The operation limitation (Constraints)
arise from the fact that the generating units, transmission lines,
transformers, phase shifts can not be loaded beyond the capacity. The
constraints are equality and inequality constraints. In the electric utility
the OPF is applied in the area of the system operations and planning. For
operations, OPF is used for real time and study applications in the area of
planning.
OPF is used for capacitor placement studies and transmission capacity
panning. The OPF should be able to optimize the power system in the normal
state produce remedial action an any set of the selected post-contingency
states and provide preventive scheduling for base-cases and post-contingency
limit violations in a defined set of contingency cases. In the OPF, the
controls are varied such that constraints are achieved and the objective
function is either minimized or maximized. The OPF problem can be formulated
as:
Minimize F(u,x) with respect to u
Subjective
to: g(u,x)=0 , h(u,x)<0
Where u is a
vector of controllable variables and x is a vector of dependent or stable
variables.
In this OPF formulation, power flow
equations are the equality constraints and the limits on designated quantities
are inequality constraints.
1-Objective Functions
The objective functions commonly used for operations and planning are as follows:
1.
Minimum const of operation.
2.
Minimum increase in cost.
3.
Minimum deviation or minimum control shift.
4.
Real power loss minimization.
5.
Reactive power loss minimization.
6.
Minimize the number of controls.
7.
Minimize the cost of installation of new capacitors and
reactors.
8.
Minimize the cost of MVAR Supplied.
9.
Maximize MW transfers.
10.
Minimize the time to correct the violations.
11.
Minimize the total emission.
2-Constraints
The equality constraints are ones that must be exactly satisfied to have a feasible solution. In most of the optimization problems, the bulk of the effort and complexity are due to the inequality constraints. The inequality constraints must satisfy either an upper or lower bound on the value of the function to have a feasible solution.
At the
optimum feasible point, all inequality constraints will be in the following two
states. In state one, the inequality constraints will be fixed at the upper or
lower limit and they are referred to as the active of binding inequality
constraints. In second state, the constraints will be within the limits and
they are referred to as inactive or not binding constraints is the most
difficult part of solving the optimization problem. No direct method without
any iterative process to find the active set is published.
Instead, various iterative methods are
employed to gradually identify the correct
active set over the course of optimization.
During the optimization, the following
constraints are enforced:
1-Control limits.
2-Ampere flow
limits on circuits.
3-Bus high and low
voltage magnitude limits.
4-Generator MVAR outputs.
5-Active power
import/export limits.
6-MW and MVAR
reserve limits.
7-Reactive power
import/export limits.
8-Sum of flows in
any group of lines.
9-Control action
response time.
10-Bus voltage angle
differences.
11-Post contingency
limits.
12-Environmental constraints.
3-Classification Of System Variables
1-Fixed:
Such
as parameter of the transmission lines ( R , XL , Xc )
2-State (X) :
Such as bus angles at all buses
bus voltage at load buses
Where:
n = Total no. of buses.
ng = No. of generation buses.
3-Control (U) :
4-Controls (Control Variables)
The elimination of constraints violations and achieving the objective can be accomplished through the use of controls. The operator may use the controls freely within their limits to achieve operational goals. A list of controls commonly used is shown below:
1-Generator MW output.
2-Voltage-ratio transformers taps.
3-Phase shifter taps.
4-MW interchange.
5-HVDC link power.
6-Load shedding.
7-Generator voltage set points.
8-Shunt reactor and capacitor.
9-Static VAR compensatory.
10-Synchronous
condenser.
5-OPF Applications
The OPF can be used for many applications in electric utilities. Some of the major applications of OPF are listed below and followed by a brief description. For further detailed information, readers are encouraged to read the reference listed:
1-Security constrained economic dispatch.
2-MW or MVAR loss minimization.
3-Constraints priority optimization.
4-Hierarchical control.
5-Remedial actions or corrective
rescheduling.
6-Preventive scheduling or contingency
constrained dispactch.
7-Maximum transfers capacity.
8-Closed loop control.
9-Capacitor and reactor placement.
10-Switching action.
11-MVAR cost optimization.
12-Voltage control.
13-Sensitivity
analysis.
14-Transaction
evaluation.
15-Dispatcher
training simulator scenario building.
16-Shadow price
evaluation.
17-Wheeling loss
computation.
18-Nodal price for
power.
19-Environmental
dispatch.
20-Constraint
costing.
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