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1 power systems eehlaboratory

5 c b c s D e f h l p q r T u u x

9 N { 1,..., N } t q (t) t e (t) q (t) = 1 t (e (t) e (t 1)), t q mn e mn q (t) q max e (t) e max.

10 c b (t) c s (t) p (t) := l (t) r (t) + q (t), l (t) r (t) t { c s (t)p (t) p (t) 0 g (p (t), t) := c b (t)p (t) p (t) < 0. t x (t) := (q (t), e (t)). t x(t) := (x 1 (t),..., x N (t)). t 1 t f(x(t), x(t 1)) := (p(t) p(t 1)) 2, p(t) := N p (t) =1 π Π [ ] π 1 T N T = argmn lm E g (p (t), t) + γ f(x(t), x(t 1)). π Π T T t=0 =1 t=0

11 { 1,..., N } t q (t) = e (t) e (t 1), t q mn q (t) q max, e mn e (t) e max. π t q(t) = π(x(t 1),... x(0), l(t),..., l(0), r(t),..., r(0)), l r

12 { } t t,..., t + T t x h(x) A x = b, = 1,..., N, x mn x x max, = 1,..., N, x x x := (x (t),..., x (t + T )) h f γ ḡ

13 h(x) = N t+t t+t g (p (τ), τ) + γ f(x(τ), x(τ 1)). =1 τ=t } {{ } ḡ (x ) τ=t } {{ } f(x) C := { x A x = b, x mn x x max } { }, 1,..., N. x (k) = x (k) α k ξ (k) ( ) x (k+1) = Π C x (k) ξ (k) δh(x (k) ) h x (k) Π C ( ) 1 Π C (y) := argmn z C 2 z y 2 2, 2

14 α k α k =, lm α k = 0. k k x (k) = x (k) α k f(x (k) ) x (k+1) = prox αk ḡ ( x (k) ) prox αk ḡ ( prox αk ḡ (y) = argmn α k ḡ (z) + 1 ) z C 2 z y 2 2. α k α k =, αk 2 <. k ḡ ( ( )) x (k+1) = Π C x (k) α k x f(x (k) ) + x ḡ (x (k) ). x (k+1) k ( (k) := ηk ḡ x η k x f(x (k) ) ). }{{} y ξ ḡ x (k)

15 ḡ ( ) 1 prox ηk ḡ (y) = argmn z C 2 z y η kξ z ( 1 = argmn z C 2 z z y z + 1 ) 2 y y + η k ξ z ( 1 = argmn z C 2 z z (y η k ξ ) z + 1 ) 2 y y ( ) = argmn z (y η k ξ ) 2 2 z C η k = α k k = Π C (y η k ξ ) ( ) = Π C x (k) η k x f(x (k) ) ηξ = Π C ( x (k) η k ( x f(x (k) ) + ξ )), x f(x (k) )

16 Fgure 3.1: Coordnaton between the dfferent consumers.

17 D R NT l := (l (t),..., l (t + T )) { 1,..., N } D = {l } N =1. ϵ δ D D { 1,..., N } l l 1 δ l j = l j j δ M ϵ S (D, D ) Pr[M(D ) S] exp(ϵ) Pr[M(D) S]. T 1 : D T 1 (D) ϵ k 2 T k : (D, s 1,..., s k 1 ) T k (D, s 1,..., s k 1 ) C k ϵ (s 1,..., s k 1 ) Π k 1 j=1 C j D D S Π k j=1 C j P ((T 1,..., T k ) S) e kϵ P ((T 1,..., T k ) S), P D ϵ

18 b Lap(x b) = 1 ( ) x 2b exp. b φ : R n R m M m (D, φ, ϵ) = φ(d) + (Y 1,..., Y m ), Y Lap(x /ϵ) φ ϵ f x f(x (k) ) ϵ k ϵ k (k) ϵ k ϵ k K K k=1 ϵ k ϵ K b = (k) ϵ k = K (k) ϵ.

19 K K { x (1) } N =1 p (k) = l r + q (k) ĝ (k) := x f(x (k) ) + w (k) w (k) Lap(x b (k) ) ˆp (k) := p (k) + w (k) w (k) Lap(x b (k) ) ĝ (k) := x f(ˆp (k) ) ˆx (k+1) x (k+1) θ [0, 1] = prox αk ḡ = (1 θ)x (k) ( x (k) α k ĝ (k)) + θˆx (k+1) ϵ (k) f x N x f(x) = 2γP q D (u j + P q x j ), u j = (l j (t) r j (t),..., l j (t + T ) r j (t + T )) l 1 (k) j=1

20 Fgure 3.2: Communcaton for the case of a trustworthy medator. Fgure 3.3: Communcaton for the case of an untrustworthy medator.

21 prox f (x) prox f (y) x y. x y x := prox f (x) ȳ := prox f (y) ζ x f x prox f (x) prox f (y) (ζ x + x x) (ȳ x) 0 (ζȳ + ȳ y) ( x ȳ) 0. ζ x ζȳ f(ȳ) f( x) ζ x (ȳ x) f( x) f(ȳ) ζ ȳ ( x ȳ). 0 (ζ x ζȳ) (ȳ x). ((x y) + (ζȳ ζ x ) + (ȳ x)) (ȳ x) 0. (ȳ x) (ȳ x) (y x) (ȳ x) + (ζ x ζȳ) (ȳ x). (ȳ x) (ȳ x) (y x) (ȳ x). ȳ x 2 y x ȳ x, prox f (y) prox f (x) y x. D D C j = C j j { 1,..., N }

22 ĝ(x (1) ), ĝ(x (2) ),..., ĝ(x (k 1) ) l 1 x (k) k = 1 x (k) (D ) x (k) (D) = 0. x (k) (D ) x (k) (D) = 0, k > 1 j { 1,..., k 1 } v (k) (D) := x (k) (D) α k x f ( ) x (k) (D). ( ) ( x (k) (D ) x (k) (D) = prox αk 1 ḡ v (k 1) (D ) prox αk 1 ḡ v (k 1) (D ) v (k 1) (D). x f(x (k 1) ) v (k 1) (D ) v (k 1) v (k 1) (D)) (D) = x (k 1) (D ) x (k 1) (D) ĝ(x (1) ), ĝ(x (2) ),..., ĝ(x (k 1) ) l 1 x f x (k) (k) = 2γ Pq D 1 δ δ (k) x f(x (k) (D)) x f(x (k) (D )) 1 N ( = 2γP q D uj (D) + P q x (k) j (D) ) 2γPq D = 2γP q 2γP q D D j=1 N j=1 N j=1 ( lj (D) r j + P q x (k) j (D) ) ( lj (D ) r j + P q x (k) j (D ) ) N j=1 ( uj (D ) + P q x (k) j (D ) )

23 = 2γP q D N ( lj (D) l j (D ) ) + 2γ P N q DP q j=1 j=1 ( x (k) j (D) x (k) j (D ) ) 1 = 2γ P q D 1 l (D) l (D ) 1 2γ P q D 1 δ. k γ p p (1), p (2),..., p (k 1) l 1 p (k) (k) = δ (k) p (k) (D) p (k) p (k) (D ) 1 x (k) l (k) l (k) (D) x (k) (D ) 1 + l (k) (D) l (k) (D ) 1. (D) l (k) (D ) δ p (k) p (k) (D) p (k) (D ) δ s (k) (ω x (k), ω (k) l ) x ˆf(x (k) ) = x f(x (k) ) + 2γP q D N =1 ( ω (k) l ) + P q ω x (k) } {{ } ω m (k).

24 l (k) x (k) Kδ ϵ D ω m (k) N ( Var(ω m (k) ) = Var 2γPq (k) Dω l + Pq DP ) q ω x (k) j=1 ( = 4γ 2 N Var Pq (k) Dω l + Pq DP ) q ω x (k) ( ) = 4γ 2 N 2 Var(ω (k) l ) + 2 Var(ω x (k) ) = 8γ 2 N (2 K2 2 ϵ K2 2 ) ϵ 2 = 32γ 2 N K2 2 ϵ 2, = δ ω (k) Var(ω (k) ) = 2 K2 2 ( ϵ 2 = 2 2γ Pq = 32γ 2 K2 δ 2 ϵ 2, D ) 2 K 2 1 δ ϵ 2 Pq D 1 = 2 D P q N

25 PFNET MP-PFNET CVXPY ECOS ScPy pyparallel ALPG PVWatts

26 q mn q max e mn e max Table 4.1: Battery specfcatons. Table 4.2: Consumer types Load Power [kw] Tme [h] Fgure 4.1: Aggregate load and solar profles. deg deg Table 4.3: Renewable generator specfcatons.

27 Energy Prce [EUR/kWh] :00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 Tme of Day Fgure 4.2: Energy buyng prce c b used for the numerc experments. ECOS γ Net Consumpton [kw] Battery Chargng [kw] Energy Prce [ /MWh] w/o smoothng 2 wth smoothng 4 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Tme of Day w/o smoothng wth smoothng 6 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Tme of Day Energy Prce 20 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Tme of Day Fgure 4.3: Net power profle wth smoothng objectve.

28 T t γ T t γ α k 1/k θ K c s /c b { } 0, 0.8 Table 4.4: Smoothng experment parameters. Table 4.5: Performance experment parameters. c s /c b 10 1 c s /c b = c s /c b = Error 10-1 Error Iteratons Dstrbuted Projected Gradent Iteratons Dstrbuted Proxmal Gradent Fgure 4.4: Algorthm performance wth dfferent energy cost functons.

29 ϵ log(3) log(10 9 ) K K γ θ α k { 1/k } ϵ log(3), log(5), log(10), { log(1e2), } log(1e4), log(1e9) K 0,..., 30 c s /c b Table 4.6: Parameters for the performance/prvacy trade-off experment. ϵ = log(10) ϵ =

30 30.0 ɛ = log(3) ɛ = log(5) ɛ = log(10) 25.0 h(x)/h(x ) ɛ = log(10 2 ) ɛ = log(10 4 ) ɛ = log(10 9 ) 25.0 h(x)/h(x ) Iteratons: K Iteratons: K trustworthy medator untrustworthy medator Iteratons: K Fgure 4.5: Performance/prvacy trade-off. γ θ α k { 1/k } ϵ log(10), K K c s /c b T Table 4.7: Parameters for the MPC smulatons.

31 Energy [kwh] Power [kw] Power [kwh] Battery SOC Net Consumpton Load Consumpton 0 5:00 7:00 9:00 11:00 13:00 15:00 17:00 19:00 21:00 23:00 Tme of Day Power [kw] Cost - Cost w/o Smoothng [ ] Prce [ /MWh] Battery Chargng Accumulated Cost Energy Prce 10 5:00 7:00 9:00 11:00 13:00 15:00 17:00 19:00 21:00 23:00 Tme of Day ɛ = log(10) (trustworthy) ɛ = w/o smoothng ɛ = log(10) (untrustworthy) Fgure 4.6: MPC performance.

32 ϵ

34 x f f(x(τ), x(τ 1)) = ( N p (τ) p (τ 1) ) 2 =1 ( N ) 2 = (u (τ) u (τ 1) + q (τ) q (τ 1)), =1 u (τ) := l (τ) r (τ) u q τ { t,..., t + T } u = (u (t),..., u (t + T )) q = (q (t),..., q (t + T )). f f(x) = γ N (Du + Dq ) 2 2, =1 D T (T +1) D :=

35 (T +1) 2N P q x f(x) = N (Du + DP q x ) 2 2. =1 u := N =1 u D := D D f(x) N = Du + DP q x k 2 2 k=1 = ( N ) ( Du + DP q x k Du + DPq k=1 N = u D Du + 2u D DP q x k + k=1 = u Du N + 2u DPq x k + = u Du + 2u D N k=1 N k=1 P q x k + k N m=1 k=1 m=1 N x m ) k=1 m=1 N x k P q x k P q m + 2 k x k P q DP q x + x P q DP q x. N x k P q D DP q x m DP q x m DP q x m f x x f = 2P q Du + 2P q DPq x k + 2Pq DP q x = 2P q Du + 2P q D k N (P q x k ) k=1 = 2Pq D ( N (u k + P q x k ) ). k=1 f h x x x h(x) = { x f} + x ḡ (x ), x ḡ (x ) ḡ x

37 gthub.com/martnzellner/mp-pfnet

6 Supplementary Materials

6 Supplementary Materials 6 Supplementar Materals 61 Proof of Theorem 31 Proof Let m Xt z 1:T : l m Xt X,z 1:t Wethenhave mxt z1:t ˆm HX Xt z 1:T mxt z1:t m HX Xt z 1:T + mxt z 1:T HX We consder each of the two terms n equaton