## Unified theorys active and reactive powers

Energy2green Wind And Solar Power System

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Fundamental positive-sequence complex power supplied by the wind generator showed in fig. 1 is expressed as:

S+ = 3 Va+ ■ 1A + = 9VA +• (Ye + % • Y/) = P+ + Q+ (15)

Positive-sequence active power (P+) is the real part of the above quantity and it characterizes the direct torque applied to the axis of the wind-generator. This quantity has two components, due to the active loads (Pa+ ) and caused by the unbalances (Pu+ ):

P+ = 3 Va+ • 1Aa+ = 9VA +• (Ge +Su • Yi • cos (a+ - a- + a) = Pa++ Pu+

Pu+= 3 Va+ • 1Aau + = 9 Su ■ • cos (a+ -a+a) •

Pa+ is the positive-sequence active power supplied by the wind-generator under positive-sequence balanced voltages; thus, it may be defined as the positive-sequence active power due to the load consumptions. This quantity measures the active power which is transformed under the best efficiency and power quality conditions. Pu+ represents the increasing of the positive-sequence active power produced by the voltage and load unbalances. Last quantity identifies the poor power quality in the power system, since it occurs when there are voltage unbalances, and it may have the same or different sign that Pa+ , so it increases or decreases the total positive-sequence active power (P+ ). Positive-sequence reactive power ( Q+ ) is the module of the imaginary part of the positive-sequence complex power. Expressed in complex notation, this quantity has the following formulation:

Q+ = 3 VA+ • ^Ar + = j 9VA+ ■ (Ye ' sin ae + Su ' Yi ' sin(a+ - a- + ai )) = (17)

= j 9VA+ • (±Be +Su • Yt • sin(a+ -a-+ai)) = Qr + + Qu+

Positive-sequence reactive power characterizes the main magnetic field of the windgenerator and it holds two components, due to the reactive loads (Qr + ) and caused by the unbalances (Qu+ ):

Qr + = 3 VA+ • lArr + = j 9 Ye • sin ae • VA+ = ±j 9 Be VA+

Qu+ = 3 VA+ ' hru+ = j 9 Su ■ Yi • sin(a+ -a-+ai ) • VA+

Qr + is the positive-sequence reactive power supplied by the wind-generator under positive-sequence balanced voltages. This quantity determinates the reactive power established under the best efficiency and power quality conditions. Qu+ defines the increasing of the positive-sequence active power produced by the voltage and load unbalances. This quantity identifies the poor power quality in the power system, since it occurs when there are voltage unbalances, and it may have the same or different character (inductive or capacitive) that Qr +, and thus it can increase or decrease the positive-sequence reactive power , Q+ . 