The power network is a complex system, having three parts performing generation, transmission and distribution of electric power. It is required from the ISO to operate the system by using minimum resources incorporating stability and security constraints.
6.2.1 Optimal Power Flow
Optimal Power Flow (OPF) problem formulation is concerned with the optimal setting of control variables for the steady state performance study of power system with respect to a predefine objective function subjected to various equality and inequality constraints [213]. The main goal of the OPF problem is to minimize the total fuel cost while satisfying all the equality and the inequality constraints.
Mathematically, OPF problem may be represented as:
Minf(i, j) (6.1)
subject to : m(i, j) = 0 n(i, j)≤0
)
Where the above objective function f(i, j) is taken as the total production cost, the equality constraints m(i, j) are taken as the power flow equations and the in- equality constraints n(i, j) are taken as static and dynamic constraints as transmis- sion line loading constraints, generator capacity constraints, security and transient stability constraints.
The vector of dependent variablesi may be represented by 6.2.
iT = [PGslack, VL1, ..., VLN L, QG1, ..., QGN G, SL1, ..., SLN T] (6.2) WherePGslackis the slack bus power,VLis the load bus voltages, QG indicates the reactive power outputs of the generators, SL indicates the transmission line flows, NL is the number of load buses, NG is the number of generator buses and N T L is the number of transmission lines. Similarly, the vector of control variables j may be written by Equation 6.3.
jT = [PG2, .., PGN G, VG1, .., VGN G, T1, .., TN T, QC1, .., QCN C] (6.3) WhereN T is the number of tap changing transformers and N C is the number of shunt VAR compensators, VG is the terminal voltage at the generator buses, PG is the active power output of the generators, Ti is the tap setting of the tap changing transformers and QC is the output of shunt VAR compensator.
6.2.2 Objective Function
The objective function can be described by the concepts such as production cost, social welfare, and fuel cost. In an interconnected power system, the production cost is given by the fuel cost curve approximated as the quadratic function of generated
active power output. In this case, the total production, minimization is taken in to consideration as the objective of TSSCOPF problem and it is expressed mathemat- ically in Equation 6.4 [123].
F(PG) =
NG
X
x=1
axPGx2 +bxPGx+cx
(6.4)
Where F(PG) is the total generating cost, ax, bx and cx are the fuel cost coeffi- cients of the generatorx,PGx is the active power generation of the unitx,NG is the total number of generator buses.
6.2.3 Constraints in OPF Problem with Security and Tran- sient Stability
Security and Transient Stability constrained OPF can be considered as a conven- tional OPF with additional inequality constraints imposed by the transmission line loading limits and the rotor angle limits. The power flow should meets the steady state constraints related to solution of the conventional OPF problem and dynamic constraints imposed on the rotor angle during the transient period under undesirable conditions. The OPF problem has two categories of constraints (viz. the equality constraint and the inequality constraint). These two types of constraints are, se- quentially, described below:
6.2.3.1 Equality Constraints (Power Flow Constraints)
The power flow equations form the equality constraints as shown in Equation 6.5.
PGx −PDx −Vx
NB
P
y=1
Vy(Gxycosδxy +Bxysinδxy) = 0 QGx+QCx −QDx −Vx
NB
P
y=1
Vy(Gxysinδxy −Bxycosδxy) = 0
(6.5)
Where NB is the total number of system buses; PGx and QGx are active and reactive power outputs, respectively of the generator x. PDx and QDx are total active and reactive power loads respectively of bus x. Vx and Vy are the voltage of
the buses x and y respectively. QCx is shunt reactive source at bus x. Gxy, Bxy and δxy, respectively, are the transfer conductance, the susceptance and the phase difference of voltage, between buses x and y.
6.2.3.2 Inequality Constraints (Static and Dynamic Constraints)
• Generator Constraints
Generator voltage, active power outputs and reactive power outputs of bus x should lie between their respective lower and upper limits, as shown in equation 6.6.
VGminx ≤VGx ≤VGmaxx , x= 1,2, . . . , NG PGmin
x ≤PGx ≤PGmax
x , x= 1,2, . . . , NG QminG
x ≤QGx ≤QmaxG
x , x= 1,2, . . . , NG
(6.6)
where VGxmin; VGxmax are the minimum and the maximum generator voltage, respectively, of bus x, PGxmin; PGxmax are the minimum and the maximum active power output, respectively, of busxandQminGx ;QmaxGx are the minimum and the maximum reactive power output, respectively, of bus x.
• Transformer Constraints
Transformer tap settings are bounded by their respective upper and lower limits as shown in Equation 6.7 .
Txmin ≤Tx≤Txmax, x= 1,2, . . . , NT (6.7) where Txmin; Txmax are the minimum and the maximum tap setting limits, respectively, of transformerxandNT is the number of regulating transformers.
• Shunt Compensator Constraints
Reactive power injections at buses are restricted by their respective maximum and minimum limits as shown in Equation 6.8.
QminCx ≤QCx ≤QmaxCx , x= 1,2, . . . , NC (6.8)
where QminCx ; QmaxCx are the minimum and the maximum VAR injection lim- its, respectively, of the shunt compensator x and NC is the number of shunt compensators.
• Security Constraints
These include transmission line loadings and voltages at load buses are repre- sented by the Equations 6.9 and 6.10 respectively.
VLminx ≤VLx ≤VLmaxx , x= 1,2, . . . , NL (6.9) SLx =
q
PL2x +Q2Lx ≤SLmaxx , x= 1,2, . . . , NT L (6.10) where VLxmin ; VLxmax are the minimum and the maximum load voltage, respec- tively, of load busx,SLx;SLxmaxare the apparent power flow and the maximum apparent power flow limit, respectively, through branch x, NL is the number of load buses and NT L is the number of transmission lines.
6.2.4 Transient Stability Assessment and Constraints
TSA of power system is required to find whether system can maintain synchronism during disturbance or not [283]. This decision is taken by monitoring the trajectories of rotor angles during a perturbation period. The swing equation shows the transient behavior of the system. If the trajectories of rotor angle of either single generator or a group of generators are found to have continuous increment without limit with reference to remaining machines, then the system is unstable. Another phenomenon, if rotor angles of all participating system generators remain bounded within their respective permissible limits, then system is stable [33, 102]. The transient stability constraints of TSSCOPF problem constitute a set of differential algebraic equations [33] that may be solved by time-domain technique. The swing equation set of the generator x can be represented by the Equations 6.11, 6.12 and 6.13.
dδx
dt = ∆ωx (6.11)
d∆ωx dt = 1
Mx(Pmx −Pex−Dx∆ωx) (6.12)
Mxd2δx
dt2 = (Pmx−Pex) (6.13)
In the Equation 6.12, Mx is moment of inertia of generator x, Pmx and Pex are mechanical power input and electrical power outputs, respectively of the generator x. Dx and ∆ωx are damping coefficient and rotor speed deviation, respectively of the generator x. δx is rotor angle of generatorx.
The generator rotor angle deviation with respect to the rotor angle of slack gen- eratorδslackis expressed in the form of inequality constraints, as stated in Equations 6.14 and 6.15 [121].
δmin ≤ |∆δx| ≤δmax (6.14) where
∆δx =|δx−δslack| x= 1,2, ...,(NG−1) (6.15) For this work maximum allowable value of relative rotor angle for secure operation δmax is taken as 1200 [271–273].
6.2.5 Formulation of TSSCOPF Problem
The formulation of the TSSCOPF problem is summarized according to the equality and inequality constraints which are defined in sections 6.2.3.1 and 6.2.3.2 respec- tively.
For the equality constraints, the power balance based on Newton-Raphson method is met by the power flow constraints in Equation 6.5, while the swing equation in Equations 6.11 and 6.12 are satisfied by the time-domain simulation based on the Euler method.
For the inequality constraints, the penalty function is adopted to deal with all operating limits as given in Equations 6.6 to 6.10. The penalty function for limit violation of any variable can be defined as follows [284]:
h(λi) =
λi−λimax λimin−λi
0
if if if
λi > λimax λi < λimin λimin ≤λi ≤λimax
(6.16)
Where h(λi) represents the penalty function of variable λi: λimin and λimax are the lower Eand upper limits of the variableλi. λirepresent the variables expressed in Equations 6.6 to 6.10, but the penalty value for Equation 6.14 is kept constant. The penalty functions reflect the violation of the inequality constraints and is assigned a high cost of penalty for a candidate point far away from the feasible region.
In order to enforce all inequality constraints mentioned above, the objective func- tion as shown in Equation 6.4 is added with the penalty functions of active power generation of slack bus, reactive power generation, load bus voltage magnitude, tran- sient stability limit, and transmission line loading. The fitness value is calculated by Equation 6.17.
Fi =F(PG) + P (6.17)
Where P for penalty describes as
P =Kp[h(Pslack)] +KQ NG
X
i=1
h(Qgi) +KV Nl
X
i=1
h(VLi) +KT NG
X
i=1
h(∆δi) +KS Nline
X
i=1
h(Si) Where h(Pslack),h(Qgi),h(VLi),h(∆δi) andh(Si) are penalty functions of active power output of slack bus, the reactive power output of the generator, load bus voltage magnitude, relative rotor angle and transmission line loading respectively, Ks are the corresponding penalty weights. Nl is the total number of load buses.