Optimization

The basic objective of any optimization method is to find the values of the system state variables and/or parameters that minimize some cost function of the system. The types of cost functions are system dependent and can vary widely from application to application and are not necessarily strictly measured in terms of dollars. Examples of engineering optimizations can range from minimizing

• the error between a set of measured and calculated data,

• active power losses,

• the weight of a set of components that comprise the system,

• particulate output (emissions),

• system energy, or

• the distance between actual and desired operating points,

to name a few possibilities. The basic formulation of any optimization can be represented as minimizing a defined cost function subject to any physical or operational constraints of the system:

minimize f(x, u) x ∈ Rn (6.1) u ∈ Rm

subject to

g(x, u) = 0 equality constraints (6.2)

h(x, u) = 0 inequality constraints (6.3)

where x is the vector of system states and u is the vector of system parameters. The basic approach is to find the vector of system parameters that, when substituted into the system model, will result in the state vector x that

Electric

In many physical systems, the system operating condition cannot be determined directly by an analytical solution of known equations using a given set of known, dependable quantities. More frequently, the system operating condition is determined by the measurement of system states at different points throughout the system. In many systems, more measurements are made than are necessary to uniquely determine the operating point. This redundancy is often purposely designed into the system to counteract the effect of inaccurate or missing data due to instrument failure. Conversely, not all of the states may be available for measurement. High temperatures, moving parts, or inhospitable conditions may make it difficult, dangerous, or expensive to measure certain system states. In this case, the missing states must be estimated from the rest of the measured information of the system. This process is often known as state estimation and is the process of estimating unknown states from measured quantities. State estimation gives the “best estimate” of the state of the system in spite of uncertain, redundant, and/or conflicting measurements. A good state estimation will smooth out small random errors in measurements, detect and identify large measurement errors, and compensate for missing data. This process strives to minimize the error between the (unknown) true operating state of the system and the measured states. The set of measured quantities can be denoted by the vector z, which

may include measurements of system states (such as voltage and current) or quantities that are functions of system states (such as power flows). Thus

ztrue = Ax (6.4)

where x is the set of system states and A is usually not square. The error vector is the difference between the measured quantities z and the true quantities:

e = z − ztrue = z −Ax (6.5) Typically, the minimum of the square of the error is desired to negate any effects of sign differences between the measured and true values. Thus a state estimator endeavors to find the minimum of the squared error, or a least squares minimization:

minimize ‖e‖2 = eT · e = m∑ i=1

⎡ ⎣zi −

aijxj

⎤ ⎦ 2

(6.6)

The squared error function can be denoted by U(x) and is given by

U(x) = eT · e = (z −Ax)T (z −Ax) (6.7) =

( zT − xTAT ) (z −Ax) (6.8)

be equivalently as

zTAx = ( zTAx

)T = xTAT z

Therefore, the squared error function is given by

U(x) = zT z − 2xTAT z + xTATAx (6.10)

The minimum of the squared error function can be found by an unconstrained optimization where the derivative of the function with respect to the states x is set to zero:

∂U(x)

∂x = 0 = −2AT z + 2ATAx (6.11)

Thus

ATAx = AT z (6.12)

Thus, if b = AT z and Aˆ = ATA, then

Aˆx = b (6.13)

which can be solved by LU factorization. This state vector x is the best estimate (in the squared error) to the system operating condition from which the measurements z were taken. The measurement error is given by

e = zmeas −Ax (6.14)

Example 6.1

A set of measurements for the circuit shown in Figure 6.1 is given by

Ammeter 1 z1 4.27 A Ammeter 2 z2 –1.71 A Voltmeter 1 z3 3.47 V Voltmeter 2 z4 2.50 V

where R1 = R3 = R5 = 1.5Ω and R2 = R4 = 1.0Ω. Find the node voltages V1 and V2.