Enhancing Infrastructure Resilience by Using Dynamically Updated Damage Estimates in Optimal Repair Planning: The Power Grid Case

Repair planning is characterized by the following features:

Each line can be accessed by the two buses it connects; therefore, $4·2$ travel combinations exist between lines $ij$ and $mn$. Hence, for every additional damaged line to set $K$, $4·2·K$ travel combinations are added and thus, the set $K$ has a total of $8·(K·(K-1)/2)$ travel combinations. The crew routing is described by the variable $v_{ij,mn}(k)\in \{0,1\}$, which takes the value 1 if the crew travels from bus $j$ of line $ij$ to bus $m$ of line $mn$, performs the repair of line $mn$ and completes the task at bus $n$ during step $k$. Thus, the time estimate $c_{ij,mn}$ associated with this repair schedule is:where $a_{jm}$ = shortest path travel time between bus $j$ and $m$, and $t_{r_{mn}}$ = repair time estimate of line $mn$. Accordingly, $\mathit{v}(k)$ and $\mathit{c}$ are vectors of the $4·K·(K-1)$ values of $v_{ij,mn}(k)$ and $c_{ij,mn}$.

To complete the repair, all damaged transmission lines must be visited. The following constraint prevents that a line gets visited twice in opposite directions:

Eq. (5) is formulated as an inequality to accommodate for the planning horizon $\stackrel{\sim }{K}\le K$ and is an equality in the OL case. Here, we assume one repair crew which can travel one path at a time. Thus,holds for all permutations of $(ij)$, $(mn)\in K$. Furthermore, a crew can only depart from and arrive to a line, which is formulated by the constraint:

The time difference between the steps $k$ and $k+1$ is represented by $dt(k)$ as illustrated in Fig. 5. To exclude the case that no damaged transmission lines are available for repair because the damage assessment is still ongoing, the waiting time $t_{wr}(k)\ge 0$ is introduced. With the constraint, Eq. (6) which allows only one entry of $\mathit{v}(k)$ to be 1, the time difference $dt(k)$ can be calculated by:

The MPC receding horizon scheme and the quantification of the performance of a repair plan require that the scheduling and completion of repair actions be related to a common reference time. To this aim, the total elapsed time at step $k$ is introduced:

$T_0$ is initially set as the completion of the aerial survey time. In the receding horizon, $T_0$ needs to be set for every new planning iteration to $T(k=1)$ of the previous planning iteration to keep account of the elapsed time. Due to the logistics issue of spare parts, the repair of a transmission line cannot start before its ground survey completion time $t_{f_{mn}}$. This is enforced by the constraint:

To indicate that a line has already been repaired at time step $k$, the indicator variable $r_{ij}(k)\in \{0,1\}$, $r_{ij}(k)\le r_{ij}(k+1)$, is introduced as:and $r_{ij}(k)=r_{ji}(k)$. For the repair planning of the MPC approach, the transmission lines already repaired are removed from the set of the damaged lines $K$, which is updated at each planning iteration. The repaired lines are not included in the remaining planning.

In the power grid, any bus $i$ can host loads and generators, and is characterized by the active power injection $P_i$ and reactive power injection $Q_i$. The active power flow from or into the bus $i,P_i(k)$, is calculated by:where $p_{ij}$ = active power flow on the transmission line connecting buses $i$ and $j$. For the reactive power flow, $Q_i(k)={\sum }_{j\ne i}{q}_{ij}(k)$. If $P_i>0$, then the bus represents a net generator bus. The net power injections by generators and the withdrawal by loads of active and of reactive power at bus $i$ are characterized, respectively, by the lower bounds P_i and Q_i and by upper bounds P_i and Q_i. Further, the electric properties of a bus are described by its bus bar voltage and phase angle. The transmission lines are described by their conductance, susceptance, and thermal line flow limit ${\overline{S}}_{ij}$.

Because the grid is in the recovery state and not at its normal point of operation, the DC power flow approximation does not match the AC solution sufficiently as it neglects reactive power flow and bus bar voltages (Coffrin and Van Hentenryck 2015). Therefore, the AC power flow equations are approximated by a linear programming approach which is known as linear programming AC (LPAC) (Coffrin and Van Hentenryck 2014). This approach approximates the cosine function of the phase angle difference in the power flow equations as LPAC ${\widehat{\cos }}_{ij}$ with the selection of the best linear functions as approximations by solving a linear optimization problem.

To schedule grid operations, two sets of decision variables are considered, i.e., (1) which lines are in operation, and (2) how to shed load and how to set the generator feed-in. The former set accounts for the Braess’s paradox, i.e., a repaired line should be kept out of operation if this leads to a larger power supply. This is implemented by introducing the variable $z_{ij}(k)\in \{0,1\}$ indicating if the line $ij$ is operational with the constraints

The second set of variables constrains the power injections $P_i$ and $Q_i$ to minimize the ILOS and ensure safe operations. Therefore, transmission lines can only transmit power when in operation and must not be overloaded, i.e., the apparent power must stay within its thermal limit ${\overline{S}}_{ij}={\overline{S}}_{ji}$:

The nonlinear constraint in Eq. (15) is replaced by eight linear constraints, forming a piece-wise linear approximation as explained in Coffrin et al. (2012a).

After introducing the objective function, damage assessment, repair planning and power flow, the whole optimization problem is formulated as the following mixed integer problem:

The objective function in Eq. (2) and the constraints presented in the section “MPC Approach for Repair Planning of Power Grids” result in mixed integer nonlinear programming. As the cosine functions of the power flow equations are linearized by the LPAC approximation and the apparent power limitation constrains are replaced by a piece-wise linear approximation as explained in the subsection “Power Grid Model,” only the objective function term minimizing the ILOS remains nonlinear. This allows precomputing the power flows according to every possible repair action over the $K$ steps as linear optimization problems. Eventually, the best sequence can be obtained by assessing the objective function terms which minimize the ILOS.

The poorer the quality of the aerial survey, the more likely the estimated repair time deviates from the real repair time and deteriorates the prediction of the prospective ILOS. In this case, the replanning likely results in an improved repair sequence. If the aerial survey estimates of the damage severity are exact, the MPC approach obtains the same repair sequence as the OL approach. The difference in ILOS is determined, hereby, by the waiting time until the ground survey has been completed and the initial MOP resulting from the damage induced topology. Moreover, the modification of the repair sequence might lead to an increase of the travel time and, thus, accrue the ILOS as compared with the theoretical optimal order achieved by the OL approach. This happens if the travel time is comparable to the repair time, if the damage is widespread or if the number of crews is limited.

Fig. 10 illustrates the ILOS (upper row) and exceedance probability (EP) (lower row) for two different severity levels and quality aerial surveys of two equally damaged topologies. As expected, in both cases the perfect aerial survey yields the best performance. In the case of the high severity damage, the difference between the EPs for a given ILOS value increases, and a poor aerial survey causes a maximum range of ILOS which is 15% larger as compared with the perfect aerial survey.

If the travel time increases by $\mathrm{\Delta }dt(k)$ due to the replanned sequence, this also increases the prospective ILOS by the cost of replanning:

The increase of the number of damaged components increases the variance of the repair time estimates and, thus, the likelihood of suboptimal replanning. This is due to the cumulative effects of uncertainties in the components’ damage levels identified by the imperfect aerial survey. Moreover, the damage level of a component also influences the quality of the repair time estimates based on the aerial survey. Therefore, there may be a misclassification of the damage severity and the associated repair times as detailed in the Appendix under “Damage Assessment.” Fig. 11 shows the distribution of the actual and of the estimated repair times per line resulting from a poor aerial survey for 6,000 transmission lines. Fig. 11(a) shows the histogram for the case of light damage severity and Fig. 11(b) for high damage severity. A low severity with a maximum of 10 components per line with no damage or light component damage leads to a similar performance of the algorithm given perfect or poor aerial survey quality. This does not hold true anymore for a high severity with up to 50 components per line of light and heavy component damage level. For the light severity, the histogram shows less discrepancy between aerial survey and real repair times as for the high severity.

If the quality of an aerial survey is perfect, the MPC approach has access to the correct planning data from the beginning which results in the same repair sequence as the OL approach. By definition, the OL approach starts the repair work not before the last line’s ground survey has been finished, whereas the MPC approach is capable of starting the first repair work by the time the first line’s ground survey has been finished. The time difference of these two different starting times is the waiting gap $T_g$ which is the time advantage of the MPC approach. Given the initial MOP after the damage with $MO{P}_{T0}$, the integral loss of service waiting gap ($ILO{S}_w$) quantifies the advantage upper bound of the MPC approach and is calculated by

Given that the OL approach is provided with the full ground survey information, it is possible to compute the optimal repair sequence. The MPC approach, however, relies on a mix of aerial survey and already completed ground survey data. Hence, only in a best-case scenario the MPC approach finds the same sequence with an ILOS of $ILO{S}_{MPC}=ILO{S}_{OpenLoop}-ILO{S}_w$. This sets the $ILO{S}_w$ as an upper boundary of over-performance against the OL approach.

Replanning based on updated information might improve a suboptimal trajectory during the overlap of ground survey and repair. Because the ILOS is composed of time and LOS, additional travel time results in a higher additional ILOS in the beginning when the MOP is particularly low. Thus, if many repair actions are scheduled during the overlap, it is more likely for them to be rescheduled. Furthermore, the more information and completed ground survey results are obtained within the overlap, the more likely is the case of replanning. Given these considerations, the MPC can especially play its advantage in the case of large-scale damage in topology and severity causing ground surveys of long duration. If the overlap is long, the waiting gap will increase, and thus the benefit of starting early increases. However, after the whole ground survey has been completed, no replanning will happen anymore because information will not change.

Especially the first repair actions are depending on the ground survey scheduling because repair can only start on those lines which damage assessment has been completed. Thus, the ground survey and its scheduling are critical for both the OL and the MPC approach. Whereas for the OL approach only the duration of the whole damage assessment is of interest because it determines the waiting gap; it has a direct impact on the repair sequences being determined in the MPC approach. As explained in the subsection “Waiting Gap and Replanning,” the durations of repair actions in the beginning have a higher impact on the resulting ILOS than the repair actions in the end when the MOP is already close to 100%. If the quality of the aerial survey is increased, the MPC approach will be less prone to suboptimal planning due to deviating information. On the other hand, if the duration of the ground survey gets shortened, the ILOS of OL approach and the MPC approach will converge because the advantage of starting early vanishes, and information will be fully available after the short ground survey. This last point leads to the future perspective of the damage assessment process. As recently reported (Pasztor 2017), drone inspection will be used in the future for the damage assessment as it has been successfully tested. Depending on the number of drones available and due to their speed compared with ground survey crews, this might blur the line between aerial and ground survey which might be combined eventually. Due to a smaller distance to the damaged components, the damage classification is likely to be more precise. This will allow to decrease the waiting gap to a minimum and improve the repair planning data quality.