(Adds APRA quotes, context) SYDNEY, July 11 (Reuters) -...

(Adds APRA quotes, context)

SYDNEY, July 11 (Reuters) - Australia's prudential regulator said on Thursday it has told three of the country's biggest banks to set aside an additional A$500 million ($348 million) each until they have strengthened risk management and reimbursed customers wrongly charged fees.

The Australian Prudential Regulation Authority (APRA) said it wrote to Westpac Banking Corp (WBC) , Australia and New Zealand Banking Group Ltd (ANZ) and National Australia Bank Ltd (NAB) to tell them of additional capital requirements.

"Australia's major banks are well-capitalised and financially sound, but improvements in the management of non-financial risks are needed," APRA Chair Wayne Byres said in a statement.

"This will require a real focus on the root causes of the issues that have been identified, including complexity, unclear accountabilities, weak incentives and cultures that have been too accepting of long-standing gaps."

The country's top lender, Commonwealth Bank of Australia (CBA) , was slapped with an additional A$1 billion capital requirement last year after it was accused of thousands of breaches of anti-money laundering protcols.

APRA then asked dozens of other financial services companies to report on their own their risk assessment systems, which it said on Thursday confirmed that "many of the issues" identified in its inquiry into the money-laundering scandal were "not unique to CBA".

The companies' self-assessments showed that "they have fallen short in a number of areas, and APRA is therefore raising their regulatory capital requirements until weaknesses have been fully remediated," Byres said.

APRA added that it may impose more capital requirements on the sector without specifying which company.

ANZ said the directive represented a 0.18 percentage point impact on its Common Equity Tier 1 capital ratio from 30 September 2019.

($1 = 1.4364 Australian dollars)