# New Approaches for Very Short-term Steady-State Analysis of An Electrical Distribution System with Wind Farms

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## Abstract

**:**

## 1. Introduction

- frequency regulation (a few seconds to about a minute);
- load following (a few minutes to a couple of hours);
- unit commitment (from several hours to one or more days into the future).

- Physical methods – methods that use physical information about the site (e.g., wind conditions at the height of turbine hubs and weather conditions). These methods are typically characterized by significant computational complexity and are mainly used for long-term forecasting.
- Statistical methods—methods that can forecast either a wind-speed/power value (“point-forecast” methods) or a wind-speed/power probability density function (“pdf-forecast” methods). Both are obtained from statistical analyses of time series from past data, and they usually employ recursive techniques. They can furnish useful information with reduced computational efforts for short-term forecasting (a few hours ahead).
- Artificial neural network methods—methods that are designed to determine the relationship between wind power and the time series from past data.
- Hybrid methods – methods that are combinations of the previous methods.

## 2. Very Short-Term, Steady-State Analysis of a Distribution System with Wind Farms

- (i)
- Step 1: forecast “at hour t” the wind power “at hour t + k” at the sites where the wind farms are installed; and
- (ii)
- Step 2: perform the very-short-term (VST), steady-state analysis of the electrical distribution system.

## 3. Wind Power Forecast Methods

#### 3.1. “Point-forecast” Methods [5]

_{k}is the correlation coefficient between ${P}_{t}$ and ${P}_{t+k}$; this coefficient is calculated using the following relationship:

#### 3.2. “Pdf-forecast” Methods

- a.
- Bayesian Method

_{0}and α

_{1}are the coefficients of the model.

- b.
- Markov Method

_{ij}is the probability of a transition between states i and j.

_{ij}is the number of transitions from state i to state j encountered in the measured wind speed data.

^{th}row of the matrix to find the next state; in particular, if x is lower than (or equal to) the first element in the row ($x\le {p}_{i1}$), the next state is state 1. If x is greater than the first element in the row and lower than (or equal to) the sum of the first two elements in the row (${p}_{i1}<x\le {p}_{i1}+{p}_{i2}$), the next state is state 2. Thus, in general, if x it is greater than the sum of the first g-1 elements in the row and lower than (or equal to) the sum of the first g elements in the row ($\sum _{j=1}^{g-1}{p}_{ij}}<x\le {\displaystyle \sum _{j=1}^{g}{p}_{ij}$), the next state is state g. The process is then repeated to find further states, using the latest state as the initial state.

- weight the generic element of the wind power time series with the coefficient (1-${c}_{k}$), where ${c}_{k}$ is the correlation coefficient defined in Equation 4; and
- add the results obtained in 1. to the last observed power P
_{t}weighted with the correlation coefficient ${c}_{k}$.

## 4. Load flow equations for distribution systems with wind farms

#### 4.1. Deterministic Load Flow

_{load}are load busbars and those from n

_{load}+1 to n

_{bus}are generator busbars. (The last busbar is the slack busbar.) In the load busbars, the active and reactive powers are assigned (PQ busbars), while in the generator busbars without the slack, active power and voltage amplitude are assigned (PV busbars). A single-phase representation of the system is adequate, and the steady state of the system is described by the following, non-linear, equation system (load-flow equations):

_{i}, δ

_{I}are the voltage magnitudes and arguments, respectively; ${G}_{ij},{B}_{ij}$ are the conductance and susceptance of the I - j term of the admittance matrix, respectively; n

_{bus}, n

_{load}are the system bus number and the load bus number, respectively; ${\theta}_{ij}$ = ${\delta}_{i}-{\delta}_{j}$.

- fixed speed WTGUs (induction generators directly connected to the distribution system that are driven by wind turbines with either a fixed turbine blade angle or a pitch controller to regulate the blade angle);
- semi-variable-speed WTGUs (induction generators with a rotor-resistance converter); and
- variable-speed WTGUs (doubly-fed induction generators or synchronous/induction generators with full-scale static converters).

- a.
- Fixed-speed WTGUs

_{r}and stator X

_{s}leakage reactances, X

_{m}is the magnetizing reactance, X

_{c}is the reactance of the capacitor bank used for power factor improvement, R is the sum of the stator R

_{s}and rotor R

_{r}resistances, and s is the slip.

_{m}(wind power forecast, Step 1, Subsection 2.1), if active power losses are neglected.

- b.
- Semi-variable speed WTGUs

_{m}of the WTGU. With reference to the reactive power, it should be noted that the rotor resistance is variable and unknown, as is the machine slip; to take these facts into account, in [15], a very simple procedure is suggested that reduces the two unknowns (slip and rotor resistance) to only one unknown by considering that the expressions of active and reactive power outputs can be written as a function of only one unknown R

_{eq}= R

_{r}/s, obtaining:

_{eq}is known, the reactive power output can be calculated with the following equation:

- c.
- Variable-speed WTGUs

_{m}of the WTGU. For the reactive power output, the following expression applies:

^{sp}and $\mathrm{cos}{\left(\mathsf{\phi}\right)}^{sp}$ are the specified values of reactive power and power factor, respectively.

#### 4.2. Probabilistic Load Flow

## 5. Experimental Section

- -
- 1.0 MW, semi-variable-speed WTGU at bus #9;
- -
- 1.0 MW GBBC, variable-speed WTGU at bus #13;
- -
- 1.0 MW DFIG, variable-speed WTGU at bus #15;
- -
- 1.0 MW stall-regulated, fixed-speed WTGU at bus #17.

**Figure 3.**17-busbar, balanced, three-phase test distribution system [17].

**Figure 5.**BM-predicted values (minimum, mean, and maximum values) and measured values of the hourly wind speed at the site where the WTGU is connected to busbar #17.

**Figure 6.**BM and MM mean values; GPM, PM, NM predicted values; and measured values of the hourly WTGU power at busbar #13 (a) and at busbar #17 (b).

## 4. Conclusions

- The steady-state analysis of a distribution system is mandatory in order to determine the voltage profile and the losses, which are strongly influenced by the presence of the wind farm.
- All wind-forecasting methods, with the exception of the Generalized Persistence Method, furnished similar acceptable results in terms of both voltage profile and losses.
- In the case of the probabilistic approaches, the mean value seems to be the most reliable statistical figure to take into account.
- Probabilistic approaches seems to be the most useful, since they are capable of taking into account the unavoidable time-varying nature of wind speed and of the loads on the steady-state behavior of the distribution system.

## Acknowledgments

## References and Notes

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**MDPI and ACS Style**

Bracale, A.; Carpinelli, G.; Proto, D.; Russo, A.; Varilone, P.
New Approaches for Very Short-term Steady-State Analysis of An Electrical Distribution System with Wind Farms. *Energies* **2010**, *3*, 650-670.
https://doi.org/10.3390/en3040650

**AMA Style**

Bracale A, Carpinelli G, Proto D, Russo A, Varilone P.
New Approaches for Very Short-term Steady-State Analysis of An Electrical Distribution System with Wind Farms. *Energies*. 2010; 3(4):650-670.
https://doi.org/10.3390/en3040650

**Chicago/Turabian Style**

Bracale, Antonio, Guido Carpinelli, Daniela Proto, Angela Russo, and Pietro Varilone.
2010. "New Approaches for Very Short-term Steady-State Analysis of An Electrical Distribution System with Wind Farms" *Energies* 3, no. 4: 650-670.
https://doi.org/10.3390/en3040650