CHAPTER 2 GENERATION REDISPATCH DURING CYBER ATTACKS
2.4 Resilience-oriented OPF
We present an automated procedure to design RAS for a given power system topology and its state vector. The generated RAS logic attempts to keep the power system safe from all potential up- coming contingencies. First, contingency analysis is performed to identify the list of incidents that drive the power system to the emergency state. For each credible contingency, a remedial action is calculated and developed that brings the system back to its normal safe state. In this section, a generation redispatch technique is developed through reformulating the OPF to maximize the system security.
First, we give a quick review on optimal power flow (OPF). OPF minimizes the operation cost subject to the power flow and other constraints:
min f (x, u) = X
i∈UG
Ci(Pi)
h(x, u) ≤ 0 (2.1)
where u is the control variable and x is the state variable, which includes the voltage phasors (magnitudes and phase angles) on individual power buses. The voltage magnitude of the PV buses (generators) and the voltage magnitude and angle of the slack bus are excluded since their values are known. The control variables are the generator MW output set-points, settings of the flexible alternating current transmission system (FACTS) devices, phase shifting transformers, and static VAR compensators. For simplicity and without loss of generality, only the generator MW output may be considered as control variable and the objective function may be written as:
f (x, u) = X
i∈UG
Ci(Pi) (2.2)
where Ci(Pi) is the cost of operating generator i with the MW output of Pi, and UG is the set of
generators.
The equality constraint g(x, u) corresponds to the power flow equations and ensures that the active and reactive power at the PQ buses (loads) and the active power at the PV buses (generators) match their given values. The inequality constraint h(x, u) may include the line flow limits, the voltage magnitude limits and the generators output limit as given by:
Vimin ≤ Vi ≤ Vimax i ∈ UP Q
Pimin ≤ Pi ≤ Pimax i ∈ UP V
Qmini ≤ Qi ≤ Qmaxi i ∈ UP V
Pi,j ≤ Pi,jmax (i, j) ∈ I (2.3)
where Vi, Pi and Qi are respectively the voltage magnitude, the active power and the reactive
power at bus i; and UP Q and UP V are the set of PQ and PV buses, respectively. Pi,j is the active
power on the line between buses i and j, Pmax
i,j is the flow limit of this line, and I is the set of all
(i, j) for which there is a line connecting bus i to bus j. Note that the generator output limit is a physical constraint and cannot be violated at any time. On the other hand, the voltage limit and line flow limits are operating constraints that relate to system reliability, and these may be formulated
as soft constraints.
2.4.1
Security Constrained OPF
The first step in SCOPF is to determine the list of contingencies to be considered. The list includes contingencies which are likely to occur and violate at least one of the network constraints. In- significant or infrequent contingencies are ignored. SCOPF determines the optimal control which minimizes the objective function for the base case and satisfies the power flow feasibility and the network constraints for the base case and each contingency case as expressed in:
min f (x(0), u) s.t g (j)(x(j), u) = 0 h(j)(x(j), u) ≤ 0 j = 0, 1, · · · , K (2.4)
where x(j), g(j), and h(j)represent the state, the power flow feasibility, and the network constraints for the contingency case j, respectively. K is the size of the contingency list. The pre-contingency or base case is denoted by j = 0 as expressed in
j = 0 base case 1 ≤ k ≤ K contingency case k (2.5)
SCOPF ensures that the system remains in the normal state and does not transition to the emer- gency state when a contingency occurs. It increases the system security and eliminates the need for remedial action schemes. However, this is achieved with the expense of less economical operation. The additional constraints for the contingency cases reduce the size of the feasible solution space. Therefore, the solution from SCOPF has a higher cost than the one from OPF as illustrated in Fig. 2.3.
Figure 2.3: Comparison of the SCOPF and OPF solutions.
2.4.2
Proposed Resilience-oriented OPF
The optimal power flow is reformulated in the context of security control [76]. This new formula- tion is termed as the resilience-oriented OPF (ROPF) and recovers the system from the emergency state to the normal state after a contingency. Conventional OPF minimizes the operation cost subject to the power flow and other constraints. When a contingency occurs, retaining system op- eration becomes the first priority rather than the operation cost. Hence, the objective function of ROPF optimizes the security instead of cost.
Similar to the conventional OPF, the equality constraints of ROPF correspond to the power flow equations and the inequality constraints relate to the generator output limits, voltage con- straints, and line flow limits. The voltage constraints and line flow limits may be formulated as soft constraints since they are operating constraints, and unlike the physical constraints, they can be violated. These constraints are modeled by:
Vi ≤ Vimax+ ti i ∈ UP Q
− Vi ≤ −Vimin+ ri i ∈ UP Q
Pi,j ≤ Pi,jmax+ sij (i, j) ∈ I
0 ≤ ti, ri, sij (2.6)
where ri and ti are respectively the slack variables for the voltage upper and lower limits at bus i
the soft constraints and are penalized in the objective function. The objective function enforces the voltage constraints and the line flow limits as expressed in:
f (x) = X i∈UP Q V ¯V (2ti+ t2i) + X i∈UP Q VV(2ri+ ri2) + X (i,k)∈I VI(2sik+ s2ik) (2.7)
where V ¯V , VV and VI are the weighting parameters chosen with respect to the desired importance of each term.
Solving the proposed ROPF is computationally expensive and might not be practical for a larger system. In the case of a credible contingency, it is crucial to take an effective control action as quickly as possible. Hence, finding a solution that is less accurate, but faster to compute is preferable. Motivated by this, the optimization problem is simplified to reduce the computational complexity. First, the equality constraints associated with the power flow equations are linearized and similar to DC-OPF, DC-ROPF is defined [77]. Second, the inequality constraints associated with the voltage limit are relaxed. The objective function is modified accordingly to exclude the terms associated with the voltage limits:
f (x) = X (i,j)∈I (2sij + s2ij) − X i∈N ui (2.8a) s.t : Pimin ≤ Pi ≤ Pimax i ∈ UP V (2.8b) Qmini ≤ Qi ≤ Qmaxi i ∈ UP V (2.8c) Bij(θi− θj) = Pij (i, j) ∈ I (2.8d) X (i,j)∈I Pij− X (j,i)∈I Pji+ Pi− Di+ ui = 0 i ∈ N (2.8e) 0 ≤ ui ≤ Di i ∈ N (2.8f) Pij ≤ Pijmax+ sij (i, j) ∈ I (2.8g) 0 ≤ sij (2.8h)
Figure 2.4: Security-compliant generator dispatch subspace synthesis.
partial demand fulfillment at each node. The imbalance between generation and load is allowed by introducing the bounded unmet demand variable (2.8f) at each node and penalizing it in the objective function. This form of ROPF with the voltage constrains excluded is termed as relaxed ROPF throughout the chapter. Numerical results indicate that solving the relaxed ROPF is much faster, yet the solution is as effective as the regular ROPF as will be shown later.