bridge networks in areas with many bridges, which effectively solves the problem of bridge network repair decision-making when multiple bridges fail, and improves the post-earthquake resilience of bridge networks. The powerful computing power and optimization capabilities of genetic algorithms make the evaluation process more convenient and the evaluation results more accurate. In the future, it will consider adjusting the topology of the bridge network, optimizing the node distribution, and integrating strategies such as increasing the bridge traffic capacity into the model for decision-makers to choose.