Power Flow (Cable Ratings)

[SeptimusBoshoff]

The problem can be stated as follows. Given a power system, a system with sources, loads and cables, what is the power flow through the cables in steady state, and is the power limit for the cable exceeded?

The cable limit is defined by its Surge Impedance Loading (SIL) and the operating voltage (Line to Line). Typically the maximum operating voltage is taken as 1.05 p.u. A related quantity which has to be computed as an intermediary step to the SIL is the characteristic impedance, which is a function of the cable capacitance and inductance. Therefore, the problem becomes; if a network topology is given, that is, the connectivity matrix, the number of sources, and the number of loads, (including the power ratings) what should the capacitance and inductance of the cable be? This is a different formulation of the problem, when compared to how a utility would design its system, but for our goals so be it.

• https://www.electrical-technology.com/2019/10/surge-impedance-loading.html

The first thing to do would be to randomly generate the CM (with sources, loads, and cables). The next step would be to solve the power flow equations for the randomly generated network. This would employ an iterative algorithm (e.g. Newton-Rhapson, Gaus-Seidel, Decoupled, etc.). The solution would tell us what the power flows are through the cables. All the cables, for which the SIL is exceeded should be identified. Now it we have a choice, either replace all of the exceeded cables with better cables, or only replace, say the first 2 maximally exceeded cables. In either case, the power flow algorithm should be run again, and the steps repeated until there are no more violations.

1. Generate random power system (if not specified by user)
2. Calculate characteristic impedance for cables
3. Calculate SIL for cables
4. Solve power flow
5. Replace cables
6. Go to 4. until no more violations

[Webbah]

• [x] 1. Generate random power system (if not specified by user)

  • correct random generation should be done in #17
  • Here, first example: small grid (2 sources, 1 (passive) load) with underrated cables

• [x] 2. Calculate characteristic impedance Z_0 for cables

  • Based on above described SIL-"idea" (Compare also here; 10.3. p. 357 or here; 4.5 p. 236; called natürliche Leistung )
  • Z_0 = sqrt(L/C)
  • Add Z_0 and P_max to the cable parameter dict?

• [x] 3. Calculate SIL (->P_max) for cables using 2.

  • This we can take as reference to see if the parameters are sufficient for the in 4. calculated power the cable has to take

• [x] 4. Solve power flow

  1. Calulate Admittance matrix Y
  2. Solver powerflow equations unsing
     • Newton Rapson
     • Gauß (-Seidel?) Which node is the reference? What are the unknowns?

• [x] 5. Replace cables

• [x] 6. Go to 4. until no more violations

[Webbah]

Update

Problem: With power flow equations we would have too many unknowns and too less equations.

Idea: Use 'compressed Sensing' to consider the design of the cables as kind of optimization problem (Compare e.g. Data-Driven Science and Engineering chapter 3.2).

Maybe that can be extended to optimize the power it at all, too, in case of where to place a cable (could be used to construct the CM matrix if not defined)

[Webbah]

Back to powerflow

Since we would have somehow to solve the power flow equations (PFE) to figure out - depending on the power demand - which source would deliver what power and where is it flowing, let's try to solve the PFE with a lot of unknows and see where it goes.

Always fixed:

• P_load (?)
• Vbus (?) the magnitude is typically a range from 0.95 to 1.05

Sometimes fixed:

• delta (for one source/load, i.e. the reference bus)
• Q_load

Variable:

• P_sources (?)
• G (conductance; real part of Y)
• B (Suceptance; imag part of Y)
• delta (bus angle)

Notes:

• the ratio between L and C is fixed. This needs to be implemented otherwise the solver will simply set the susceptance to infinity, i.e. the capacitance of the cables set to 0.
• there might be a relationship between the droop parameters and the solution to the PFE that should be incorporated. It might not be that P_sources is not a variable, but rather fixed through a complicated function.
• the power of the loads is not necessarily fixed. Battery energy storage systems are capable of generating power and absorbing power. Surely, we want to implement variable and dynamic loads as well? One of the defining characteristics of interconnected microgrids/DER is bi-directional power flow.

Solutions:

• Try to solve with typical solver/optimizer (e.g. fmincon,...)
• Simulated Annealing (a method of solving a macroscopic system based on local interactions)
• Perhaps some probabilistic approach might be required?

[Webbah]

List to be defined in the parameter-dict in thenodeconstructorbefore cable rating funtion is called:

• parameters["grid"]
• parameters["source"][i]
  • v_pu_set
  • pwr

-parameters["load"]

• pwr -> to be added to passive loads? (maybe for specific vltg level and impendance(2 pi f_grid)

  • [x] has to be manually defined for every impedance (R, RL, RLC, LC,....)

• How to handle active source? -> would sit in sources with negative ref...

  • [x] check for ctrl_type?

To be defined by the cable rating funtion and set in the nc:

• cabel parameter X)
• power factor (https://github.com/upb-lea/dare/blob/a5d744c31a2adbad795cb55d9c2162221ce689aa/src/nodeconstructor.jl#L359)

[Webbah]

Nearly done

Added power flow equation(pfe) to Power_System_theory script Left todos:

• Lagrangians # TODO make these parameters depending on price ($)?
• https://github.com/upb-lea/dare/blob/b501c1914959f9e4b1722cbbe1685d5011e8c5fc/src/Power_System_Theory.jl#L637

Used PFE in NC and overwrite ALL cable parameters if one is not defined to ensure that it fits pfe (to be changed and keep defined cable parameters??) - https://github.com/upb-lea/dare/blob/b501c1914959f9e4b1722cbbe1685d5011e8c5fc/src/nodeconstructor.jl#L638

[SeptimusBoshoff]

The limits for the cable were calculated using the equations in the text book Power System Analysis and Design (Glover and Sarma), particularly equation number 5.5.3.

[Webbah]

todo: put this result to be included into the environment

(I1 -> limit_array[idx_I1])

[Webbah]

Final implementation notes

layout_cabels(CM, num_source, num_load, parameters) function in Power_System_Theory.jl which gets called in the nodeconstructor to fill up the parameter dict with missing cable entries.

• len has to be dinfied in beforehand (is ensured by the nodeconstructor
• if not all cable parameters are defined (R,L,C), they are calculated based on power flow equations (steady state)
• current limits added to the parameter dict of the cable (calulation based on Eq. 5.5.3 like described above)
• these current limits are in the end used in the env to fill the norm array