OPTIMAL ENERGY CONSUMPTION AND RUN TIME WITH LOSSES INTRODUCED INTO THE REGENERATIVE SYSTEM EQUATIONS

*ABSTRACT*

In the previous article the choice of an objective function constructed for minimum energy consumption was defined and mathematical methods necessary to solve for the lossless case investigated. This paper explains how to add realistic **loss functions** into the equation set, and develops heuristic numerical methods required to obtain a solution for **energy lost in a motor-coast-brake cycle**.

1. MODELLING LOSS TERMS INTO THE OPTIMISATION PROBLEM

In practice a number of efficiency factors prevent all the input energy being turned into kinetic energy, and furthermore the subsequent recovery of all the kinetic energy by regenerative braking. The cause of non-recoverable braking energy falls into three categories.

A.
The power rating of the electric motor constrains the regeneration capability of
the equipment at higher braking speeds, and results in a greater proportion of
mechanical or friction brake. Alternatively, use of a Constant Rate of Work
(CROW) braking profile can overcomes this to a large extent. The strategy is
introduced above the CROW braking speed, w_{c}, set to coincide with
the power rating of the motor in brake. The disadvantage is lower braking rates
at high speed and correspondingly longer time to brake to standstill.

B.
The power losses associated with electrical and mechanical components. The
vehicle mechanical resistance against speed was modelled as a sectionalised
loss function using Davis coefficients, A(M) + B(M)w + Cw^{2}, to
account for both rolling friction force and air drag resistance. The
coefficients A and B are functions of the train mass, M. Electrical losses
include power converter and for DC machines the brush volt drop, as well motor
and supply conductor rail heating. Motor iron losses are included by an
approximate piecewise model (Figure 1) which is also function of train
speed. The effects of field saturation and armature reaction (for DC brushed
and brushless motors) were not included.

C. Constraints on absorbing and transmitting regenerated energy. The transfer of energy from braking to powering trains is influenced by the coincidence of braking and powering periods, the extent of power network sections, and the maximum permitted voltage at the terminals of a braking train.

The first and second categories can be modelled directly into the system equations, but the third category depends on operational strategies best studied either with the aid of a network simulator that models train movements, or from observation of operational rapid transit system behaviour.

Figure 1 Three Region Piecewise-Linear Model to represent Per-Unit Motor Iron Losses

2. HEURISTIC ITERATION OF NUMERICAL INTEGRATION

The inclusion of the non-linear mechanical loss function prevents algebraic solution or manual integration of the state equations. Consequently, it is necessary to numerically integrate the equations for every run time calculation. Hence, the generalised solutions determined for the lossless case are no longer possible. The solution becomes a time consuming process because the optimum period of coasting is unknown while the station-station distance is a fixed predetermined value. The only way to resolve this type of problem is by iterating the numerical integration of the state equations, by adjusting the coasting period under energy minimisation constraints until the distance travelled converges to the specified station-station distance.

A solution paradigm was established that set the coasting period to zero on the integration first pass, then computing the distance error at the end of the last regenerative braking region, when the speed reached zero. By setting all the loss terms to zero (the lossless case), convergence could be achieved in one step, and agreement with the lossless algebraic solutions confirmed. This was achieved by setting an initial condition for the period of coasting equal to coasting speed divided by the distance error, and then the resulting integration converged exactly on the required station-station distance. The same adjustment was made with the loss terms included, and the iterative process continued until a distance convergence error of < 0.1% was achieved.

Software refinements are required to prevent the numerical optimisation failing when the minimisation routine selects a non-feasible set of variables. This results in one of three types of integration failure:

A. Integration routine can not converge to the required station-station distance

B. Region end point boundary conditions can not be satisfied

C. A very short region where the accuracy of integration is less than the specified minimum

A soft failure mechanism for each of the above failures was invoked as follows:

A. If convergence can not be achieved after several iterations, an increased value of the objective function is returned to the minimisation routine to move it away from a possible non-feasible solution.

B. This failure occurs if the required track distance coincides with the end of a region with the train non-stationary. The software misses the region end point and continues the integration until a bound error occurs. The iteration is ignored and the previous objective function value returned to the minimisation routine with updated initial conditions. This proves acceptable for this rare failure mode and only results in slower convergence.

C. The accuracy condition is temporarily relaxed. This is tolerable, since from the nature of the fault the time interval concerned is short. The integration is continued as if no failure has occurred.

Occasionally one failure would precipitate another and a string of faults result. When this string is of length 10, then the integration is abandoned and the first soft failure mechanism operated.

Once the integration has converged and the results established for the multi-objective of run time and energy consumption minimisation, the solution is re-run to include the integration of auxiliary equations to determine heating and iron energy losses not explicit to the original optimised numerical solution.

3. ENERGY EQUATIONS AND BOUNDS

The distribution of lost energy in a single motor to coast to brake cycle is shown in the energy flow diagram (Figure 2).

Figure 2 Energy Flow diagram showing Motor-Coast-Brake cycle Losses

The total energy used is calculated from the relationship,

E_{L}
= peak kinetic energy + mechanical energy lost + electrical energy lost + iron
energy lost - regenerated energy

where peak kinetic energy =
0.5Mw_{3max}^{2} .. equation 4.1

Mechanical
energy lost = ∫^{Tra }(Aw
+ Bw^{2} + Cw^{3})dt .. equation
4.2

=
∫^{Tra }(ePm[w_{2}^{2}/w^{2}]
+ qPm[w_{2}/w])dt for w > w_{2} .. equation 4.3b

Parameters e and q are the proportions of heat and copper energy lost respectively, compared to motor power at base speed, Pm. The iron energy lost was modelled as the summation of hysteresis and eddy current loss; therefore the power lost relationship is of the form,

P_{I} =
a_{1}VФ^{½} + a_{2}V^{2}

where V is the motor voltage and Ф the field flux.

The iron loss curve (figure 1) is approximated by a 3 region graph assuming the flux, Ф, is constant in region 1, inversely proportional to speed in regions 2, and takes a lower fixed value in region 3, while the voltage, V, is constant apart from in region 1. Parameter d is the proportion of iron energy lost compared to motor power at base speed. The approximate iron energy lost equations simplify to,

Iron energy lost
= ∫^{Tra} dP_{m}[w_{1}/w]dt for 0 ≤ w ≤ w_{1} .. equation 4.4a

=
0.5∫^{Tra} dP_{m}(1+[w_{1}/w]^{½})dt for
w_{1} < w ≤ w_{2} .. equation 4.4b

=
0.5∫^{Tra} dP_{m}(1+[1/K]^{½})dt for
w > w_{2} .. equation 4.4c

A general expression for
regenerative energy was deduced from the inter-relationship of the motoring and
braking characteristic (see second article). This was reduced
according to one of three possible modes of operation, depending on values of
K, K_{1}, w_{1}, β and n, where n is the proportion of
tare train weight on the motored axles. Hence,

regenerated
energy = Mw_{1}^{2}K_{1}(K(1 + loge[1/βK]) - K_{1}/2n) for K_{1} ≤ K.n .. equation 4.5a

= 0.5nMw_{1}^{2}/β^{2} for
K_{1} > n/β^{2}K .. equation 4.5b

=
Mw_{1}^{2}K_{1}K(0.5 + loge[n/β^{2}K_{1}K]^{½}) otherwise .. equation 4.5c

With a state equation integration approach to establishing the value of the objective function it is essential to keep the search within the feasible space. This is difficult when the solution is on a constraint boundary, hence the routine is only allowed to take very small steps if the result is close to a constraint. This prevents excursions beyond the bounded region. In addition to the inequality constraint equations (equations 3.4 to 3.6, third article) used in the lossless case, three additional fixed bounds stop unnecessary deviations into the non-feasible space. These fixed bounds are,

β ≤ 1/K w_{1} ≤ w_{3max}/K w_{1}
> r.P_{m}/(hMg)

where r is the proportion of train axles motored, similar but not necessarily identical to n.

*In the next article we will look at how realistic losses
affect the numerical solutions, and the application of the resulting optimal
driving policies to reducing energy consumption in operational metro systems.*

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