Dynamic Resilience Quantification of Hydropower Infrastructure in Multihazard Environments

Under hazard exposure, system performance $P(t)$ usually deviates from its initial level $P_0(t)$. Such deviation of the system performance indicates the system performance losses $\rho (t)$, where $\rho (t)$ represents the area bounded by $P(t)$ and $P_0(t)$ during the system’s down period [Fig. 3(a)], starting from $t_0$ (i.e., hazard impact time) until the system bounces back to its target performance level $P_T(t)$ at $t_r$.

Within the system’s down period ($t_0$ to $t_r$), the system performance typically passes through three different periods. These include the deterioration period representing the deterioration in the system performance while exposed to a hazard, starting at $t_0$ and ending at $t_1$. Within the deterioration period, $P(t)$ deteriorates gradually by a deterioration rate (${\theta }_d$) until it reaches its minimum performance level $[P_{\min }(t)]$, indicating the system’s robustness. As the system robustness increases (e.g., through the deployment of risk mitigation plans), the deterioration rate decreases (i.e., ${\theta }_d^1<{\theta }_d^2$), and subsequently $P_{\min }(t)$ increases [i.e., $P_{\min }^1(t)$], as shown in Fig. 3(b). Moreover, preparedness plans can be deployed to increase system performance prior to hazard impact [i.e., $P_0^1(t)>P_0(t)$], so the $P_{\min }(t)$ corresponding to the hazard impact is shifted upward [i.e., $P_{\min }^1(t)$], as shown in Fig. 3(c). As $P_{\min }(t)$ increases, the time needed for the system to return to its target (or original) performance level decreases (i.e., $t_r^1<t_r$) and subsequently the system rapidity improves.

The system reaching its $P_{\min }(t)$ triggers the recovery process to restore system performance within the necessary repair time. However, typically, the system first requires a repair preparation time (i.e., pre-repair period) starting at $t_1$ and ending by $t_2$ until the recovery procedure can be initiated (Almufti and Willford 2013, 2014), as shown in Fig. 3(a). The pre-repair period starts at $t_1$ and ends by $t_2$, including the impeding factors and the utility disruption that delays the start of the recovery process. Impeding factors consider the delay time in initiating the repair and recovery process, which may result, for example, from the requirement of expert inspections, access logistics, and service mobilization. The utility disruption reflects the time required for setting up the tools used in the repair process (e.g., refueling generators, initiation of the backup system). As the pre-repair time decreases (i.e., $t_2^1<t_2$), the delay in the recovery process initiation decreases, and subsequently the system recovers faster (i.e., $t_r^1<t_r$), as shown in Fig. 3(d).

At the end of the pre-repair time, the recovery period starts at $t_2$ with a recovery rate (${\theta }_r$), reflecting the system resourcefulness, until it reaches its target performance level $P_T(t)$ at $t_r$. In the context of hydropower operations, $P_T(t)$ may be higher [i.e., $P_H(t)$] or lower $[P_L(t)]$ than the initial performance level depending on system recovery standards (Simonovic and Arunkumar 2016), as shown in Fig. 3(e). As the recovery rate increases (i.e., ${\theta }_r^2<{\theta }_r^1$) (e.g., resources are added to accelerate the recovery process), the system recovers faster (Bristow 2019), and system rapidity improves (i.e., decreases), as shown in Fig. 3(f). Fig. 3(g) also shows that enhancing the system redundancy can further improve the system rapidity, where adding a backup system within a period of time (i.e., $t_2$ to $t_3$) (e.g., the backup battery can be used in case of dam grid failure) can improve system performance and decrease the recovery period. In the context of hydropower systems, $P(t)$ may propagate nonlinearly through the deterioration and recovery processes due to the nonlinear relationships between the system components. The value of ${\theta }_r$ is usually smaller than ${\theta }_d$, where ${\theta }_d$ may be equal to 90° as the operations may halt essentially immediately when a hazard occurs (e.g., an earthquake may lead to instant tripping or forced outage of the hydropower station).

In multihazard environments, the primary hazard can trigger one or more secondary hazards. Fig. 4 shows a general representation of the system performance under two consecutive hazard events. In Fig. 4(a), the secondary hazard event occurs at time $t_0^2$ after the system fully recovers from the primary hazard impact at $t_r^1$. Each hazard impacts the system with different deterioration rates $({\theta }_d^1,{\theta }_d^2)$ and may reach different minimum system performance levels $[P_{\min }^1(t),P_{\min }^2(t)]$. However, the secondary hazard impact may start at different initial performance levels $[P_0^2(t),P_0^3(t)]$, depending on the target performance level of the primary hazard impact, as shown in Fig. 4(a). As explained previously, the target performance level may be higher or lower than the initial performance level $[P_0^1(t)]$. Accordingly, the $P_{\min }(t)$ corresponding to the secondary hazard impact depends on the system’s initial/normal performance level [e.g., $P_{\min }^3(t)$, $P_{\min }^4(t)$]. However, the recovery rates $({\theta }_r^1,{\theta }_r^2)$ may not change within the two hazard impacts because the recovery rates relate mainly to system resourcefulness and not only to the type or extent of hazard impact. However, the system’s pre-repair time may vary, depending on the hazard type (i.e., $t_1^1$ to $t_2^1$for Hazard 1 and $t_1^2$ to $t_2^2$for Hazard 2). For example, flood events typically cut off roads, increasing the time to access the dam site, subsequently delaying the recovery process. After initiating the system recovery process, $P(t)$ bounces back to its target performance level. The target performance under the secondary hazard can be equal to, or larger or smaller than, its initial performance level according to the system recovery strategies to overcome the losses in system outputs (e.g., hydropower energy) that occurred during system downtime. Fig. 4(b) shows the system performance curve $P(t)$, where the secondary hazard occurs during the deterioration period of the primary hazard. As such, the deterioration rate increases $({\theta }_d^3={\theta }_d^1+{\theta }_d^2)$ during the overlap period starting from $t_0^2$ to $t_1^1$. Subsequently, the system reaches its minimum performance level $(P_{\min }^2)$ at the end of the secondary hazard duration, $t_1^2$. The pre-repair period is defined based on the secondary hazard type between $t_1^2$ and $t_2^2$. The system starts the recovery process from both hazards at $t_2^2$ with the recovery rate $({\theta }_r^2)$ until it reaches its target performance level at $t_r^2$. Fig. 4(c) represents the system performance, where the secondary hazard event impacts the system at $t_0^2$ during the pre-repair period pertaining to the primary hazard (between $t_1^1$ and $t_2^1$). The recovery process is delayed until the secondary hazard duration and its re-repair period ends at $t_2^2$. Subsequently, the system recovers under the secondary recovery rate $({\theta }_r^2)$ until it reaches its target performance level at $t_r^2$. Fig. 4(d) shows the system performance, where the secondary hazard impacts the system with the deterioration rate ${\theta }_d^2$ within the recovery period from the primary hazard (i.e., $t_0^2>t_2^1$). The pre-repair time and the subsequent recovery process period are set based on the secondary hazard event.

Using the SD model, the system performance curve and its parameters $[P_0(t),{\theta }_d,{\theta }_r,P_T(t),P_{\min }(t),t_0,t_1,t_2,t_r]$ can be constructed. Subsequently, the system performance losses $\rho (t)$ can be quantified at each time step ($t$) by integrating the area bounded between $P(t)$ and $P_0(t)$ within the system downtime starting from $t_0$ to $t_r$, as shown in Eq. (2)

The dynamic resilience $r(t)$ can be calculated at each time step by normalizing the system performance losses over time (Simonovic and Peck 2013), as shown in Eq. (3)

In the case of two consecutive hazard impacts, where the secondary hazard impacts the systems prior to the system’s full recovery from the primary hazard’s impact [cases represented in Figs. 4(a–c)], both hazard impacts can be assumed as a composite hazard impact that starts at $t_0^1$ and ends by $t_r^2$. As such, Eq. (2) can calculate the system performance losses where $t_0$ should refer to the primary hazard impact time $(t_0^1)$, whereas $t_r$ should refer to the secondary hazard impact recovery time $(t_r^2)$. Subsequently, using the same approach, Eq. (3) can determine the dynamic system resilience $r(t)$ by normalizing $\rho (t)$ using $P_0(t)$ and $t_0^1$. However, when the secondary hazard impacts the system after the system has fully recovered from the primary hazard impacts, Eq. (2) should consider each hazard impact separately by updating $P_0(t)$, $t_0$, and $t_r$ at the beginning of each hazard impact [i.e., $(P_0^1(t),t_0^1,t_r^1)$, $(P_0^2(t),t_0^2,t_r^2)$]. Subsequently, Eq. (3) can be used to calculate the system resilience by updating $\rho (t)$, $P_0(t)$, and $t_0$after the beginning of each hazard impact [i.e., $(P_0^1(t),{\rho }^1(t),t_0^1),(P_0^2(t),{\rho }^2(t),t_0^2$]. Eqs. (4)–(9) show the $\rho (t)$ and $r(t)$ calculations for consecutive multihazards ($N$).

If ($t_0^1\le t\le t_0^n$)

If ($t_0^n<t$ and $t_0^n\le t_r^{n-1}$)

If ($t_0^n<t$ and $t_0^n>t_r^{n-1}$)

Using these equations [Eq. (3) for single hazard or Eqs. (4)–(9) for multihazard], dynamic resilience curves $r(t)$ can be generated. Fig. 5(b) shows a generic dynamic resilience curve $r(t)$ generated using the system performance curve $P(t)$, shown in Fig. 5(a), according to Eqs. (1)–(9) as explained previously. Dynamic resilience curves aim to show the system resilience value evolution pre-, during, and post-hazard impact. The system resilience starts with 1.0 during the normal operation/pre-hazard realization, and then gradually decreases due to the losses in the system performance (i.e., area $a_i$ larger than 0). Such reduction in system resilience depends on the ratio between area $a_i$, representing the system performance losses $[\rho (t_i)]$, and area $A_i$, representing the total area as if there is no reduction in system performance. As the system performance losses increase, the system resilience decreases, as shown in Fig. 5(b). System resilience continues its deterioration until it reaches its minimum value $r_{\min }$ at $t_r$ calculated using the total performance losses (i.e., area $a_{tr}$) that occurred due to hazard realization. After $t_r$, the resilience value increases gradually with time $t$, when the system normally operates and no more performance losses occur ($a_i/A_i$ decreases with time as $a_i$ does not exceed $a_{tr}$, while $A_i$ increases with time). The value of $r(t)$ continues to increase until the resilience value converges to approximately 1.0 at $t_f$, when the system losses are diminished considering the length of time span of system operations (i.e., $A_{tf}$ is much larger than $a_{tr}$) under normal conditions. As $r_{\min }$ increases (e.g., $r_{\min 1}$, $r_{\min 2}$) or $t_f$ decreases (e.g., $t_{f1}$, $t_{f2}$), this refers to a more resilient system with lower system performance losses occurring due to the hazard impacts. These two metrics ($t_f$ and $r_{\min }$) can be used to assess the efficiency of multiple system operation strategies to mitigate system performance losses due to hazard event impacts.

However, each hazard may have different impacts on the system components (e.g., earthquakes may affect both power and gate releases), and subsequently the overall system resilience $R(t)$ must consider all the individual hazard impacts ($i$). Using the previous approach, system performance curves [i.e., $P_{(i)}(t)$] can be constructed corresponding to each hazard $\mathrm{impact}(i)$ [e.g., $P_{\mathrm{gate}\mathrm{releases}}(t)=\mathrm{actual}\mathrm{gate}\mathrm{releases}/\mathrm{designed}\mathrm{gate}\mathrm{releases}$]. Thus, system performance losses ${\rho }_{(i)}(t)$ and the corresponding dynamic resilience curve per hazard impact $r_{(i)}(t)$ can be evaluated using the same approach presented previously (e.g., $r_{\mathrm{gate}\mathrm{releases}}(t)={\rho }_{\mathrm{gate}\mathrm{releases}}(t)/[P_{0(\mathrm{gate}\mathrm{releases})}(t)·(t-t_0)]$). Accordingly, the overall system resilience $R(t)$ for all-hazard impacts ($z$) can be calculated using the geometric mean as shown in Eq. (10) (Simonovic and Peck 2013). The geometric mean is preferred for $R(t)$ quantification due to its efficiency in dealing with outliers and extreme values more than the arithmetic mean, which is very sensitive to outliers (Clark-Carter 2010). In addition, the geometric mean is usually used for calculating rates (e.g., recovery rates) because it considers the compounding that occurs from period to period