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Model description

Relevant temperatures

Inside a transformer, there are many different components, each with its own thermal behavior. This makes a transformer thermally a very complex system. To say something about the thermal behavior of a transformer, it can be simplified by only looking at the oil temperature and the winding temperature. With this simplifications there are a few specific temperatures and parameters that are important to consider:

  • Top-oil temperature - temperature of the oil at the very top of the transformer. As this is generally also the highest oil temperature in the transformer, it is one of the two temperatures that is modelled with this package.

  • Oil temperature outflow windings - temperature of the oil at the top of the windings.

  • Average oil temperature - temperature of the oil in the middle of the tank. It is assumed that there is a linear relationship between the height in the tank and the oil temperature. For this reason, the temperature of the oil in the middle of the tank is equated to the average oil temperature in the tank.

  • Oil temperature inflow windings - temperature of the oil at the bottom of the windings.

  • Average winding temperature - winding temperature in the middle (in terms of height) of the transformer. This temperature is often determined by a resistance measurement.

  • Gradient - temperature difference between the average oil temperature and the average winding temperature.

  • Hot-spot temperature - Temperature of the windings at the hottest point. This point is usually just below the top of the windings. Together with the top-oil temperature, this is one of the two temperatures that is modelled with this package.

  • Hot-spot factor - It is assumed that the difference between the top-oil temperature and the hot-spot temperature is greater than the difference between the average oil temperature and the average winding temperature (i.e., the gradient). This is quantified with the Hot-spot factor. This factor, multiplied by the gradient, indicates the difference between the Hot-spot temperature and the top-oil temperature.

The thermal model can model the top-oil temperature and the hot-spot temperature based on the thermal properties of the transformer. These are the most indicative temperatures because they indicate the highest temperatures for the oil and the windings. Additionally, the standard (IEC 60076-7) also prescribes temperature limits for these two quantities.

Model equations

As discussed in the previous section, there are different specific locations in the transformer where the temperature can be determined. This package focuses on modelling the top-oil and hot-spot temperature as these indicate the highest oil and winding temperatures, respectively. The equations below describe how these two temperatures are modelled. In essence, package uses a recursive method where the temperature at the next time step is calculated using the current temperature. This means an initial value has to be set (or the package initializes the transformer at ambient temperatures).

Top-oil temperature

The model calculates the top-oil temperature at time \(t\) as follows:

\[ \theta_\text{o}[t] = \theta_\text{o}[t-1] + \left[ \theta_\text{a}[t] + \Delta\theta_\text{or} \cdot \left[ \frac{1+R \cdot {K[t]}^2}{1 + R} \right]^x - \theta_\text{o}[t-1] \right] \cdot \left[ 1 - \exp(- \frac{\rm d t}{\tau_\text{o}k_{11}}) \right], \]

where:

  • \(\theta_{\rm o}[t]\): top-oil temperature at time step t;
  • \(\theta_{\rm o}[t-1]\): top-oil temperature at the previous timestep;
  • \(\theta_{a}[t]\): ambient temperature at time step t;
  • \(Δθ_{or}\): top-oil temperature rise (from the temperature-rise-test);
  • \(R\): ratio of short-circuit and no-load loss;
  • \(K[t]\): load level as a percentage of nominal at time step t;
  • \(x\): oil exponent;
  • \(dt\): time step size in minutes;
  • \(τ_{\rm o}\): oil time constant;
  • \(k_{11}\): oil constant.

The equation for the top-oil temperature consists of several parts. First, the top-oil temperature from the previous time step is taken as the starting point. Then, it is determined what the temperature of the oil would converge to under the current load level and ambient temperature. The difference between this final temperature and the top-oil temperature at the previous time step is calculated in the middle section of the formula (between the large square brackets). Because the transformer does not immediately reach the final temperature but reacts with a certain inertia (according to Newton's heating/cooling law), a delaying part is added to the equation in the rightmost square brackets. This inertia is dependent on the oil time constant.

Hot-spot temperature

The hot-spot temperature rise above the top-oil temperature at time t can be calculated as follows:

\[ \Delta\theta_{\text{h}}[t] = \Delta\theta_{\text{h}}[t-1] + \left[ H \cdot gr \cdot K^{y}[t] - \Delta\theta_{\text{h}}[t-1] \right] \cdot \left[ k_{21} \left( 1 - e^{-\frac{dt}{\tau_{\text{w}}k_{22}}} \right) - \left( k_{21} - 1 \right) \cdot \left( 1 - e^{-\frac{dt}{\tau_{\text{\rm o}}/k_{22}}} \right) \right], \]

where:

  • \(\Delta\theta_{\text{h}}[t]\): hot-spot temperature rise at time step t;
  • \(\Delta\theta_{\text{h}}[t-1]\): hot-spot temperature rise at the previous time step;
  • \(H\): hot-spot factor;
  • \(gr\): winding-oil gradient (from the temperature-rise-test);
  • \(K[t]\): load level as a percentage of nominal at time step t;
  • \(y\): winding exponent;
  • \(k_{21}\): winding constant;
  • \(k_{22}\): winding constant;
  • \(dt\): time step size in minutes;
  • \(\tau_{\text{w}}\): winding time constant;
  • \(\tau_{\text{\rm o}}\): oil time constant.

Similar to the equation for the top-oil temperature, the temperature rise at the previous time step is used as the starting point for the calculation. In the middle section, the difference between the rise at the previous time step and the rise to which the hot-spot converges under current conditions is determined. The right part of the formula represents the delaying effect. In this case, the delaying effect consists of two parts:

  • The winding effect indicates the inertia with which the windings themselves heat up (without the influence of the oil).
  • The oil effect A slowing effect caused by the large thermal mass of the oil surrounding the windings.

When the hot-spot temperature rise and the top-oil temperature at time t are known, the hot-spot temperature can simply be determined as:

\[ \theta_{\text{h}}[t] = \theta_{\text{\rm o}}[t] + \Delta\theta_{\text{h}}[t] \]

The mentioned equations have a memory up to 1 time step back in time. This means that for a given load profile and ambient temperature profile, the temperature profile of the top-oil and hot-spot are calculated sequentially.