  The energy efficiency and power of a quantum thermoelectric system with
multiple electric currents and only one heat currents are studied. The system
is connected to the hot heat bath with one terminal but the cold bath with
multiple terminals or vice versal. We find that the cooperative effects can be
a potentially useful tool in improving the energy efficiency and output power
in multi-terminal mesoscopic thermoelectric systems. As an example, we show
that the cooperation between the two thermoelectric effects in three-terminal
thermoelectric systems leads to markedly improved performance of heat engine
within the linear response regime using the Landauer-B\"{u}tiker formalism.
Such improvement also emerge in four-terminal thermoelectric heat engines with
three output electric currents. Cooperative effects in these multi-terminal
thermoelectric systems can significantly enlarge the physical parameter region
with high efficiency and power. For refrigeration, we find that the energy
efficiency can also be substantially improved if multi-terminal configurations
are considered, suggesting a useful scheme toward electronic cooling. Our study
illustrates cooperative effects as a convenient approach toward
high-performance thermoelectric energy conversion in multi-terminal mesoscopic
systems.
