Transmission line switching is an important control action in electrical power systems, and has generated increasing attention in recent years. Opening and closing transmission lines change the topology of the grid. It is a useful tool to redistribute power flows and change the operational state of the system. Changing the topology could potentially save 10%, or even up to 25% of the economic cost [59, 60, 25]. It also provides opportunities to eliminate congestion and avoid violating operational constraints [61]. Line switching is also an important tool in power systems restoration.
Significant research has been devoted to designing algorithms for Optimal Transmission Switching (OTS) [25]. The goal in OTS is to find the best subset of lines to switch off in order to minimize generation costs. This line of research almost exclusively focuses on analyzing power flow under steady-state before and after the switching. From a mathematical standpoint, the OTS problem is a non-convex Mixed-Integer Non-Linear Program (non-convex MINLP), which is computationally challenging. For this reason, most OTS studies replace the non- convex AC power flow equations by the linear DC power flow equations [25, 62, 63, 64, 65]. This reduces the computational complexity, as the DC-OTS problem can be modeled as a Mixed-Integer Linear Program (MILP). The optimal solution, with integer variables fixed to the optimal values, can be used as a starting solution point and fed into an ACOPF solver to convert into an AC feasible solution. Unfortunately, there is no guarantee that the resulting solution can be transformed into an AC-feasible solution [66, 67]. To overcome this limita- tion, recent work has advocated the use of AC formulations (AC-OTS), or focusing on tighter approximations and relaxations [59, 60, 68].
ing. Without loss of generality, we assume the system is a transmission system withSbbase =
100 MVA andVb = 230kV. In this example, we have three buses, three transmission lines, and
only one load at bus 3 drawing 1 p.u. of active power (i.e. 100MW) and 0 p.u. reactive power. The buses are assumed to be operating within the range of [0.9, 1.1] p.u. voltage magnitude. Line (1,2) and line (2,3) both have negligible resistance and 0.05 p.u. reactance (i.e. z23=z12 = 0 + i 0.05). Line (1,3) has a larger resistance and reactance of both 0.10 p.u. (i.e.z13= 0.10 + i 0.10). For simplicity, we neglect line chargelcnmand bus shuntsgns andbns. There are two generators in the system: one cheap distant generator at bus 1 and one expensive generator at bus 3 located directly on the bus with the load. The cheapest solution is to deliver as much power as possible from generator 1 to avoid using generator 3. For simplicity, we assume the cost function of both generator are linear: c(1,p1g) =p1gandc(3,p3g) =10p3g. We first ignore voltage bounds and line thermal/power limiting constraints. Closing line (2,3) will result in cheaper generation costs of $101 comparing to opening line (2,3) with a cost of $110. Costs in both cases are higher than $100 due to line losses. Adding lines to the network decreases the aggregate network resistance, and therefore resulting in a cheaper generation costs to sup- ply the load. We now consider two congestion cases: 1) imposing a tight thermal/power limit ofgSnm=1 p.u. (MVA) on line (2,3), and 2) reducing the voltage range to [0.98, 1.02] p.u..
Table 2.2 reports the OPF solution with various power flow equations [59]. With the origi- nal AC power flow equations, opening line (2,3) reduces the generation costs by more than 80% for case 1, and increases the generation costs for case 2. The results are further matched by two relaxation techniques: SDP relaxation and QC relaxation (with 5 deg phase bounds). Since the linearized DC power flow model are a coarse approximation omitting reactive power and voltages, the model is not capable to handle voltage congestions in case 2 with significant errors.
Table 2.2: OPF solution: Two different congestion settings on four power flow models
Power Flow Con. case 1 Con. case 2 Model Line open Line close Line open Line close
AC-OPF 110 985 655 102
SDP-OPF 110 972 655 102 QC-OPF (5 deg) 110 772 655 102
Figure 2.4: 3-bus power transmission system: Line switching example
Transmission Line Switching Model LetLsbe the subset of transmission lines(n,m)∈
Lwheren<m, andLrbe the subset of transmission lines inLwheren>m. We now show the general line-switching model in Model 4 for switching-offktransmission lines based on a pre- vious formulation [59]. The model introduces extra binary variablesznmfor every transmission line(n,m)∈Lsto indicate whether it should be switched-off (znm=0) or not (znm=1). The objective (O1) and power flow balance constraints (C2.x) are the same as in Model 1. Major modifications are on the AC power flow equations and thermal limit bounds (C3.2 - C3.7). We addznm to these equations to enforce: case 1) no power flow if the line is switched-off (i.e.,znm=0), or case 2) AC power flow within feasible bounds if the line is not going to be switched-off (i.e.,znm=1). We avoid re-defining an extra set of line switching variablesznm for the reverse lines inLr, by linking their power flows to the line switching variables of their counterparts in (C3.5 - C3.7). C3.1 restricts the model to search for solutions switching onlyk lines. We can remove C3.1 if we aim for solutions switching any number of lines.
Model 4AC Optimal Transmission Switching Model Inputs:
P=hN,L,G,Oi Power network input
k Number of lines to be switched off
Variables:
Vn∈[Vn,Vn],∀n∈N Voltage magnitude θn∈(−π,π),∀n∈N Voltage phase angle
png∈[png,pn
g],∀n∈G Active power dispatch
qng∈[qng,qn
g],∀n∈G Reactive power dispatch
pnm∈[pnm,pnm],∀(n,m)∈L Active power flow
qnm∈[qnm,qnm],∀(n,m)∈L Reactive power flow
znm∈ {0, 1},∀(n,m)∈Ls Line switching variable Minimize