. (25 points = 5 50 ) Check convexity of sets and functions.

(a) Check convexity of the set S1 = {(x, t) : t ? x3 , x ? R+, t ? R}.

(b) Let V = Vr + jVi denote a complex voltage. Consider the set S2 = {(Vr, Vi) : V ? |V | ? V , Vr ?

0, Vi ? 0}, check convexity of the set S2 for two cases: i) V = 0.95, V = 1.05; and ii) V = 0,

V = 1.2.

(c) Let vi

denote the squared voltage magnitude at bus i, and `ik

the squared magnitude of the

current flflow from bus i to bus k. Pik and Qik are the active and reactive power flflows from bus i

to bus k. Is the set S3 = (vi , `ik, Pik, Qik) ? R4

: vi`ik = P

2

ik

+ Q2 ik convex?

(d) Is the set S4 = (vi , `ik, Pik, Qik) ? R4

: vi`ik ? P

2

ik

+ Q2 ik convex? Can you rewrite S4 in the

form of a second-order cone?

(e) Let p = [p1, p2, . . . , pn]> ? Rn collect the power outputs of n generators. Is the generation cost

function C(p) = loge(ep1 + + epn ) convex on Rn+?

1. (25 points = 5 x 5') Check convexity of sets and functions. (a) Check convexity of the set 81 :{( :1: ,t). ' t > x3, :1: E R+,t E IR}. (b) Let V: V + jV denote a complex voltage Consider the set 52 {(V r, [J V; 2 0} check convexity of the set 52 for two cases: i) K 035,17 7 = 1.2. (c) Let 'Ug denote the squared voltage magnitude at bus 5', and Egg: the squared magnitude of the current ow from bus 5' to bus k. P215 and Qik are the active and reactive power ows from bus 5' to bus 15.13 the set 53: {(15,355, P,,,, Q55) 5 124 : 15355 Pk + (23,} convex? (d) Is the set 84 = {(v,k,Pk,Qk) E R4 : 115551,, > P33 + Q2215} convex? Can you rewrite 84 in the form of a second-order cone? 1%)\"va =105,