A new donor-accepter-donor-accepter-donor (D-A-D-A-D) type 2,1,3-benzothiadiazole-thiophene-based acceptor unit 2,5-di(4-(5-bromo-4-octylthiophen-2-yl)-2,1,3-benzothiadiazol-7-yl)thiophene ($DTBTTBr_2$) was synthesized. Copolymerized with fluorene and indeno[1,2-b]fluorene electron-rich moieties, two alternating narrow band gap (NBG) copolymers PF-DTBTT and PIF-DTBTT were prepared. And two copolymers exhibit broad and strong absorption in the range of 300-700 nm with optical band gap of about 1.75 eV. The highest occupied molecular orbital (HOMO) energy levels vary between -5.43 and -5.52 eV and the lowest unoccupied molecular orbital (LUMO) energy levels range from -3.64 to -3.77 eV. Potential applications of the copolymers as electron donor material and $PC_{71}BM$ ([6,6]-phenyl-$C_{71}$ butyric acid methyl ester) as electron acceptors were investigated for photovoltaic solar cells (PSCs). Photovoltaic performances based on the blend of PF-DTBTT/$PC_{71}BM$ (w:w; 1:2) and PIF-DTBTT/$PC_{71}BM$ (w:w; 1:2) with devices configuration as ITO/PEDOT: PSS/blend/Ca/Al, show an incident photon-to-current conversion efficiency (IPCE) of 2.34% and 2.56% with the open circuit voltage ($V_{oc}$) of 0.87 V and 0.90 V, short circuit current density ($J_{sc}$) of $6.02mA/cm^2$ and $6.12mA/cm^2$ under an AM1.5 simulator ($100mA/cm^2$). The photocurrent responses exhibit the onset wavelength extending up to 720 nm. These results indicate that the resulted narrow band gap copolymers are viable electron donor materials for polymer solar cells.