Date:30 November 2015
For submission to: Journal of Power Sources
Control of microbial fuel cell voltage using a gain scheduling control strategy
Hitesh C. Boghania, Iain Michiea, Richard M. Dinsdalea, Alan J. Guwya, Giuliano C. Premiera*
a Sustainable Environment Research Centre (SERC), Faculty of Computing, Engineering and Science, University of South Wales, Pontypridd, Mid-Glamorgan, CF37 1DL, UK.
Supplementary Information
*Corresponding Author:
Prof. Giuliano C. Premier
Phone: +44 (0)1443 482333
Fax: +44 (0)1443 482169
Email:
Email Addresses for first and remaining authors:
Hitesh Boghani –
Iain Michie –
Richard Dinsdale –
Alan Guwy –
Table S1: Parameters values of identified models for MFC1, according to Eq. 1 in the manuscript, with n = 4.
Step (Ω) ↓ / z1 / z2 / z3 / p1 / p2 / p3 / p4 / k16.4 – 20 / -0.06646 / -0.000970927175743 + 0.018658785455779i / -0.000970927175743 - 0.018658785455779i / -22.1635 / -0.06215 / -0.000960076468046 + 0.018750546838707i / -0.000960076468046 - 0.018750546838707i / 0.020708
20 – 24.8 / -0.01127 / -0.000277794139102 + 0.001398692678334i / -0.000277794139102 - 0.001398692678334i / -0.35993 / -0.00845 / -0.000380851355548 + 0.001357563074127i / -0.000380851355548 - 0.001357563074127i / 2.27E-04
24.8 – 29.5 / 1.588358 / -0.00388 / 4.11E-05 / -44.9943 / -0.06873 / -0.00406 / -2.88E-04 / -0.01176
29.5 – 34.3 / 0.000399423265971 + 0.019242023147467i / 0.000399423265971 - 0.019242023147467i / -9.40E-04 / -0.2202 / -0.000000000000002 + 0.019501913155205i / -0.000000000000002 - 0.019501913155205i / -4.07E-04 / 7.83E-05
34.3 – 39 / -0.6387 / -0.02622 / -5.09E-04 / -1.05059 / -0.41915 / -0.0263 / -4.58E-04 / 3.20E-04
39 – 43.8 / -0.47401 / -0.00162 / 8.89E-05 / -69.5044 / -0.05999 / -0.00159 / -3.98E-05 / -0.00515
43.8 – 48.5 / -0.00539282074576872 + 0.00856984337948585i / -0.00539282074576872 - 0.00856984337948585i / -0.00025 / -0.36504 / -0.00548619357246040 + 0.00821616598444972i / -0.00548619357246040 - 0.00821616598444972i / -0.00015 / 6.93E-05
48.5 – 53.3 / -0.000137455118682987 + 0.00948640235352109i / -0.000137455118682987 - 0.00948640235352109i / -0.00147 / -0.37404 / -0.000194920699416794 + 0.00921478303964381i / -0.000194920699416794 - 0.00921478303964381i / -0.00102 / 3.99E-05
53.3 – 58 / 0.000350768052659603 + 0.00506585484930879i / 0.000350768052659603 - 0.00506585484930879i / -0.00192 / -0.38452 / -5.70040976227704e-15 + 0.00509358913989431i / -5.70040976227704e-15 - 0.00509358913989431i / -0.00137 / 3.36E-05
58 – 67.5 / -1.42299150857825e-05 + 0.0230699642099397i / -1.42299150857825e-05 - 0.0230699642099397i / -0.00164 / -0.21723 / -1.15235598840968e-12 + 0.0230733780997512i / -1.15235598840968e-12 - 0.0230733780997512i / -0.00127 / 2.69E-05
67.5 – 77.1 / 0.000259384985713992 + 0.0191181811423901i / 0.000259384985713992 - 0.0191181811423901i / -0.000905274151950564 / -0.206345346867620 / -3.92580933039621e-14 + 0.0188535282321289i / -3.92580933039621e-14 - 0.0188535282321289i / -0.000579497952439002 / 1.67E-05
77.1 – 86.6 / -0.000767213876899625 + 0.0152169561348589i / -0.000767213876899625 - 0.0152169561348589i / -0.000865128690812889 / -0.486491061386214 / -0.000797092466626511 + 0.0140517032647474i / -0.000797092466626511 - 0.0140517032647474i / -0.000525486932502761 / 2.65E-05
86.6 – 96.1 / 0.000516906949144214 + 0.0180631733080859i / 0.000516906949144214 - 0.0180631733080859i / -0.00251808835131152 / -0.375869669149574 / -8.98054200448861e-13 + 0.0181748230007590i / -8.98054200448861e-13 - 0.0181748230007590i / -0.00173592380900896 / 2.03E-05
96.1 – 129.4 / -0.02382 / -0.00298 / -2.51E-05 / -0.93425 / -0.02412 / -0.00298 / -2.17E-11 / 0.000125
129.4 – 157.8 / -0.18742 / -0.01973 / 0.000255 / -52.3734 / -0.28866 / -0.00873 / -4.47E-05 / -0.00136
157.8 – 191 / -0.000118406762226836 + 0.0156171031919089i / -0.000118406762226836 - 0.0156171031919089i / -0.00240093439583416 / -0.199827964620187 / -0.00144209609725790 + 0.0148022227250947i / -0.00144209609725790 - 0.0148022227250947i / -0.00150028634346985 / 6.82E-06
191 – 219.5 / 3.52866538707518e-05 + 0.0191240976120928i / 3.52866538707518e-05 - 0.0191240976120928i / -0.00629259979356677 / -0.188725161226624 / -0.00150740964907445 + 0.0195712818690719i / -0.00150740964907445 - 0.0195712818690719i / -0.00296587966111817 / 3.41E-06
219.5 – 252.8 / 0.000640356751826445 + 0.0172034142856679i / 0.000640356751826445 - 0.0172034142856679i / -0.00277223539334138 / -0.0407330892988842 / -1.98625837999344e-16 + 0.0169044102515516i / -1.98625837999344e-16 - 0.0169044102515516i / -0.00174604246844657 / 8.38E-07
252.8 – 351.3 / 0.0137699342705209 / -0.000381828104315011 + 0.00697487926453176i / -0.000381828104315011 - 0.00697487926453176i / -0.0404249729449553 / -0.00186375052450130 + 0.0102775960140833i / -0.00186375052450130 - 0.0102775960140833i / -0.00299347855520455 / -1.50E-07
351.3 – 451.6 / 0.000414875137975067 + 0.0117271653728726i / 0.000414875137975067 - 0.0117271653728726i / -0.00411715158589697 / -0.171873028501780 / -0.000301148482865212 + 0.0117033821111401i / -0.000301148482865212 - 0.0117033821111401i / -0.00165659462447535 / 2.03E-06
Table S2: Parameters values of identified models for MFC2, according to Eq. 1 in the manuscript, with n = 4.
Step (Ω) ↓ / z1 / z2 / z3 / p1 / p2 / p3 / p4 / k15.2 – 20.4 / -0.010473549722635 + 0.005733954706474i / -0.010473549722635 - 0.005733954706474i / 1.36E-05 / -0.33003 / -0.010775141137404 + 0.005599124870447i / -0.010775141137404 - 0.005599124870447i / -7.41E-14 / 5.20E-04
20.4 – 25.1 / -0.02311 / -0.01046 / -0.00124 / -0.21631 / -0.010369352885877 + 0.007551715729665i / -0.010369352885877 - 0.007551715729665i / -0.00131 / 1.74E-04
25.1 – 29.8 / -0.07287 / -0.00646 / -2.28E-04 / -0.52664 / -0.02317 / -0.00922 / -2.39E-04 / 2.22E-04
29.8 – 34.5 / -0.10431 / -0.00776 / -7.87E-04 / -0.3406 / -0.04137 / -0.00845 / -8.01E-04 / 1.00E-04
34.5 – 39.3 / -0.10336 / -0.000330233241911 + 0.064421420599996i / -0.000330233241911 - 0.064421420599996i / -0.3508 / -0.000323439590272 + 0.064502994758404i / -0.000323439590272 - 0.064502994758404i / -0.05152 / 7.35E-05
39.3 – 44.1 / -0.08076 / -0.00504049908854160 + 0.0529490875059527i / -0.00504049908854160 - 0.0529490875059527i / -0.496 / -0.00427623340199861 + 0.0530726129253312i / -0.00427623340199861 - 0.0530726129253312i / -0.03974 / 7.35E-05
44.1 – 48.8 / -0.0176675465633231 + 0.0579325149341771i / -0.0176675465633231 - 0.0579325149341771i / -0.03914 / -0.43816 / -0.0142631575443453 + 0.0546240743787865i / -0.0142631575443453 - 0.0546240743787865i / -0.02422 / 4.87E-05
48.8 – 53.5 / -0.8535 / -0.06611 / 0.000457 / -1.61351 / -0.36475 / -0.04499 / -0.00054 / -9.83E-05
53.5 – 58.2 / -0.00147854562551642 + 0.0353015656431178i / -0.00147854562551642 - 0.0353015656431178i / -0.02392 / -0.21797 / -0.00123101060474088 + 0.0351447605179721i / -0.00123101060474088 - 0.0351447605179721i / -0.01687 / 2.11E-05
58.2 – 67.7 / -0.0381266328692741 / -0.00221735483213273 + 0.000566790443658286i / -0.00221735483213273 - 0.000566790443658286i / -0.355520927650897 / -0.0393694507342397 / -0.00220833131398370 + 0.000568873246649037i / -0.00220833131398370 - 0.000568873246649037i / 7.31E-05
67.7 – 77.2 / -0.0139015597845682 + 0.0224477879441385i / -0.0139015597845682 - 0.0224477879441385i / -0.00255291917846869 / -0.178161238905499 / -0.0140707833707010 + 0.0225402414104923i / -0.0140707833707010 - 0.0225402414104923i / -0.00255415255424419 / 2.39E-05
77.2 – 86.7 / -0.0409744623397162 / -0.00224289591701242 + 0.000389194871222342i / -0.00224289591701242 - 0.000389194871222342i / -0.463215059860078 / -0.0420335401896967 / -0.00223176199658457 + 0.000397965405100548i / -0.00223176199658457 - 0.000397965405100548i / 4.96E-05
86.7 – 96.2 / 0.798122 / -0.0068 / -4.66E-05 / -3.90002 / -0.05786 / -0.00702 / -0.00013 / -7.31E-05
96.2 – 129.4 / -0.07927 / -0.00601 / -0.00014 / -0.49539 / -0.04394 / -0.00522 / -7.48E-05 / 2.99E-05
129.4 – 157.7 / -0.09775 / -0.00525 / -7.19E-05 / -0.7445 / -0.05043 / -0.00436 / -1.37E-17 / 1.99E-05
157.7 – 190.8 / -0.55316 / -0.00379 / -0.00011 / -1.09163 / -0.06113 / -0.00372 / -2.20E-05 / 1.22E-05
190.8 – 219.1 / -0.1351 / -0.00453 / -0.00027 / -1.3347 / -0.06048 / -0.00355 / -2.49E-17 / 1.13E-05
219.1 – 252.4 / -0.09409 / -0.00473 / -0.00083 / -0.45564 / -0.04522 / -0.00391 / -0.00015 / 5.30E-06
252.4 – 350.6 / -0.000711008558091492 + 0.0110798596720228i / -0.000711008558091492 - 0.0110798596720228i / -0.00227369542929591 / -0.00205271687110053 + 0.0107608154071386i / -0.00205271687110053 - 0.0107608154071386i / -0.00369971914693036 / -8.29359257923469e-05 / 1.58E-08
350.6 – 450.7 / -0.91848 / -0.00436 / -0.00055 / -1.89856 / -0.08628 / -0.00401 / -1.02E-12 / -1.11E-06
In Table 1 and Table 2, the parameter values highlighted in gray are very small compared to other dominating values in the same model. They have time constant of at least about 140 sec (1/0.007) which is relatively large compared to other time constants, which are of the order of a few seconds. Highlighted values can reasonably be ignored because their effect will appear as a slow drift and the effective model can be approximated by a 1st order function.
Table S3: Parameters of 1st order models for MFC1 and MFC2 as per Eq. 2.
Time Constant, (s) / Steady state gain, (V/Ω × 0.001)Step (tap no.) / MFC 1 / MFC 2 / MFC 1 / MFC 2
15.2 – 20.4 / 6 / 6 / 5.63 / 5.86
20.4 – 25.1 / 6 / 11 / 4.49 / 6.02
25.1 – 29.8 / 6 / 13 / 4.91 / 5.72
29.8 – 34.5 / 8 / 13 / 4.83 / 4.88
34.5 – 39.3 / 8 / 13 / 3.5 / 3.4
39.3 – 44.1 / 8 / 15 / 3.03 / 2.74
44.1 – 48.8 / 8 / 17 / 2.14 / 2.2
48.8 – 53.5 / 8 / 19 / 1.86 / 1.98
53.5 – 58.2 / 8 / 19 / 1.74 / 1.76
58.2 – 67.7 / 8 / 19 / 1.2 / 1.44
67.7 – 77.2 / 8 / 19 / 0.93 / 1.13
77.2 – 86.7 / 9 / 12 / 0.87 / 1.02
86.7 – 96.2 / 9 / 15 / 0.82 / 0.86
96.2 – 129.4 / 12 / 22 / 0.51 / 0.56
129.4 – 157.7 / 12 / 22 / 0.45 / 0.46
157.7 – 190.8 / 18 / 30 / 0.35 / 0.4
190.8 – 219.1 / 27 / 38 / 0.3 / 0.52
219.1 – 252.4 / 27 / 38 / 0.26 / 0.6
252.4 – 350.6 / 290 / 300 / 0.17 / 0.26
350.6 – 450.7 / 120 / 100 / 0.13 / 0.16
Oscillating artefactevident in the MFC voltage during continuous control.
As seen in the Figure 5 (between hour 26 and 30).
It is necessary to explain the oscillatory artefacts in the data in order to eliminate them from our understanding of the data. These oscillations were caused by alternate switching of the digital potentiometer from high resolutionpotentiometer (HRP) mode, to low resolutionpotentiometer (LRP) mode, in a particular circumstances. Steps were missed by the coded algorithm when the potentiometers were driven from higher to lower loading. This was due to an error in the coding that was rectified after examining the results. Figure S3a shows the oscillations occurring due to inappropriate tap positions on the loading arranged as in Figure S3b (comprising LRP, HRP and a relay). Considering the circumscribed data in Figure S3a, a demand change from -2 to -1 (red, open circles), which represented a load change from 59.9 Ω (corresponds to tap number 1 on LRP and 0 on HRP with both pots in series) to 56.9 Ω (corresponds to tap number 0 on LRP and 6 on HRP with both pots in series), the pot seemed to jump 7 tap positions (corresponds to tap number 0 on LRP and 29 on HRP with only HRP in the circuit) and this corresponded to a load of 47.2 Ω. This large change in the load perturbed the system and caused oscillations. In fact, the same was true in the case of Figure 5 where the oscillations occurred in the same loading regions. The control was relatively smooth once it increased above this region. This error in the algorithm was easily rectified once diagnosed, by the addition of a case statement to account for stepping the potentiometers downward, which had been omitted.