<PAGE>
 
                                                            HECO Exhibit 10.6(b)
                                                            --------------------


                                                          CONTRACT NO: C06931(B)


              Amendment No. 2 to "Purchase Power Contract Between
          Hawaiian Electric Co., Inc. and City and County of Honolulu
                             Dated March 10, 1986"


By mutual agreement, sections 2(c) and 2(d) of the subject Contract are replaced
by sections 2(c), 2(d), 2(e), and 2(f) of this Amendment No. 2 to reflect a
modification of the deadlines for billing and payment for energy and capacity
received by Hawaiian Electric Company.

(c)   By the fifth working day (i.e. excluding Saturdays, Sundays and legal
      holidays of either the federal government or the Hawaii state government)
      of each calendar month, the Company shall provide the Seller or its
      designated agent with the appropriate data for the Seller to compute the
      energy charge for electricity delivered to the Company in the preceding
      calendar month as determined in accordance with this Contract.

(d)   By the tenth working day of each calendar month, the Seller shall submit
      to the Company an invoice that contains (1) an energy charge for energy
      purchased by the Company in the preceding month; (2) a capacity charge for
      the available capacity in the preceding month; and (3) the metering charge
      as set forth in Section 5 of this Contract.  Each invoice shall include
      the Seller's backup data for the computation of the capacity charge.

(e)   By the twentieth working day of each calendar month (but no later than the
      last working day of that month if there are less than twenty working days
      in that month), the Company shall make payment on such invoice, or provide
      to the Seller an itemized statement of its objections to all or any
      portion of such invoice and pay any undisputed amount.

(f)   Notwithstanding all or any portion of such invoice in dispute, any payment
      not made to the Seller after the twentieth working day of each calendar
      month (or the last working day of that month if there are less than twenty
      working days in that month), shall accrue interest at the maximum rate
      allowed by law until the outstanding interest and invoiced amounts are
      paid in full.  Partial payments shall be applied first to outstanding but
      unpaid interest and then to outstanding but unpaid invoice amounts.

This Amendment shall become effective on the Effective Date of the Firm Capacity
Amendment to Purchase Power Contract Dated March 10, 1986, dated April 8, 1991.

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<PAGE>
 
This Amendment covers billing and payment dates only and is in no way intended
to affect the pricing mechanisms established in the Contract.


HAWAIIAN ELECTRIC COMPANY, INC. ("Company")


By   /s/ Edward Y. Hirata        4/28/92
  --------------------------    ---------
 Its Vice President        Date



By   /s/ George T. Iwahiro       4-28-92
  --------------------------    ---------
 Its Vice President        Date



CITY AND COUNTY OF HONOLULU ("Seller")



By   /s/ Glen S. Nonaka          JUN 04 1992
  ---------------------------    -----------
 For Its Director of Finance    Date



By   /s/ C. Michael Street       MAY 27 1992
  ---------------------------    -----------
 Its Director-Chief Engineer,    Date
 Department of Public Works

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