Eastern European countries are preparing for a possible wave of Ukrainian refugees as Russia targets power and heating plants ahead of winter, with President Volodymyr Zelenskiy saying about 4 million people are already without power.

Zelenskiy said 14 regions plus the capital Kyiv were without power and Ukraine's electrical grid operator Ukrenergo said scheduled hourly power outages would affect the whole of the country on Wednesday.

Russian forces have targeted Ukraine's energy infrastructure with missiles and drone strikes in the run up to winter, when mean temperatures typically drop to several degrees below zero Celsius (32 Fahrenheit), with lows of minus 20C.

Some 6.9 million people are believed displaced internally within Ukraine and east European countries such as Slovakia and Hungary are preparing for an influx in coming months.

"An increase in numbers is being felt, and is expected. It is currently up 15%," said Roman Dohovic, an aid coordinator for the eastern Slovak city of Kosice.

Ukrainian forces have been on the offensive in recent months while Russia is regrouping to defend areas of Ukraine it still occupies, having called up hundreds of thousands of reservists over the past month.

Zelenskiy said his forces would not yield "a single centimetre" in battles for the eastern Ukrainian region of Donetsk while Russian-installed officials said Ukrainian forces were moving into a southern town with tanks.

The focal points of the conflict in the industrial region of Donetsk are around the towns of Bakhmut, Soledar and Avdiivka, which have seen the heaviest fighting since Russian forces invaded Ukraine in late February.

"The activity of the occupiers remains at an extremely high level - dozens of attacks every day," Zelenskiy said in his nightly video address late on Tuesday.

"They are suffering extraordinarily high losses. But the order remains the same - to advance on the administrative boundary of Donetsk region. We will not yield a single centimetre of our land," he said.

The region is one of four Russia said it annexed in September. Fighting had been going on there between Ukrainian military and Russian proxy forces since 2014, the same year Russia annexed Crimea in the south.

Kyiv-based military analyst Oleh Zhdanov said on Tuesday that 21 Russian conscripts had surrendered to Ukrainian forces around Svatove in Luhansk region.

"These poor mobilised men - really poor, they had had nothing to eat or drink in three days - of course they decided to surrender," Zhdanov said on his YouTube channel.

FIERCE FIGHTING IN SOUTHERN TOWN

A Russian-installed mayor in the town of Snihurivka, east of the southern city of Mykolaiv, was cited by Russia's RIA news agency as saying residents had seen tanks and that fierce fighting was going on.

"They got into contact during the day and said there were tanks moving around and, according to their information, heavy fighting on the edge of the town," the mayor, Yuri Barabashov, said, referring to the residents.

"People saw this equipment moving through the streets in the town centre," he said.

Kirill Stremousov, deputy head of the Russian-installed administration in the Kherson region, said on the Telegram messaging service that Ukrainian forces had tried to advance on three fronts, including Snihurivka.

Vitaly Kim, the Ukrainian governor of Mykolaiv region, apparently quoting an intercepted conversation between Russian servicemen, suggested that Ukrainian forces had already pushed the Russians out of the area.

"Russian troops are complaining that they have already been thrown out of there," Kim said in a statement on his Telegram channel.

Reuters was unable to verify the battlefield reports.

There was no official word on the situation in the town from military officials in either Ukraine or Russia.

Giving an update on the situation in the neighbouring southern region of Kherson, the Ukrainian military on Tuesday evening accused Russian troops of more looting and destruction of infrastructure. A showdown has been looming for weeks in Kherson city, the only regional capital Russia has captured intact since its invasion.

"A convoy of trucks passed over the dam of the Kakhova hydroelectric station loaded with home appliances and building materials," the military said.

Russian forces were dismantling mobile phone towers and taking equipment, it said, adding that near the city of Beryslav, Russian forces "blew up a power line and took equipment from a solar power station".

In Kherson city, it said Russian troops removed exhibits, furniture and equipment from a museum devoted to the painter Oleksiy Shovkunenko.

Kherson is one of four partially occupied Ukrainian provinces that Russia says it annexed. It controls both the only land route to the Crimea peninsula and the mouth of the Dnipro, the river that bisects Ukraine.

The U.N. General Assembly is due to vote next week on a draft resolution recognising that Russia must be responsible for reparation in Ukraine for the injury, including any damage, caused by "internationally wrongful acts". The text has been put forward by Ukraine, Canada, Guatemala and the Netherlands.

Three-quarters of the 193-member General Assembly denounced Russia's invasion in a vote in March, and in October condemned its self-proclaimed annexation of parts of Ukraine.

(Reporting by Reuters bureaux; Writing by Grant McCool; Editing by Bill Berkrot, Michael Perry and Simon Cameron-Moore)