Analysis of transients in linear electrical circuits

The purpose of the work: To conduct a study of a linear circuit containing several electric energy storage devices in the Engee program.

The content of the work:

create a system of equations for a given electrical circuit;
develop a program for calculating currents in circuit branches and voltage drops on capacitive elements;
present the results of the program calculation in the form of graphical dependencies;
create a model of an electrical circuit in Engee;
compare the simulation results with the calculation results of the developed program.

Engee and the DifferentialEquations.jl package are used as the software used to solve the system of equations. The calculation is carried out using the ode45 method, a one-step explicit Runge-Kutta method of the 4th (5th) order.
Here is an example of calculating a linear electrical circuit.:

1. Application of the DifferentialEquations.jl package

Package [DifferentialEquations.jl](https://diffeq.sciml.ai/stable /) provides a large number of well-optimized solvers of differential equations of various types:

ordinary differential equations
differential algebraic equations
stochastic differential equations
differential equations with a delayed argument
stochastic differential equations with a delayed argument
differential equations with parametric uncertainty
and others .

using .DifferentialEquations, .Plots

2. Solving the differential equation

The general procedure is as follows:

1. Set the function

f(u, p, t) 
f(du, u, p, t)

where u - variables, p - equation parameters, du - derivatives .

2. Define the task

prob = ODEProblem(f, u0, tspan)
prob = ODEProblem(f, u0, tspan, p)

where u0 - initial conditions, tspan - estimated time interval ()

3. Numerical solution

sol = solve(prob, alg; kwargs)

where alg - numerical algorithm, kwargs - additional arguments.

For example, for users familiar with MATLAB/Python/R, there are translations of standard library methods.:

ode23 → BS3()
ode45/dopri5 → DP5(), although in most cases Tsit5() is more effective
ode23s → Rosenbrock23(), although in most cases Rodas4() is more effective

You can read more about solvers in документации.

4. Analysis of the results

sol(t)
plot(sol)

where the function is used plot the Plots.jl package. When using other graphics packages

plot(sol.t, sol')

3. A program for calculating an electrical circuit

A system of differential equations describing an electrical circuit:

where Uc1, Uc2 are the voltage across the capacitances that can be found from solving the equations

In order for the system of equations to be solvable with respect to unknown quantities standing under the sign of the differential, it is necessary to express the currents of branches that do not contain inductance in terms of the desired parameters. To do this, using Kirchhoff's laws, we formulate the coupling equations (1.2) and supplement (1.1):

# Ввод параметров электрической цепи
R1 = 10.0
R2 = 15.0
R3 = 20.0
R4 = 10.0
f  = 50.0
C1 = 10e-6
C2 = 30e-6
L1 = 15e-3
L2 = 20e-3

# Определение дифференциальной системы
function circuit!(du, u, p, t)
    U, R1, R2, R3, R4, C1, C2, L1, L2 = p
    Uc1, Uc2, i1, i4 = u
    
    # Алгебраические уравнения связи
    i2 = (u[1] + u[2] + (u[3] - u[4]) * R4) / (R2 + R4)
    i3 = u[3] - i2
    i5 = i3 - u[4]
    
    # Дифференциальные уравнения
    du[1] = i3 / C1
    du[2] = i5 / C2
    du[3] = 1 / L1 * (U(t) - i2 * R2 - u[3] * R1)
    du[4] = 1 / L2 * (u[2] + i5 * R4 - u[4] * R3)

end

# Начальные условия для u
u0 = [0.0, 0.0 , 0.0, 0.0]  

# Диапазон времени для расчета, пусть будет два периода промышленной частоты 50 Гц
tspan = (0.0, 0.04) 

# Задание параметров
U(t) = 100 * sin(2 * π * f * t)
params = (U, R1, R2, R3, R4, C1, C2, L1, L2)

# Создание объекта ODEProblem с передачей параметров
prob = ODEProblem(circuit!, u0, tspan, params)

# Решение дифференциальных уравнений
sol = solve(prob, Tsit5());

# Построение графиков
plot(sol, label = ["Uc1(t)" "Uc2(t)" "i1(t)" "i4(t)"], layout = 4)

4. Comparison of calculation results with simulation results

Let's load the model located in /user/start/examples/power_systems/lab_work/lab_work.engee.

modelName = "lab_work";
model = modelName in [m.name for m in engee.get_all_models()] ? engee.open( modelName ) : engee.load( "$(@__DIR__)/$(modelName).engee");

Let's run the simulation:

results = engee.run( modelName );

Let's give convenient names to the output variables of our model (those that are marked as logged signals):

t = results["i1"].time;
i1 = results["i1"].value;
i4 = results["i4"].value;
Uc1 = results["Uc1"].value;
Uc2 = results["Uc2"].value;

Let's build graphs based on the simulation results.:

plot(t, [Uc1 Uc2 i1 i4], label = ["Uc1_sim" "Uc2_sim" "i1_sim" "i4_sim"], layout = 4)

Let's display the simulation results and calculation results on one graph.:

plot(t, [Uc1 Uc2 i1 i4], label = ["Uc1_sim" "Uc2_sim" "i1_sim" "i4_sim"], layout = 4)
plot!(sol, label = ["Uc1" "Uc2" "i1" "i4"], layout = 4)

Conclusions on the work: in the course of the work, a system of algebraic differential equations was compiled, a program for calculating currents in branches and voltage drops on capacitive circuit elements was developed, graphical dependences of the laws of current and voltage variation on time are shown. The results of Engee modeling are compared with the results of the developed calculation program. In general, the laboratory work presents an analysis of an electrical circuit containing several electric energy storage devices.