Question

In power system operations, to simplify real-time computations, approximations are made One example is DC state estimation (SE). In DC SE, an m x 1 measurement vector z can be written as follows: z Hxe where H is the m xn system Jacobian matrix; x is the n x 1 state vector; e is an m x 1 vector of errors in measurements, assumed to be independent of x and H, and the ith entry of e is Gaussian distributed with zero mean and variance a (e~N(0, a7)) Let be the estimate of x. The residual (state estimation error) vector can be written as r =z- H (a) What is the equation to compute from z using MMSE (minimize mean-square error) method? (Hint: minimizelr|2) (b) A bad data detector (BDD) is used to detect bad data in DC SE. The BDD compares the l2-norm ofr with a threshold T. If ||r|| = ||z - H3| < T, no bad data exists, and we say can bypass the BDD Suppose a measurement vector z can bypass the BDD. Now assume a malicious attacker who is able to change the measurements from z to z + a. What is the condition of a so that the attack is unobservable to the BDD (i.e., z + a can also bypass the BDD)?

Answer 1

(b) Br z-H 1 Maliaous atteacken is able to change z to (ス+a), be caue o change of Hea surement thee coill be change. c ehe stati d +C. feam to Madihed ll z+a-H(+é) . Noco ib 1rMalibed l ril,then a-He => |ス+a -H-He 11. z -HA thus vnobsevable to BDD.

z H+e z-HA n beom ス Comput VSmg HMSE. r= スーH We knoco that USng HMSE, Mininisma n ri r(2-H(2-H) 1lri = = 7z-TH- z- H -HTz-4T2 - a > -zai' ...The equotien 1Sed to calulat xbem z is 2. -(2