For years, public housing authorities in large cities such as Washington have been telling the federal government that it wasn't providing enough money to maintain the housing projects it has built.

And now, with 10 of the 22 largest housing authorities in the red, the U.S. Department of Housing and Urban Development concedes that it has made a mistake.

A new report prepared for HUD by a private consultant says that HUD's formula for allocating operating funds to the projects has shortchanged housing authorities in large cities by failing to take into account their particular problems in struggling with crime, vandalism and old, decaying buildings.

"We do recognize that (the large authorities) have been underfunded and disadvantaged" by the formula, said Clyde McHenry, deputy assistant secretary of HUD for housing programs. "HUD is going to have to come up with a system that adequately funds them."

"People are screaming bloody murder because of that formula," said Sidney Glee, the District's public housing administrator. The District's public housing authority is now $20 million in the red.

The formula, the Performance Funding System, divvies up nearly $1 billion a year in operating grants. It has been criticized since it was instituted in 1975.

Housing authorities that have fewer than 1,200 units to manage -- such as those in the Washington suburbs -- fared fairly well under the formula because their operating costs were reflected more accurately, according to the new study, prepared by Abt Associates Inc. of Cambridge, Mass.

But some of the larger authorities ran into trouble.

First, HUD refused to pay all the actual operating costs of an authority when they were higher than those of a model, well-managed agency (as defined by the government) when the formula was first used. Though the funds were adjusted for inflation over the rest of the decade, the 1975 figure was always used as the base and the authorities could never catch up, the report says.

In addition, the formula, designed by the Urban Institute, a private non-profit research group, pegged inflation adjustments to the salary increases of unionized city employes, at a time when those salaries were rising much slower than the overall inflation rate.

The report also says that the formula failed to account for the larger authorities' extra expenses for maintenance, security and social services for residents.

The report found that it now costs a public housing authority with more than 6,600 units $139 a month to operate each of its units on average, compared with $86 a month for authorities with fewer than 500 units.

The utility costs for the largst authorities are $33 per unit higher than for the smaller ones, and their maintenance costs are $18 per unit more. They spend nearly $1 more each month per unit on security to fight crime and vandalism and $3.50 more for job counseling, recreation, health and other social services for a population that is primarily women with three or four children on public assistance.

The financial pressure on public housing, though, dates back even further -- to 1969 when Congress stipulated that tenants could not pay more than 25 percent of their income as rent. Since that time, the rental income for all housing authorities has increased only 49 percent, while operating costs have gone up 131 percent.

The government provided $12.6 million in 1969 to help pay operating costs. Within four years, the figure had ballooned to $348.4 million, and has increased by an average of $75 million every year since then.

HUD officials are now working to revise the formula for distributing the operating subsidies "to make it more equitable for the largest (authorities)," McHenry said. He said "a substantial increase" in the subsidy would be needed, but the department has no estimate of what the amount would be. The increase would have to be approved by Congress, at a time when it is looking for places to cut the federal budget.

Congress did appropriate $13.8 million this year to help the 10 authorities that are in the most financial trouble. It also appropriated $741.5 million in operating subsidies for all the authorities this year.