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

for Power

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.