# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 4 Instantaneous/Average Compensation  
In previous sections, general power terms are derived within the GA framework. In addition, those expressions are used to decompose the current vector into two orthogonal components. In this section, different compensation strategies and compensation currents are established from the point of view of the exposed development. From the new formulation, two compensation approaches are presented: time instantaneous compensation (TIC) and time average compensation (TAC). Regarding the second approach, different specific compensation objectives can be proposed: constant supply active power, a unity power factor, and current in phase with the fundamental component of the supply voltage. In addition, the operative expressions of the reactive power variables allow the imposition of other compensation objectives [1; 2; 3; 4; 5; 6]. Figure 3 presents the schematic of a compensated power system. The supply, the load, and the shunt compensation system are highlighted. Thus, according to the fact that the instantaneous power multivector is additive, it follows that:  
\[\mathbf{s}_{\mathrm{S}}=\mathbf{s}_{L}+\mathbf{s}_{\mathrm{C}} \tag{75}\]  
where \(\mathbf{s}_{\mathrm{S}}\) is the source instantaneous power multivector, \(\mathbf{s}_{\mathrm{L}}\) the load instantaneous power multivector, and \(\mathbf{s}_{\mathrm{C}}\) the compensator instantaneous power multivector. If the load instantaneous power multivector, \(\mathbf{s}_{\mathrm{L}}\), is considered to be:  
\[\mathbf{s}_{\mathrm{L}}=\mathbf{u}\mathbf{i}_{\mathrm{L}}=p_{L}+\mathbf{q}_{L} \tag{76}\]  
then the load current takes the expression:  
\[\mathbf{i}_{L}=\frac{\mathbf{u}}{u^{2}}(p_{L}+\mathbf{q}_{L})=\mathbf{i}_{Lp} +\mathbf{i}_{Lq} \tag{77}\]  
The time instantaneous compensation process (TIC) is a compensation procedure without energy storage and corresponds to reducing \(\left|\mathbf{i}_{\mathrm{q}}\right|\) or \(\left|\mathbf{q}\right|\), without altering the instantaneous power transfer. Figure 3: Multi-phase power system with a shunt compensator. **Definition 7**. : _The time instantaneous compensation method, TIC, consists of connecting a compensator without energy storage in shunt with the load, so that it transfers the instantaneous reactive power of the load with the opposite sign._  
**Theorem 7**. : _In a multi-phase system consisting of a load with an instantaneous power multivector, \(\mathbf{s_{L}}\), and a shunt compensator without energy storage with an instantaneous power multivector, \(\mathbf{s_{C}}\), the time instantaneous compensation process assumes that:_  
\[\mathbf{i_{S}}=\frac{\mathbf{u}}{u^{2}}p_{L};\quad\mathbf{i_{C}}=\frac{- \mathbf{u}}{u^{2}}\mathbf{q}_{L} \tag{78}\]  
In fact, the TIC means that the compensator transfers all the instantaneous reactive power; that is, for the references in Figure 3:  
\[\mathbf{s_{C}}=-\mathbf{q}_{L} \tag{79}\]  
then the compensator current is:  
\[\mathbf{i_{C}}=\frac{\mathbf{u}\mathbf{s_{C}}}{\left\|u\right\|^{2}}=\frac{ \mathbf{u}(-\mathbf{q}_{L})}{\left\|u\right\|^{2}} \tag{80}\]  
The application of the Kirchhoff Current Law (KCL) to the system in Figure 3 allows the source current to be determined after compensation:  
\[\mathbf{i_{S}}=\mathbf{i}_{L}+\mathbf{i_{C}}=\frac{\mathbf{u}\mathbf{s}_{L}} {\left\|u\right\|^{2}}+\frac{-\mathbf{u}\mathbf{q}_{L}}{\left\|u\right\|^{2} }=\frac{\mathbf{u}}{\left\|u\right\|^{2}}(\mathbf{s}_{L}-\mathbf{q}_{L})= \frac{\mathbf{u}p_{L}}{\left\|u\right\|^{2}}\equiv\mathbf{i}_{Lp} \tag{81}\]  
which matches with the load instantaneous power current. Thus, the source current only transfers the load instantaneous power. The time average compensation process (TAC) corresponds to a reduction in the losses averaged over time, without altering the active power from source to load.