Management of Power Systems With In-line Renewable Energy Sources (Mid-term presentation) Z I WEI YU M ENTOR: JA R ED M CBRIDE

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1 Management of Power Systems With In-line Renewable Energy Sources (Mid-term presentation) HAOIANG SHI HAOZ HE XU XIAOYU SHI Z I WEI YU M ENTOR: JA R ED M CBRIDE

2 Project Distribution Ziwei Yu / Haoliang Shi: collect and analyze the resources Haozhe Xu / Xiaoyu Shi: establish the math modeling Ziwei Yu / Haoliang Shi / Haozhe Xu / Xiaoyu Shi: Graph drawing of the DistFlow ODE system Find the similar management of power system with renewable sources forward report writing for each section

3 BACKGROUND o How Electricity produced by in-line renewable resources, e.g. Hydraulic, solar, wind, and nuclear energy, flows to the users

4 Pros & Cons of renewable energy Renewable energy Pros Cons Solar Energy Wind Energy Hydraulic Energy Geothermal Energy ocation-convenient Weather limitation ow carbon emission ow converting rate High efficiency ocation limitation Durable in a long period of time unsustainable High generating capacity Ecosystem unfriendly Multifunction(Irrigation) ocation limitation High efficiency ow rate of reproduction Persistent ocation limitation

5 MOTIVATION Aim 1: Based on the characteristics of the renewable sources, analyze and develop algorithms that optimize the usage of renewable energy sources into the power grid. Aim 2: Apply the mathematical model on the renewable energy sources (low parametric ODE model)by fixing variables in the model. Aim 3: Efficient management renewable energy to maximize the utility of generation in per length and minimize the reactive power consumption in the in-line power grid.

6 Theory-- Ohm s aw The current through a conductor between two points is directly proportional to the voltage across the two points I = V R V: voltage I: currents R: resistance

7 Theory -- Kirchhoff s aw Kirchhoff s Circuit aw: n k=1 i k = 0 Kirchhoff s Voltage aw: V k = 0 n k=1

8 Theory -- power Real Power: The power that can be used to do work The power that has the net transfer of energy in one direction over a complete cycle P = I 2 R I : current R : resistance Reactive Power: Imaginary power consumption averaging between power generator and loads The power that is consumed by the reactive components Q = I 2 X I : current X : reactance Apparent Power: The combination of the real power and the reactive power S = P + jq

9 Variables and Parameters Variables: P k, Q k -input real/reactive power p k, q k real/reactive power consumption V k voltage at node k I current r resistance over the feeder line x inductance over the feeder line total length of the feeder line l k individual length between node k-1 to k k cumulative distance from 0 to node k-1 N number of loads z relative position along the feeder line Parameters: s dimensionless parameter of the feeder line functions ρ / τ dimensionless parameter of changing from the flowing real/reactive power density ϑ dimensionless parameter of changing from the voltage v

10 Formulas P k+1 P k = p k r k ( P k 2 +Q k 2 v k 2 ) Q k+1 Q k = q k x k ( P k 2 +Q k 2 v k 2 ) v k+1 2 v k 2 = 2(r k P k + x k Q k ) (r k 2 + x k 2 ) P k 2 +Q k 2 v k 2

11 Analysis on a particular point on feeder line Analyze on node k: N>>1, N r k = r l k, x k = x l k set z = k where k = k 1 i=0 F k = F(z) + F( k )/N ( P.Q.V. function) l i Analyze the position with respect to feeder line: p(z) =p k, q(z) = q l k k l k P(z) =P k, Q(z) = Q l k k l k

12 Derivation of discrete consumption w.r.t. z RHS Analysis: P k+1 -P k +r k I 2 =P k Multiply l k on both side: (P k+1 P k ) l k (P k+1 P k ) l k + r k l k I k 2 = p k l k (1) + r( P k 2 +Q 2 k v 2 ) = p k k l k Where P k+1, P k, v k can be converted to P(z), P(z ), v(z) P(z ) P(z) + r P(z)2 +Q(z) 2 l k v(z) 2 = p k l k Then eq.1 HS can be converted to p(z), and, l k, r, are physical terms with constant values Then we have left hand side in terms of consumption of power Hence we conclude that p(z) = p k l k

13 Fundamental derivation of function F k+1 F k F (z) l k F(z+h) F(z) h Set h = l k F( k + l k ) = F( = F k+1 Therefore, F k+1 F k l k = F (z)relating difference to derivative = F (z) k 1 i=0 l k + l k )

14 Analysis F k+1 F k F (z) l k d dz P Q V = p r P2 +Q 2 v 2 q x P2 +Q 2 v 2 xq+rp V The DistFlow ODEs

15 Formulas (Rescaling and Nondimensionalizing) Set s= p r v() ρ s = r p τ(s) = r ϑ(s) = v(z) v() d ds ρ τ θ ( z) as dimensionless parameter P z v Q(z) p v() = ρ = sign(p) p2 +τ 2 θ 2 A B p2 +τ 2 θ 2 ρ+bτ θ r / ρ P / θ Ω/ θ 2 /Ω (θ 2 /Ω)/V = Ω/V V/Ω = 1 The DistFlow ODEs

16 Analysis (for example)

17 Future Project Plan solve the Cauchy problem to get the following formula: = s θ(s ) p r v() = 1 θ(s ) P(0) = ρ(s ) p r θ(s ) Q(0) = τ(s ) p /r θ(s ) Rescaled time: s: 0 s * Analyze the first case study result where p = -1, q = 0.5 Use the analyzed result to find out the similar relationship of renewable sources forward in the rescaled time

18 References A very grateful thanks to our mentor Mr. Jared McBride! 1) 2) K. Turitsyn D. Wang and M. Chertkov. \DistFlow ODE: Modeling, analyzing and controlling long distribution feeder." In: Proceedings of the 51st IEEE Conference on Decision and Control (2012). url: 3) M. Baran and F. Wu, Optimal sizing of capacitors placed on a radial distribution system, Power Delivery, IEEE Transactions on, vol. 4, no. 1, pp , jan 1989.

19 Thank you!