Reac%ve power pricing strategy

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1 Department of Informa-on Engineering University of Padova, ITALY Reac%ve power pricing strategy Giulia Maso, Mar-no Minella, Anna Polidoro

2 Smart Grids Outline Model Reac-ve power compensa-on Pricing Tools Lagrange mul-pliers (centralized) Gradient es-ma-on (distributed) Pricing Strategy Shadow prices Storage strategy

3 Smart grids

4 Smart grids

5 Smart grids Model 0 0 generator s loads PCC directed graph G = ( V, E,σ,τ ) matrix of incidence A voltages and currents y( t) = y 2 sin( ω0t + y)

6 Ini$al model equa$ons (KLC) and (KVL) laws Α T Αu ξ + i = 0 + Ζξ = PCC u 0 = U 0 0 generic node u v i v = s v u U v Smart grids Model 0 η constant voltage generator v, v V exponen-al model \{0}

7 Approximate model(1) Laplacian matrix, L = Α T Ζ 1 Α Green matrix, X sa-sfying: XL = I 11 X10 = 0 T 0 Solu-on for the currents: i = Lu

8 Approximate model(2) The system can be rewrioen in the following form: u u 1 v T = i i v Xi = = 0 + U s v u U 0 v 1 0 η v, v V \{0}

9 Smart grids Compensa;on Reac$ve power flows contribute to: power losses on the lines voltage drop grid instability Problem of op-mal reac-ve power compensa-on Bolognani, S., and Zampieri, S. (2011) Distributed control strategy for op-mal reac-ve power compensa-on in smart microgrids.

10 J ' = i T Smart grids Compensa;on 1 T 1 T 1 ~ Re( x) i = p Re( x) p + q Re( x) q + J U U U 0 where it has been used the Taylor approxima-on of 0 ( ) U 0 0 ( U, s) U i for large 0 Being ~ J, ( U s) 0 infinitesimal, we have an uncoupling phenomenon QUADRATIC LINEARLY CONSTRAINED PROBLEM

11 Smart grids Compensa;on C V v s q q to subject q x q q J where q J v v T T q \ ) Im( 0 1, ) Re( 2 1 ) (, ) ( min = = = QUADRATIC LINEARLY CONSTRAINED PROBLEM

12 Reac;ve power Service compensa;on Pricing Shadow prices Lagrange Mul;pliers

13 Lagrange Mul;pliers Theory(1) Duality theory Impossibile visualizzare l'immagine. La memoria del computer potrebbe essere insufficiente per aprire l'immagine oppure l'immagine potrebbe essere danneggiata. Riavviare il computer e aprire di nuovo il file. Se viene visualizzata di nuovo la x rossa, potrebbe essere necessario eliminare l'immagine e inserirla di nuovo. Primal min f 0 ( x) s.t f ( x) 0, h ( x) = i i 0 Lagrangian g ( λ, ν ) = inf L( x, λ, ν ) Dual max g( λ, ν ) s. t λ 0 Primal+dual *,ν * λ

14 Lagrange Mul;pliers Theory(2) λ p (0,0) i = u i p ui Primal solu-on Constraint perturba-on Op;mal Lagrange mul;pliers LOCAL SENSITIVITIES

15 Lagrange Mul;pliers Applica;on(1) Objec-ve func-on J (q) Constraints on loads and compensators q = Ql, q Q c Lagrangian T L( q, λ) = J( q) + λ ( q Q) Solu-on λ * 2 1 l = D Q 2 l u0

16 Lagrange Mul;pliers Applica;on(2) Solu-on requires global knowledge of network topology Centralized algorithm Centralized unit High computa-onal cost with large number of nodes Broadband communica-on system Change in network topology

17 Lagrange mul;pliers Graphic Exploi-ng the perturba-on theory of duality we iden-fy the losses parabola Working point

18 Working point Graphic λ

19 Working point Graphic λ

20 Gradient es;ma;on Beginning from the Lagrangian with constraints on both loads and compensators: Solving for λ: Es;ma;on

21 Gradient es;ma;on (1) Using PCC voltage Each node receives informa-on from the PCC PCC λ( u) = e 2cosθ Im[ jθ ( u u u 0 0 1) ] Θ phase shi_ constant between node and PCC overes-mate results

23 Distributed Gradient es;ma;on (2) Two nodes exchange informa-on between each other PCC p λ c = λ p + 2 cosθ 2 u 0 Im[ e jθ ( u p + u c )( u p u c ) c Θ phase shi_ Between the two nodes results underes-mate

25 Valida;on test for λ es;ma;on Real λ analysis Es-mate λ analysis

26 Test Grid Aligned nodes PCC Symmetric bifurca;on PCC Load Load Load Com Load Load Load Load Load Com Com

27 1 observa;on Branch with no power flowing (λ 0) Why? The only ac;ve node

28 2 observa;on Linear increasing of λ in a branch with constant power flowing Linear increasing The only ac;ve node

29 3 observa;on Superposi-on of the various effects Power flow of nodes 5 and 10 Power flow of node 10 Ac;ve node No power flow

30 Performance of the λ es;ma;on Simple grid Different absorp;ons Many compensators Not homogeneus grid Depth tree

31 Hypothesis of no homogeneity If we have different impedance phase angles: X e jθ X then the assump-on done on Re(X) isn t true and we can t extract the phase angles from X Anyway,the homogeneus hypothesis is a realis-c assump-on in the smart microgrid

32 Depth tree The error increases with the increase of depth tree. This is due to two errors: λ 2 cosθ.. λp u c. = 2 0 Im[ e jθ ( u p + u c )( u p u c ) from parent λ from upgrade Nevertheless : we can consider compensator informa-on generally, we have levels of depth

33 Parent child communica;on strategies Ordered Randomized Centralized Right upgrades Time: # levels 1 Wrong upgrades Time: # nodes 1 Only voltage needed Time: 1 step

34 Shadow prices interpreta;on λi tells us approximately how much more profit the process could make for a small increase in availability of resource qi Lagrange Mul;plier Shadow Price However, the cost func-on f0(x) doesn t consider all the benefits related to reac-ve power Lagrange Mul;plier Shadow Price

35 Storage strategy f0(x) doesn t consider: voltage constraints conges-on constraints We define a storage strategy in order to evaluate what is truly convenient for a compensator: Sell or Store ac-ve power?

36 Compensator limita;ons A compensator provides an apparent power A that is func-on of the ac-ve power P and of the reac-ve power Q. These quan--es are linked by Boucherot theorem: Where bars indicate the available ac-ve power or the reac-ve power required.

37 The inverter installed on the compensator is characterized by an apparent power limita-on M. Thus, we have two cases: No limita-on Limita-on is ac-ve We must decide whether to sell or to store ac;ve power

38 What to do in case of limita;on? Time t=0 At the ini-al -me we suppose to provide the whole ac-ve power and produce a quan-ty Q(t=0) of reac-ve power The corresponding inverter limita-on is: Where the reac-ve power tends to zero.

39 Time t=1 Suppose to increase the reac-ve power to: Due to ac-ve limita-on, we must decrease the ac-ve power sold and store a certain quan-ty: For any produc-on increment of reac-ve power we can consider an equal storage quan-ty:

40 Cost func;on storage isn t free considering the storage inefficiency η Where is the unit price of ac-ve power

41 Gain func;on using λ For an infinitesimal varia-on of reac-ve power we assume a linear increasing of ac-ve power From which we obtain the gain func-on: Where is the unit price of ac-ve power

42 Profits inequality To have profits we need to comply with: Solving the inequality we obtain: Never true λ < 0.05 η = We have a confirma-on that λ hasn t all the necessary informa-on for pricing

43 Gain func;on with Q price Assuming to know the correct reac-ve power price, we can define the gain: Solving the profit inequality, we obtain: It is plausible that in some cases reac-ve power importance is very high even higher than the ac-ve power one

44 λ shadow price for Voltage limits Risk of conges-on Conclusions Pricing Storage strategy Future developments Define a new cost func-on Build different Lagrangians Abandon Lagrange theory

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